AN ABSTRACT OF THE THESIS OF
Brian Christopher Dietel for the degree of Doctor of Philosophy in Mathematics
presented on April 28, 2009.
Title: Mahler’s Order Functions and Algebraic Approximation of p-adic Numbers
Abstract approved:
Mary E. Flahive
If P is an integer polynomial denote the degree of P by ∂(P ) and let H(P ) be
the maximum of the absolute value of the coefficients of P . Define Λ(P ) = 2∂(P ) H(P )
and for a fixed prime p let Cp denote the completion of the algebraic closure of the p-adic
numbers. We generalize the order function of Mahler to the p-adic numbers by associating
each θ ∈ Cp to the function O(u|θ) = max log |P 1(θ)| , where | ∗ | denotes the p-adic absolute
value and the maximum is taken over all P (x) ∈ Z[x] satisfying Λ(P ) ≤ u and P (θ) 6= 0.
1
Similarly, define O∗ (u|θ) = max log |θ−α|
where the maximum is now taken over the set of
all algebraic numbers α 6= θ with minimal polynomial m satisfying Λ(m) ≤ u. Placing a
partial order and equivalence relation on both O and O∗ induces two corresponding
partial orders and equivalence relations on Cp .
In this thesis we demonstrate several results concerning O and O∗ . In particular, it
is proved that if θ is algebraic over Qp then θ must satisfy O∗ (u|θ) log u and if θ, η ∈ Cp
are algebraically dependent over Q then O(u|θ) O(u|η). Also, under certain conditions, given a function g we construct θ ∈ Cp such that g(c0 (log log u)1/2 ) O∗ (u|θ) and
O∗ (u|θ) g(c log log u), where c and c0 are positive constants. The transcendence type
considers the limiting behavior of O and O∗ and will be used to prove that under certain
√
conditions O and O∗ behave similarly. Finally, given τ ≥ (3 + 5)/2 we demonstrate that
it is possible to construct elements of Cp with transcendence type equal to τ .
c
Copyright by Brian Christopher Dietel
April 28, 2009
All Rights Reserved
Mahler’s Order Functions and Algebraic Approximation of p-adic Numbers
by
Brian Christopher Dietel
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
Presented April 28, 2009
Commencement June 2009
Doctor of Philosophy thesis of Brian Christopher Dietel presented on April 28, 2009
APPROVED:
Major Professor, representing Mathematics
Chair of the Department of Mathematics
Dean of the Graduate School
I understand that my thesis will become part of the permanent collection of Oregon State
University libraries. My signature below authorizes release of my thesis to any reader
upon request.
Brian Christopher Dietel, Author
ACKNOWLEDGEMENTS
Academic
I am indebted to my advisor Mary Flahive for her commitment and dedication. I
also thank my teachers and committee members, David Finch, Dennis Garity, Thomas
Schmidt, Holly Swisher, and Todd Palmer.
Personal
I wish to thank my parents Sharon and Wallace Dietel, and my brother Kevin Dietel.
I cannot express my gratitude for the love and support they have given me throughout
my life. I also thank my grandparents, Ruby Bull, Richard Dietel, and Warna Dietel for
their love and strength.
TABLE OF CONTENTS
Page
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1
Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
p-adic Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3
4
5
6
Properties of O on Cp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2
Equivalence of Order Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Results on O∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1
A Lower Bound for O∗ on Elements of Qp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2
Elements of Cp for Which O∗ Grows Slowly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Transcendence Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.1
Comparing τ and τ ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2
Constructing Numbers With a Given Transcendence Type . . . . . . . . . . . . . . 52
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
TABLE OF CONTENTS (Continued)
Page
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Dedicated to the memory of Harlan Bull
MAHLER’S ORDER FUNCTIONS AND ALGEBRAIC
APPROXIMATION OF P -ADIC NUMBERS
1
INTRODUCTION
When studying transcendental numbers a natural question is to consider how well
a given transcendental number can be approximated by algebraic numbers, or if there
exist integer polynomials which are close to zero when evaluated at a given transcendental
number. The algebraic numbers are dense in C and therefore any element of C can be
approximated with arbitrary precision by an element in the algebraic closure of Q. Thus it
is necessary to place a restriction on the set of polynomials or algebraic numbers which are
being used to approximate. One method to measure a polynomial is to consider a function
depending on the degree of the polynomial and the absolute value of the polynomial’s
coefficients. First let | ∗ |∞ denote the standard absolute value on the complex numbers
and suppose Q(x) = an xn + · · · + a0 ∈ Z[x]. Denote the degree of Q by ∂(Q) and the
length of Q by L(Q) = |an |∞ + · · · + |a0 |∞ . Define Λ0 (Q) = 2∂(Q) L(Q). Moreover, if α ∈ Q
let Mα (x) ∈ Q[x] denote the minimal polynomial of α over Q and define mα (x) ∈ Z[x] to
be aMα (x), where a is the least common multiple of the denominators of the coefficients
of Mα . Define ∂(α) = ∂(mα ), H(α) = H(mα ), and Λ0 (α) = Λ0 (mα ). Note that given
u ∈ N there exist only finitely many integer polynomials Q and algebraic numbers α
which satisfy Λ0 (Q) ≤ u and Λ0 (α) ≤ u. In a 1971 paper Mahler [22] associated to each
∗ , and then used the order functions to construct
θ ∈ C two “order functions” OM and OM
a classification of C based on algebraic approximation. Note that this classification is
different from Mahler’s [20] 1932 classification of R based on algebraic approximation in
which he divided R into the A, S, T, and U numbers. An overview of the results on this
2
earlier classification can be found in Bugeaud [6].
∗ (∗|θ) : N \ {1} → R by
Definition 1.1. Let θ ∈ C. Define OM (∗|θ) : N → R and OM
OM (u|θ) = 0max log
1
|P (θ)|∞
∗
OM
(u|θ) = 0max log
1
.
|α − θ|∞
Λ (P )≤u
P (θ)6=0
Λ (α)≤u
α6=θ
∗ measures approxThe function OM measures polynomial approximations and OM
imations by algebraic numbers. Thus elements of C for which OM grows quickly have
∗ grows quickly for some θ ∈ C
“good” polynomial approximations relative to Λ0 and if OM
then there are algebraic numbers that are “good” approximations of θ relative to Λ0 . By
placing a partial order on the set of order functions Mahler [22] constructed two classifications of the elements of C.
Definition 1.2. Let a(u) and b(u) be non-decreasing functions from N to R which are
positive for sufficiently large u. Define the partial order by a(u) b(u) (or equivalently
b(u) a(u)) if there exist c, u0 ∈ N and γ ∈ R+ such that a(uc ) ≥ γb(u) for all u ≥ u0 .
Define the equivalence relation by a(u) b(u) if a(u) b(u) and b(u) a(u).
∗ to be
Since Λ0 (1) = 1 it follows that OM (u|θ) ≥ 0 for all θ. It is possible for OM
negative, but it must be positive for all sufficiently large u because the algebraic numbers
∗ induces partial orders on
are dense in C. Applying the partial order to OM and OM
∗ (u|θ) O ∗ (u|η).
θ, η ∈ C given by θ η if OM (u|θ) OM (u|η) and θ ∗ η if OM
M
Likewise the equivalence relation on C is defined by θ η if OM (u|θ) OM (u|η) and
∗ (u|θ) O ∗ (u|η). Mahler [22]
the equivalence relation ∗ on C is defined by θ ∗ η if OM
M
proved the following results on OM .
Theorem 1.3.
(i) If θ ∈ C is algebraic and θ is not an algebraic integer in an imaginary quadratic field
then OM (u|θ) log u.
3
(ii) If θ is transcendental then OM (u|θ) (log u)2 .
(iii) If θ and η are algebraically dependent and transcendental then θ η.
Durand [10] gave different proofs of these results. In addition, Mahler [22] posed several questions concerning OM . Durand [11] provided solutions to most of these questions
∗ .
and in doing so proved the following results on OM and OM
Theorem 1.4.
∗ (u|θ) log u.
(i) If θ is algebraic then OM
(ii) If or ∗ is used as an equivalence relation on C then there are uncountably many
equivalence classes.
(iii) The partial orders and ∗ on C are not total orders.
(iv) With respect to Lebesgue measure OM (u|θ) (log u)2 for almost all θ ∈ C.
In proving these results Durand [11] also used results due to Fel’dman [13] to prove
the following proposition.
Proposition 1.5. If τ ≥ 2 and θ ∈ C then
lim sup
u→∞
∗ (u|θ)
OM
=∞
(log u)τ
if and only if
lim sup
u→∞
OM (u|θ)
= ∞.
(log u)τ
For θ ∈ C define
T (θ) =
OM (u|θ)
τ ≥ 0 : lim sup
=∞ .
τ
u→∞ (log u)
Define the transcendence type of θ by τ (θ) = sup{τ : τ ∈ T (θ)}. Given any τ ≥ 3,
Durand [11] constructed θ ∈ C such that τ (θ) = τ . Amoroso [2] improved this result by
4
constructing θ ∈ C for which τ (θ) = τ given any τ ≥ 2. Philippon [28] also studied the
equivalence relation on C by using nonstandard analysis.
∗ , which we denote by
Our goal is to investigate the p-adic analogues to OM and OM
O and O∗ . Although no previous work has been done on this specific problem there are
numerous results concerning related problems in p-adic algebraic approximation. Mahler
[21] constructed a p-adic version of his first classification. Escassut [12] studied an analogue
to transcendence type when considering approximation by numbers algebraic over the padics. In the p-adics Adams [1] used algebraic approximation to study the transcendence of
numbers of the form αβ where α, β ∈ Q. Nesterenko [25, 26] also proved several results on
p-adic algebraic approximation in polynomials with more than one variable. Beresnevich,
Bernik, and Kovalevskaya [3] extended some metric results to the p-adics. Teulié [32]
and Morrison [24] considered approximation of p-adic numbers by algebraic numbers of
bounded degree. The results of Teulié were then extended by Zelo [35] who considered
simultaneous algebraic approximation to sets containing real and p-adic numbers. Wang
[34] studied algebraic approximations to values of p-adic functions satisfying a certain
functional equation.
Denote the completion of the algebraic closure of the p-adic numbers by Cp . In
this thesis we first set up the necessary background on polynomials and p-adic numbers
in Chapter 2. Chapter 3 then focuses on the basic properties of the p-adic order function
O. In particular, the first main result is Corollary 3.9 which states that if θ, η ∈ Cp
are algebraically dependent over Q then O(u|θ) O(u|η). In Chapter 4, Theorem 4.2
demonstrates that if α is algebraic over the p-adic numbers then O∗ (u|α) log u, and
Theorem 4.5 constructs θ ∈ Cp for which O∗ (u|θ) grows slowly. Using the same θ from
Theorem 4.5 in Theorem 4.6 we construct a lower bound for O∗ (u|θ). Chapter 5 studies
transcendence type in the p-adic numbers and Theorem 5.9 is the p-adic analogue of
Proposition 1.5. Our final main result is Theorem 5.11 which states that given any real
√
number τ ≥ (3 + 5)/2 it is possible to construct elements of Qp with transcendence type
τ.
5
2
BACKGROUND
Before beginning our study of p-adic order functions it is first necessary to go over
some background results. The primary goal of Section 2.1 is to state several definitions and
lemmas concerning polynomials. Section 2.2 provides a brief introduction to the p-adic
numbers and gives some results concerning algebraic numbers over the p-adics.
2.1
Polynomials
We first state several definitions and results from basic field theory. Then we consider
results on polynomials in Z[x]. For additional background and results see Bourbaki [5],
Dummit and Foote [9], and Mignotte and Ştefănescu [23]
For proofs and further discussion on the following results concerning fields see Chapters 13 and 14 in Dummit and Foote [9]. Let F be a field and let K be an extension field of
F . If α ∈ K is a root of a polynomial in F [x] then α is algebraic over F . If α is not a root
of any polynomial in F [x] then α is transcendental. Every element of a finite extension
field of F is algebraic over F . If α is algebraic over F then there exists a unique monic
irreducible polynomial mα,F (x) ∈ F [x] for which mα,F (α) = 0. This will be referred to as
the minimal polynomial of α over F . The degree of α over F is the degree of the minimal
polynomial of α over F . In the particular case when F = Q the minimal polynomial of α
over Z will be defined to be amα,Q (x) ∈ Z[x] where a ∈ N is the least common multiple
of the denominators of the coefficients of mα,Q . The degree of a finite extension field K
over F will be denoted by [K : F ]. If α ∈ K it can be proved that the degree of α over F
must divide [K : F ]. If K is an extension of F and α, β ∈ K then β is algebraic over α
if there exists a polynomial P (x) ∈ F (α)[x] such that P (β) = 0, where F (α) denotes the
smallest field constaining both F and α. If β is algebraic over α and α is algebraic over β
6
then α and β are said to be algebraically dependent over F .
A field K is algebraically closed if for every Q(x) ∈ K[x] there is α ∈ K for which
Q(α) = 0. In particular this implies that all polynomials factor completely in an algebraically closed field. It can be proved by Zorn’s Lemma that for every field F there exists
a field F , unique up to isomorphism fixing F, consisting of elements algebraic over F such
that for every Q(x) ∈ F [x] there is α ∈ F for which Q(α) = 0. The field F is algebraically
closed and is called the algebraic closure of F .
Given a field F and a finite extension field K let Aut(K/F ) denote the group of
field automorphisms on K which fix F . The field K is said to be a Galois extension field
of F if the number of elements in Aut(K/F ) is equal to [K : F ]. Now suppose α ∈ K
and K is a Galois extension field of F . The conjugates of α over F are defined to be
{σ(α) : σ ∈ Aut(K/F )}. Provided K is Galois the set of conjugates is independent of the
field K. If α is algebraic over a field F let {α1 , α2 , . . . , αn } be the conjugates of α and
define the norm of α over F by
Nm(α) =
Y
αi .
1≤i≤n
For all α, β ∈ F it can be demonstrated that Nm(αβ) = Nm(α)Nm(β) and Nm(α) ∈ F .
Using these definitions and results we now turn our attention to polynomials.
Definition 2.1. Let B be a ring. Let P (x) = an xn + · · · + a0 and Q(x) = bq xq + · · · + b0
be polynomials with coefficients in B with an and bq both nonzero. Define the Sylvester
matrix Ms (P, Q) to be the n + q by n + q matrix given

a
 n

 an−1 an

 ..
.. . .
 .
. an
.


..
Ms (P (x), Q(x)) = 
a1
.
 a0

..

a0
.


.

..
. ..


a0
by

bq
bq−1 bq
..
..
.
.
b0
b1
b0
..
. bq
..
.
..
.
.
..
. ..
b0

















7
where the first q columns contain the coefficients of P and the final n columns contain
the coefficients of Q with 0 in the remaining entries. The resultant R is defined to be
R(P, Q) = det(Ms (P, Q)).
When B is an algebraically closed field let P (x) = an (x − α1 )(x − α2 ) . . . (x − αn )
and Q(x) = bq (x − β1 )(x − β2 ) . . . (x − βq ) be the unique factorizations of P and Q. It can
be proved that (see Chapter 4 of Bourbaki [5])
R(P (x), Q(x)) = aqn
Y
(αk − βj ).
1≤j≤n
1≤k≤q
From this equation it is clear that in an algebraically closed field R(P (x), Q(x)) = 0 if
and only if P and Q share a root. Related to the resultant is the discriminant.
Definition 2.2. Let B be a ring and let P (x) = an xn + · · · + a0 ∈ B[x]. Define the
discriminant of P by
0
disc(P ) = (−1)n(n−1)/2 a−1
n R(P, P )
where P 0 (x) = nan xn−1 + (n − 1)xn−2 + · · · + a1 ∈ B[x] is the formal derivative of P .
If B is an algebraically closed field and P (x) = an (x − α1 )(x − α2 ) . . . (x − αn ) then
it can be proved (again see see Chapter 4 of Bourbaki [5])
disc(P ) =
a2n−2
n
n j−1
Y
Y
(αk − αj )2 .
(2.1)
j=2 k=1
Let P (x) = an xn + · · · + a0 ∈ Z[x]. The following notation will be used throughout.
Let | ∗ |∞ denote the standard absolute value. Define H(P ) = max{|an |∞ , . . . , |a0 |∞ } to
be the height of P and L(P ) = |an |∞ + · · · + |a0 |∞ to be the length of P . Let ∂(P ) denote
the degree of P and define the size of P by Λ(P ) = 2∂(P ) H(P ). Likewise, if α ∈ Q let
mα to be the minimal polynomial of α over Z and define H(α) = H(mα ), L(α) = L(mα ),
∂(α) = ∂(mα ), and Λ(α) = Λ(mα ). Note that given u ∈ N there exist only finitely many
polynomials P (x) ∈ Z[x] and α ∈ Q such that Λ(P ) ≤ u and Λ(α) ≤ u.
8
The following proposition allows us to bound the resultant R(P, Q) and discriminant
disc(P ) in terms of the height and degree.
Proposition 2.3. Let P, Q ∈ Z[x]. Then
|R(P, Q)|∞ ≤ (∂(P ) + ∂(Q))! H(P )∂(Q) H(Q)∂(P ) .
and
| disc(P )|∞ ≤ (2∂(P ) − 1)! H(P )2∂(P )−1 ∂(P )∂(P ) .
Proof. If M is an n by n matrix with entries ai,j recall the Leibniz formula for the determinant gives

det(M ) =
X
σ∈Sn
sgn(σ)
n
Y

aσ(j),j 
(2.2)
j=1
where Sn is the symmetric group on n elements. Let P, Q ∈ Z[x]. The largest entry in each
of the first ∂(Q) columns of the Sylvester matrix Ms (P, Q) is H(P ) and the largest entry
in each of the last ∂(P ) columns of Ms (P, Q) is H(Q). Moreover, there are (∂(P ) + ∂(Q))!
elements of S∂(P )+∂(Q) and (2.2) implies
|R(P, Q)| = | det(Ms (P, Q))| ≤ (∂(P ) + ∂(Q))! H(P )∂(Q) H(Q)∂(P ) .
Since H(P 0 ) ≤ ∂(P )H(P ) and ∂(P 0 ) = ∂(P ) − 1 it thus follows that
| disc(P )|∞ ≤ (2∂(P ) − 1)! H(P )2∂(P )−1 ∂(P )∂(P ) .
When considering the product of polynomials it will be useful to bound the height
by the height of the individual factors. The next proposition will be used several times to
bound the height of a product of polynomials. A proof is given in Appendix A of Bugeaud
[6].
Proposition 2.4. Let P, P1 , P2 , . . . , Pl ∈ Z[x] be polynomials such that P = P1 P2 . . . Pl .
Then
2−∂(P ) H(P1 ) . . . H(Pl ) ≤ H(P ) ≤ 2∂(P ) H(P1 ) . . . H(Pl ).
9
In particular, note that this implies
Λ(P1 )Λ(P2 ) . . . Λ(Pl ) = 2∂(P ) H(P1 )H(P2 ) . . . H(Pl ) ≤ 2∂(P ) 2∂(P ) H(P ) ≤ (Λ(P ))2 . (2.3)
Occasionally this bound will not be sufficient and instead we will use the following bound
for the height of the product of only two polynomials.
Proposition 2.5. Let P, Q ∈ Z[x]. Then
H(P Q) ≤ (min{∂(P ), ∂(Q)} + 1)H(P )H(Q).
Proof. Suppose P (x) = an xn + · · · + a0 and Q(x) = bm xm + · · · + b0 with an and bm
both nonzero. Assume without loss of generality n ≥ m. The kth coefficient of P Q is
ak b0 + ak−1 b1 + · · · + ak−m bm where any coefficient with a negative subscript is defined to
be 0. The standard absolute value of each of these summands in the kth coefficient is at
most H(P )H(Q) and since there are at most m + 1 summands the kth coefficient can be
at most (m + 1)H(P )H(Q). Thus H(P Q) ≤ (min{∂(P ), ∂(Q)} + 1)H(P )H(Q).
We will also require an upper bound on the height and degree of a polynomial
obtained by taking the determinant of a matrix of polynomials. The following result is
proved in Durand [10].
Proposition 2.6. Let {Pij (x)} for 1 ≤ i, j ≤ n be polynomials in Z[x] and let M be the
matrix

P11
P12
. . . P1n


 P21 P22 . . . P2n
M =
 ..
..
..
 .
.
.

Pn1 Pn2 . . . Pnn




.



Then the determinant of M satisfies
∂(det(M )) ≤
n
X
j=1
max ∂(Pij )
1≤i≤n
(2.4)
10
and
H(det(M )) ≤
n
n
Y
X
j=1
!
(1 + ∂(Pij ))H(Pij ) .
(2.5)
i=1
Proof. Both parts of the lemma are proved by induction on n. The base case is immediate
for the first assertion. Assume that the first part of the lemma is true for any (n − 1) by
(n − 1) matrix of integer polynomials. Then
det(M ) =
n
X
(−1)k−1 P1k det(Mk )
k=1
where Mk is the matrix obtained by deleting the first row and kth column of the original
matrix. Thus
∂(det(M )) ≤ max ∂(P1j ) + max ∂(det(Mj ))
1≤j≤n
≤ max ∂(P1j ) +
1≤j≤n
=
n
X
i=1
1≤j≤n
n
X
max ∂(Pij )
i=2
1≤j≤n
max ∂(Pij ).
1≤j≤n
The base case is likewise immediate for (2.5) also. Assume (2.5) is true for all n − 1
by n − 1 matrices. Using the notation from above it thus follows that
!
n
n
X
X
k−1
H(P1k det(Mk )).
H(det(M )) = H
(−1) P1k det(Mk ) ≤
k=1
k=1
Proposition 2.5 then implies
n
X
(1 + min{∂(P1k ), ∂(det(Mk ))})H(P1k )H(det(Mk ))
H(det(M )) ≤
k=1
≤
n
X
(1 + ∂(P1k ))H(P1k )H(det(Mk ))
k=1
n
X
≤ max {H(det(Mk ))}
(1 + ∂(P1k ))H(P1k ).
1≤k≤n
k=1
Recall the induction hypothesis implies for 1 ≤ k ≤ n
n X
n
n X
n
Y
Y
H(det(Mk )) ≤
(1 + ∂(Pij ))H(Pij ) ≤
(1 + ∂(Pij ))H(Pij ).
i=2 j=1
j6=k
i=2 j=1
11
Hence


!
n X
n
n
Y
X
H(det(M )) ≤ 
(1 + ∂(Pij ))H(Pij )
(1 + ∂(P1k ))H(P1k )
i=2 j=1
=
n X
n
Y
k=1
(1 + ∂(Pij ))H(Pij ).
i=1 j=1
Clearly H(P + Q) ≤ H(P ) + H(Q). We will also require a bound on H(α + β) in
terms of H(α) and H(β). The next result is due to Dubickas and Smyth [8] and gives
a bound for L(α + β) and thus can be used to obtain a bound for H(α + β). Note that
a stronger inequality is given by Mignotte and Ştefănescu [23], but for our purposes it is
sufficient to use this result on L and then express it in terms of H.
Proposition 2.7. If α1 , α2 , . . . , αn ∈ Q and ∂(α1 + α2 + · · · + αn ) = d then
L(α1 + α2 + · · · + αn ) ≤ (L(α1 )L(α2 ) . . . L(αn ))d
and
H(α1 + α2 + · · · + αn ) ≤
n
Y
(1 + ∂(αi ))H(αi )
(2.6)
!d
.
(2.7)
i=1
Proof. For the proof of (2.6) see Dubickas and Smyth [8]. In order to prove the inequality
(2.7) note that H(α) ≤ L(α) ≤ (1 + ∂(α))H(α) for any α ∈ Q. Thus (2.6) implies
H(α1 + α2 + · · · + αn ) ≤ L(α1 + α2 + · · · + αn )
≤ (L(α1 )L(α2 ) . . . L(αn ))d
!d
n
Y
≤
(1 + ∂(αi ))H(αi ) .
i=1
To finish this section we now briefly give four additional results from algebra and
number theory. The following theorem gives conditions which imply a polynomial is
irreducible and will be used later in Chapter 4.
Theorem 2.8. Eisenstein’s Criterion. Let P (x) = an xn + · · · + a0 ∈ Z[x]. If there exists
a prime q ∈ N such that q does not divide an , q divides an−1 , . . . , a0 , and q 2 does not
divide a0 , then P (x) is irreducible over Q.
12
Proof. The proof follows by way of contradiction. Let P (x) = an xn + · · · + a0 ∈ Z[x] and
assume there exists a prime q ∈ N dividing an−1 , . . . , a0 such that q does not divide an and
q 2 does not divide a0 . Suppose that P (x) = P1 (x)P2 (x) where P1 (x), P2 (x) ∈ Z[x] are two
non-constant polynomials. Let P1 (x) = bm xm + · · · + b0 and P2 (x) = ck xk + · · · + c0 . Then
reducing modulo q gives an xn ≡ P1 (x)P2 (x) mod q. Since an = bm ck is not divisible
by q, both bm and ck are not divisible by q. Thus P1 (x) and P2 (x) are non-constant
polynomials modulo q. It follows that x divides both P1 (x) and P2 (x) modulo q and
hence P1 (0) = b0 ≡ 0 mod q and P2 (0) = c0 ≡ 0 mod q. Since q divides both b0 and c0
it follows that q 2 divides b0 c0 = a0 , which is a contradiction.
Since we will often be considering polynomials with integer coefficients we will require the following theorem which allows us to factor an integer polynomial into the
product of integer polynomials if we can prove a factorization exists over the rationals. A
proof can be found in Chapter 9 of Dummit and Foote [9].
Theorem 2.9. Gauss’s Lemma. If Q(x) ∈ Z[x] is reducible over Q then Q(x) is reducible
over Z.
The following theorem gives that any finite extension of a field of characteristic 0
can be generated by a single element. A proof is in Chapter 14 of Dummit and Foote.
Theorem 2.10. If F is a field of characteristic 0 and K is a finite extension of F then
there exists α ∈ K such that K = F (α).
The final theorem of this section guarantees the existence of a prime number in
certain intervals and will be used later in Chapter 4. A proof can be found in Niven,
Zuckerman, and Montgomery [27].
Theorem 2.11. Bertrand’s Postulate. For every integer n ≥ 2 there exists a prime q ∈ N
that satisfies n < q < 2n.
13
2.2
p-adic Numbers
We now turn our attention to the basic algebraic and topological properties of the
p-adic numbers. The p-adic field is an extension of Q and was first introduced by Hensel
[16]. Hensel sought an analogy between the theory of analytic functions and the rational
numbers. Instead of constructing a Taylor series centered at a particular point to obtain
local information about an analytic function at the particular point Hensel wrote rational
numbers as a sum of powers of a prime number to obtain “local” information at the
prime. Gouvêa [14] gives an elementary introduction to the p-adic numbers. Koblitz [17]
and Robert [29] are also standard references on the p-adic numbers.
Definition 2.12. Let F be a field. An absolute value | ∗ | : F → R≥0 on F is a function
which satisfies |a| = 0 if and only if a = 0, and all a, b ∈ F satisfy |ab| = |a||b| and
|a + b| ≤ |a| + |b|.
Let p be a fixed prime. We will use the notation log to denote the logarithm base p.
If a ∈ Z\{0} then there exist unique m ∈ Z with gcd(m, p) = 1 and n ∈ N, n ≥ 0 such that
a = mpn . Define v(a) = n and if
a
b
∈ Q \ {0} with a, b integers then let v( ab ) = v(a) − v(b).
The p-adic absolute value on Q is defined by | ab |p = p−v(a/b) if
a
b
6= 0 and |0|p = 0. Since p
is fixed in order to simplify notation we will usually write | ∗ | = | ∗ |p . To prove | ∗ | is an
absolute value on Q first note that |a| = 0 if and only if a = 0. Moreover, if
and
c
d
a
b
1
= pn1 ( m
k1 )
2
= pn2 ( m
k2 ) with n1 , n2 , m1 , m2 , k1 , k2 ∈ Z, k1 , k2 6= 0 and m1 , m2 , k1 , k2 all relatively
prime to p then
a c n m1 n k1 a c n1 +n2 m1 m2 −(n1 +n2 )
n1
n2
p 2 = .
=p
= |p | |p | = p 1
= p
bd
k1 k2 k1 k2 b d
Now assume without loss of generality n1 ≤ n2 . Then
n
a c p 1 m1 k2 + pn2 m2 k1 n1 m1
n2 m2 +p
=
+ = p
b d
k1
k2 k1 k2
and since k1 and k2 are relatively prime to p
a c + = |pn1 m1 k2 + pn2 m2 k1 | = |pn1 ||m1 k2 + pn2 −n1 m2 k1 |.
b d
14
Since n1 ≤ n2 it follows that n2 − n1 ≥ 0 and m1 k2 + pn2 −n1 m2 k1 is an integer. From the
definition of | ∗ | any integer has p-adic absolute value at most 1. Thus
a c n a c o a c + ≤ |pn1 | = max , ≤ + b d
b
d
b
d
and | ∗ | is an absolute value on Q. Note that | ∗ | satisfies a condition stronger than
the standard triangle inequality. In particular, |a + b| ≤ max{|a|, |b|} for all a, b ∈ Q.
An absolute value that satisfies this inequality is said to be non-archimedean. Moreover,
|a − b| defines a metric on Q. A metric d defined on a space X is called an ultrametric if
d(a, b) ≤ max{d(a, c), d(c, b)} for all a, b, c ∈ X. It is clear that the metric induced by the
p-adic absolute value is an ultrametric on Q.
The p-adic numbers Qp are constructed by taking the completion of Q with respect
to the p-adic absolute value. If θ ∈ Qp then the p-adic absolute value on Q can be extended
to Qp be defining |θ| = limn→∞ |an | where {an } is a Cauchy sequence in Q converging
to θ. By considering the sequence {an } converging to θ it can be proved that | ∗ |p is a
non-archimedean absolute value on Qp (see Chapter 3 of Gouvêa [14]). Likewise define v
on θ ∈ Qp by v(θ) = − log |θ|. The p-adic integers are given by Zp = {x ∈ Qp : |x| ≤ 1}.
As with R, the completion of Q with respect to |∗|∞ , the field Qp is not algebraically
closed. For example, p1/2 ∈
/ Qp because if it were then the properties of | ∗ | would imply
|p1/2 | = p−1/2 , which is not an integer power of p. The algebraic closure of Qp will be
denoted by Qp . Since Q ⊂ Qp the field Q embeds in Qp . The p-adic absolute value can
be extended to all θ ∈ Qp as follows. If θ ∈ Qp has a minimal polynomial of degree n
then define |θ| = | Nm(θ)|1/n where Nm(θ) is the norm of θ over the field Qp given in the
previous section. It is demonstrated that this extension of | ∗ | to Qp is a non-archimedean
absolute value in Chapter 3 of Robert [29]. A direct consequence of the definition of Nm
is that if α ∈ Qp then the p-adic absolute value of α must equal the p-adic absolute value
of every conjugate of α.
In the real case after taking the algebraic closure of R we obtain C which is algebraically closed and topologically complete. However, Qp is not complete. In fact the
15
sequence {θn } ⊂ Qp which we will construct for Theorem 4.5 is an example of a Cauchy
sequence in Qp that does not converge in Qp . Define Cp to be the completion of Qp with
respect to | ∗ |. As when extending | ∗ | from Q to Qp it is possible to extend | ∗ | from Qp to
Cp to a non-archimedean absolute value by considering | ∗ | on Cauchy sequences converging to elements of Cp . Likewise v can be extended to θ ∈ Cp by defining v(θ) = − log |θ|.
The field Cp is topologically complete and algebraically closed (see Chapter 5 of Gouvêa
[14] for a proof).
The following proposition gives several basic properties of the p-adic numbers.
Proposition 2.13.
(i) Suppose a is an nonzero integer. Then 1 ≤ |a||a|∞ .
(ii) Let θ and η be elements of Cp . If |θ| =
6 |η| then |θ + η| = max{|θ|, |η|}.
(iii) If {θn } is a convergent sequence in Cp that does not converge to 0 then the sequence
{|θn |} is eventually constant.
(iv) Every nonzero θ ∈ Qp can be written uniquely in the form an pn + an+1 pn+1 + . . .
where n ∈ Z, an 6= 0, and 0 ≤ ai < p for all i ≥ n.
Proof. Statement (i) follows easily from the definition of | ∗ |. If a is a nonzero integer then
there exist a unique nonzero m ∈ Z relatively prime to p and n ∈ N such that a = mpn .
Thus |a||a|∞ = p−n pn |m| = |m| ≥ 1.
To prove (ii) without loss of generality assume |η| < |θ|. Then |θ + η| ≤ |θ| and
|θ| = |(θ + η) − η| ≤ max{|θ + η|, |η|} = |θ + η|.
Thus |θ + η| = |θ| = max{|θ|, |η|}.
To prove (iii) first assume {θn } is a sequence in Cp converging to θ ∈ Cp , θ 6= 0.
Then there exist > 0 and M1 ∈ N such that |θn | ≥ for all n ≥ M1 . Moreover,
16
there also exists M2 ≥ M1 for which |θn − θm | < for all n, m ≥ M2 . The properties
of the p-adic absolute value give |θn − θm | ≤ max{|θn |, |θm |} for all n, m ≥ M2 . Since
max{|θn |, |θm |} ≥ this implies |θn − θm | < max{|θn |, |θm |} and by (ii) it thus follows that
|θn | = |θm | for all n, m ≥ M2 .
The proof of (iv) requires a bit more work than the previous parts of the proposition.
Our proof follows that of Koblitz [17]. We will first consider the case where |θ| = 1. Note
that if θ = a0 + a1 p + a2 p2 · · · = b0 + b1 p + b2 p2 + . . . has two distinct representations of
this form then there exists k ≥ 0 such that ak 6= bk and ai = bi for all 0 ≤ i < k. Then
since ak − bk 6= 0 it follows from part (ii) of the proposition that
0 = |(a0 + a1 p + a2 p2 + . . . ) − (b0 + b1 p + b2 p2 . . . )|
= |(ak − bk )pk + (ak+1 − bk−1 )pk+1 + . . . |
= max{|(ak − bk )pk |, |(ak+1 − bk−1 )pk+1 + . . . |}
= p−k .
This is a contradiction and thus ai = bi for all i ≥ 0. Hence if such a representation exists
it must be unique.
In order to prove such a representation exists recall that we assumed |θ| = 1 and
θ ∈ Qp . Let {bi } be a Cauchy sequence in Q converging to θ. Since θ is nonzero in part
(iii) we proved there exists M2 ∈ N such that |bi | = 1 is constant for all i ≥ M2 . Without
loss of generality delete the first M2 terms in the sequence {bi } and relabel such that
bM2 +i is now bi . Since {bi } is Cauchy it is possible to define N (k) such that for all k ∈ N,
N (k) ≥ k and for all i, j ≥ N (k),
|bi − bj | ≤ p−k .
Let bN (i) =
ci
di
(2.8)
be in lowest terms. It follows that both ci and di are relatively prime
to p since |bi | = 1. In particular there exist hi , li ∈ Z satisfying hi di + li pi = 1. Let mi be
the integer such that 0 ≤ hi ci + mi pi ≤ pi − 1 and define αi = hi ci + mi pi . The sequence
17
{αi } will provide us with the desired representation of θ. Now note
αi − bN (i) = ci hi di + di mi pi − 1 = hi di − 1 + di mi pi di ci
ci
and recall that hi di + li pi = 1 so we have
di d
i
i
i
−i
αi − bN (i) = −li p + mi p ≤ p max |−li | , mi .
ci ci
Since li , di and mi are integers and ci is relatively prime to p it follows that
|αi − bN (i) | ≤ p−i .
(2.9)
We now prove αj+1 ≡ αj mod pj for all j ≥ 1. Note that
|αj+1 − αj | ≤ max{|αj+1 − bN (j+1) |, |bN (j+1) − bN (j) |, |bN (j) − αj |}.
From (2.8) and (2.9) it thus follows that
n
o
|αj+1 − αj | ≤ max p−(j+1) , p−j , p−j = p−j .
(2.10)
Since the αi are integers the definition of the p-adic absolute value implies αj+1 = αj +aj pj
for some aj ∈ Z. From the definition of each αi recall that 0 ≤ αi < pi . Thus 0 ≤ aj < p.
Letting a0 = α1 it follows that
αj+1 = αj +aj pj = αj−1 +aj−1 pj−1 +aj pj = · · · = a0 +a1 p+· · ·+aj−1 pj−1 +aj pj . (2.11)
Thus limj→∞ αj is of the desired form and all that remains to demonstrate is that
limj→∞ αj = θ. Note that if i > j then
|αi − αj | ≤ max{|αi − αi−1 |, |αi−1 − αi−2 |, . . . , |αj+1 − αj |}
n
o
≤ max p−(i−1) , p−(i−2) , . . . , p−(j+1) , p−j
= p−j .
Since {bj } is a Cauchy sequence converging to θ it suffices to prove limi→∞ |αi − bi | = 0.
Apply the previous inequality and (2.9), (2.8) to obtain for any i ≥ N (j)
|αi − bi | ≤ max{|αi − αj |, |αj − bN (j) |, |bi − bN (j) |} ≤ max p−j , p−j , p−j = p−j .
18
Thus {ai } converges to θ.
Recall that we assumed |θ| = 1. In the case where |θ| = p−n for some integer
n then p−n θ will satisfy |p−n θ| = 1 and thus have a unique representation of the form
a0 + a1 p + . . . with a0 6= 0 and 0 ≤ ai < p for i ≥ 0. It follows that a0 pn + a1 pn+1 + . . .
must be a unique representation of θ in the desired form.
The topology of Cp is substantially different from the topology of R or C. Proofs
and discussion of the topological properties of Cp can be found in Robert [29] and Gouvêa
[14]. The space Cp is totally disconnected and any ball of positive radius is both open
and closed. Moreover, given any two open (or closed) balls in Cp one is contained in the
other, or they are disjoint. Although Qp and all finite extension fields of Qp are locally
compact both Qp and Cp are not. In particular, any nonempty closed bounded ball of
positive radius in Qp and Cp is not compact. However, there is still a p-adic analogue of
the maximum modulus principle (see Chapter 6 of Robert [29] for a proof).
Proposition 2.14. Let f (x) be a power series centered at 0 with coefficients in Cp . If
r is a rational power of p and f (x) converges on the closed ball centered at 0 of radius r
then
sup |f (x)| = max |f (x)| = sup |f (x)| = max |f (x)|.
|x|≤r
|x|≤r
|x|=r
|x|=r
Now that the basic structure of the p-adic numbers has been given we state several
more results on Qp . The following theorem is proved in Chapter 3 of Robert [29].
Theorem 2.15. Given n ∈ N there are only finitely many extensions of Qp of degree n
in Qp .
Another important theorem that takes advantage of the properties of the p-adic
numbers is Krasner’s Lemma which states that if α, β ∈ Qp and β is sufficiently close to
α with respect to | ∗ | then the field generated by β contains the field generated by α.
19
Theorem 2.16. Krasner’s Lemma. Let K be a finite extension field of Qp and α ∈ Qp .
Let α = α1 , α2 , . . . αn ∈ Qp denote the conjugates of α over K. If β ∈ Qp satisfies
|β − α| < |α − αi | for all 2 ≤ i ≤ n then K(α) ⊆ K(β).
Proof. The proof follows by contradiction. Assume |β − α| < |α − αi | for all 2 ≤ i ≤ n
and α ∈
/ K(β). It follows that [K(β, α) : K(β)] ≥ 2. Let σ be a field homomorphism from
K(β, α) to Qp such that K(β) is fixed and σ(α) = αi for some 2 ≤ i ≤ n. Recall from our
definition of | ∗ | on Qp that |θ| = |σ(θ)| for any θ ∈ K(β, α). Thus
|β − α| = |σ(β − α)| = |σ(β) − σ(α)| = |β − σ(α)|.
It follows that |α − σ(α)| ≤ max{|α − β|, |β − σ(α)|} = |α − β|, which contradicts the
assumption |β − α| < |α − αi | for 2 ≤ i ≤ n.
If α ∈ Qp let mα,Qp (x) ∈ Qp [x] be the minimal polynomial of α over Qp . From
part (iv) of Proposition 2.13 each coefficient of mα,Qp (x) can be uniquely written as a
series in powers of p. Let k be the smallest power of p appearing in the expansions
of the coefficients of mα,Qp (x). The polynomial p−k mα,Qp (x) must have coefficients in
Zp = {θ ∈ Qp : |θ| ≤ 1} and will be referred to as the minimal polynomial of α over Zp .
Now let Q(x) = an xn + · · · + a0 be a polynomial with coefficients in Qp and define
kQk by kQk = max{|an |, . . . , |a0 |}. If P (x) ∈ Qp [x] then kP − Qk defines a metric on
Qp [x]. To prove this first note kP − Qk = 0 if and only if the coefficients of P equal
the coefficients of Q. Thus kP − Qk = 0 if and only if P = Q. Since |a| = | − a| for
all a ∈ Qp it is clear that kP − Qk = kQ − P k. Finally, if P (x) = bm xm + · · · + b0
and m ≥ n then the coefficients of P − Q are of the form bi if n < i ≤ m, or bi − ai if
0 ≤ i ≤ n. Thus if n < i ≤ m then |bi | ≤ kP k ≤ max{kP k, kQk} and if 0 ≤ i ≤ n then
|bi − ai | ≤ max{|bi |, |ai |} ≤ max{kP k, kQk}. Hence the triangle inequality is satisfied and
k∗k induces an ultrametric on Qp [x]. A sequence of polynomials {Qm } in Qp [x] converging
to Q with respect to k∗k must be such that the p-adic absolute value of the coefficients of
Qm − Q tend to zero as m increases.
20
The following corollary to Krasner’s Lemma demonstrates that if two polynomials
are close with respect to k∗k then they must also have roots that are close with respect
to | ∗ | and as a result the roots satisfy the hypotheses of Krasner’s Lemma. This will be
used in Chapter 4 and to prove Q is dense in Cp .
Corollary 2.17. Let α ∈ Qp and let Q(x) ∈ Zp [x] be the minimal polynomial of α over
Zp . Let n be the degree of Q and suppose {Qm (x)} ⊂ Zp [x] is a sequence that converges
to Q(x) with respect to k∗k. Denote the degree of Qm by nm and suppose nm ≤ n for all
m ∈ N. Then there exist M ∈ N and a sequence {βm } ⊂ Qp such that for all m ≥ M , we
have Qp (α) = Qp (βm ), Qm (βm ) = 0, and
|α − βm | ≤ ckQ − Qm k1/n
(2.12)
for some constant c depending only on α.
Proof. Let α, Q, n, and {Qm } be as in the statement of the corollary and suppose that
Qm (x) = bm,n xn + bm,n−1 xn−1 + · · · + bm,0 and Q(x) = an xn + · · · + a0 where it is possible
bm,n = 0. Since the sequence {Qm } converges to Q with respect to k∗k the sequence
{bm,n } must converge to an 6= 0 and hence part (iii) of Proposition 2.13 implies there
exists M1 ∈ N such that |bm,n | = |an | for all m ≥ M1 . It follows that the degree of Qm is
n for all m ≥ M1 .
From now on we will assume m ≥ M1 and let c = |an |. Let βm,1 , βm,2 , . . . , βm,n ⊂ Qp
be the (not necessarily distinct) roots of Qm . Then
|Q(α) − Qm (α)| = |Qm (α)| = |bm,n |
n
Y
i=1
|α − βm,i | ≥ c min |α − βm,i |n .
1≤i≤n
(2.13)
Let βm be a root of Qm for which the minimum is attained. Note that the p-adic triangle
inequality implies
|Q(α) − Qm (α)| = |(an − bm,n )αn + · · · + (a0 − bm,0 )| ≤ kQ − Qm k max{1, |α|n }
and the inequality (2.13) becomes
c|α − βm |n ≤ kQ − Qm k max{1, |α|n }.
21
Thus
|α − βm | ≤ c−1/n max{1, |α|}kQ − Qm k1/n .
The term c−1/n max{1, |α|} is a constant depending only on α and (2.12) follows.
It only remains to prove that Qp (α) = Qp (βm ) for all m sufficiently large. Since
{kQ − Qm k} converges to 0 there exists M2 ≥ M1 such that for all m ≥ M2
|α − βm | ≤ c−1/n max{1, |α|}kQ − Qm k1/n ≤ min{|α − αi |}
where the minimum is taken over all conjugates αi of α not equal to α. Krasner’s Lemma
implies Qp (α) ⊆ Qp (βm ) for all m ≥ M2 . Since βm is a root of Qm the degree of βm over
Qp is less than or equal to n for all m ≥ M2 . It follows that Qp (βm ) = Qp (α) for all
m ≥ M2 .
Corollary 2.18. The field Q is dense in Cp .
Proof. Since Cp is the completion of Qp it suffices to prove Q is dense in Qp . Suppose
α ∈ Qp . Let Q(x) = an xn + · · · + a0 ∈ Zp [x] be the minimal polynomial of α over Zp .
Write each coefficient of Q in its base p expansion such that ai = ai,0 + ai,1 p + ai,2 p2 + . . .
and define the sequence of polynomials {Pm (x)} ⊂ Z[x] by Pm (x) = bm,n xn + · · · + bm,0
where bm,i = ai,0 + ai,1 p + · · · + ai,m pm . The sequence {Pm } converges to Q with respect
to k∗k and the hypotheses of Corollary 2.17 are satisfied. Thus there exist M ∈ N, a
sequence {βm } ∈ Qp , and a constant c depending only on α such that for all m ≥ M ,
|α − βm | ≤ ckQ − Qm k1/n
and Qm (βm ) = 0. Since Qm (x) ∈ Z[x] it follows that βm ∈ Q for all m ≥ M . Thus {βm }
is a sequence of algebraic numbers converging to α and the result follows.
22
3
PROPERTIES OF O ON CP
Let p be a fixed prime and define log to be the logarithm to the base p. Denote the
p-adic absolute value by | ∗ | and the standard absolute value by | ∗ |∞ . Recall that we
denote the degree of an integer polynomial P by ∂(P ) and the maximum of the absolute
value of the coefficients of P by H(P ). Define Λ by Λ(P ) = 2∂(P ) H(P ). Using Mahler’s
order functions as a template we define O on Cp as follows.
Definition 3.1. Let θ ∈ Cp . Define O(∗|θ) : N → R by
O(u|θ) = max log
Λ(P )≤u
P (θ)6=0
1
|P (θ)|
where the P are polynomials in Z[x].
The maximum can be used in this definition because for any u ∈ N there are only
finitely many integer polynomials P satisfying Λ(P ) ≤ u since a bound on Λ bounds both
the degree and the height of a polynomial. Also note that when defining the complex
order functions Mahler used Λ0 (P ) = 2∂(P ) L(P ), where L(P ) is the sum of the absolute
values of the coefficients of P, instead of Λ(P ) = 2∂(P ) H(P ). This change does not affect
any of the results in the complex case or p-adic case since it can be shown that the terms
introduced by applying the inequalities H(P ) ≤ L(P ) ≤ (1 + ∂(P ))H(P ) when changing
from Λ0 to Λ are negligible. As with the complex case for a fixed θ ∈ Cp the function
O(u|θ) is a non-negative increasing function because Λ(1) = 1. We will use the same
partial order Mahler defined on the complex order functions.
Definition 3.2. Let a(u) and b(u) be non-decreasing functions from N to R which are
positive for sufficiently large u. Define the partial order by a(u) b(u) (or equivalently
b(u) a(u)) if there exist c, u0 ∈ N and γ ∈ R+ such that a(uc ) ≥ γb(u) for all u ≥ u0 .
Define the equivalence relation by a(u) b(u) if a(u) b(u) and b(u) a(u).
23
Define the partial order on Cp by θ η if O(u|θ) O(u|η) and define the
equivalence relation on Cp by θ η if O(u|θ) O(u|η).
In Section 3.1 we first prove O(u|θ) log u for all θ ∈ Cp and then show a result of
Escassut [12] implies there exist transcendental θ ∈ C such that O(u|θ) log u. The main
result of Section 3.2 is that if θ, η ∈ Cp and η is algebraic over Q(θ) then O(u|θ) O(u|η).
From this it will follow that O(u|α) log u for all α ∈ Q.
3.1
Preliminary Results
We first prove the following proposition which gives a lower bound on O.
Proposition 3.3. Let θ ∈ Cp . Then O(u|θ) log u.
Proof. For any u ∈ N, u > p let n ∈ N be such that pn ≤ u < pn+1 . Let Qn (x) be the
constant polynomial pn . Then
1
O(u|θ) = max log
|P (θ)|
Λ(P )≤u
1
≥ log
|Qn (θ)|
= log(pn ) = n > log u − 1 > c log u
where c is some positive constant. Thus O(u|θ) log u.
Recall that in the complex case all transcendental θ ∈ C satisfy O(u|θ) (log u)2 .
This does not hold in the p-adics. The following theorem is due to Escassut [12] and
will not be proved here. A direct consequence is that there exist transcendental θ ∈ Cp
for which O(u|θ) log u. Recall that kP k denotes the maximum of the coefficients of
P (x) ∈ Qp [x] with respect to the p-adic absolute value.
24
Theorem 3.4. There exist θ ∈ Cp transcendental over Qp such that for all P (x) ∈ Qp [x],
log |P 1(θ)| ≤ − logkP k + c∂(P ) where c is some constant depending on θ.
Corollary 3.5. There exist θ ∈ Cp transcendental over Q such that O(u|θ) log u.
Proof of Corollary 3.5. Let θ be as in Theorem 3.4. If θ is transcendental over Qp then
θ is transcendental over Q because Q is contained in Qp . Given u ∈ N let P (x) ∈ Z[x]
be such that O(u|θ) = log |P 1(θ)| and Λ(P ) ≤ u. Since all polynomials in Z[x] are also in
Qp [x] Theorem 3.4 implies that
O(u|θ) = log
1
1
≤ log
+ c∂(P )
|P (θ)|
kP k
(3.1)
where c is a constant depending only on θ. Note that kP k = |a| where a ∈ Z is some
coefficient of P . By part (i) of Proposition 2.13
1
1
=
≤ |a|∞ ≤ H(P ).
kP k
|a|
Since Λ(P ) = 2∂(P ) H(P ) it follows from (3.1) that
O(u|θ) ≤ log H(P ) + c∂(P ) ≤ log Λ(P ) + c
1
(log Λ(P ) − log H(P )) ≤ c0 log Λ(P )
log 2
where c0 is a constant depending only on θ. Thus O(u|θ) log u and by Proposition 3.3
O(u|θ) log u.
3.2
Equivalence of Order Functions
Recall that if θ and η are elements of Cp then θ and η are algebraically dependent
over a subfield K ⊆ Cp if θ is algebraic over K(η) and η is algebraic over K(θ). If we state
two numbers are algebraically dependent without mentioning a base field it is implied
they are algebraically dependent over Q. Our main result for this section is the following
theorem.
25
Theorem 3.6. Let θ, η ∈ Cp be nonzero and such that η is algebraic over Q(θ). Then
O(u|θ) O(u|η).
Before proving Theorem 3.6 we first demonstrate the following lemma.
Lemma 3.7. Let M = (ai,j ) be an n by n matrix. Then
| det(M )| ≤
n
Y
j=1
max {|ai,j |}
1≤i≤n
Proof of Lemma 3.7. The Leibniz formula for the determinant states


n
X
Y
sgn(σ)
det(M ) =
aσ(j),j  .
j=1
σ∈Sn
Taking the p-adic absolute value of this gives



n
n

Y
Y
| det(M )| ≤ max sgn(σ)
aσ(j),j ≤
max {|ai,j |}.

1≤i≤n
σ∈Sn 
j=1
j=1
In the complex case Mahler [22] originally proved the order functions of algebraically
dependent elements are equivalent. Later Durand [10] presented an alternative proof. Our
proof that algebraically dependent elements of Cp have equivalent order functions follows
Durand’s proof with only a few changes to take into account the p-adic absolute value and
our different definition of Λ.
Proof of Theorem 3.6. If η is algebraic over Q(θ) then there exists P (x, y) ∈ Z[x, y] such
that P (θ, η) = 0 and
P (x, y) =
n
X
ai (x)y i
i=0
with ai (x) ∈ Z[x], an (θ) 6= 0. Fix u ∈ N. Let Q(y) ∈ Z[y] be a polynomial that satisfies
1
and Λ(Q) ≤ u. Let
O(u|η) = log |Q(η)|
Q(y) =
q
X
i=0
bi y j
26
with bq 6= 0. Note that Q, q, and the bi depend on u while P, n, and the ai (x) depend only
on θ and η.
Consider P (x, y) as a polynomial in y with coefficients in Z[x]. Using the notation
from Definition 2.1

a (x)
bq
 n

 an−1 (x) an (x)
bq−1 bq


..
..
..
..
.
. . a (x)

.
.
.
.
n


.
..
Ms (P (x, y), Q(y)) = 
b0 b1
 a0 (x) a1 (x)

..

a0 (x)
.
b0


..

..
.

.

a0 (x)

..
. bq
..
.
..
.
.
..
. ..
b0








.








Also define the polynomial T (x) ∈ Z[x] to be R(P (x, y), Q(y)) = det(Ms (P (x, y), Q(y))),
the resultant of P (x, y) and Q(y). Since the polynomials ai depend only on η and θ and
the bi are constant polynomials in x applying Proposition 2.6 to Ms (P (x, y), Q(y)) implies
there exists a constant c1 ≥ 1 depending only on η and θ such that
∂(T ) ≤
q
X
j=1
max ∂(ai ) ≤ q max (∂(ai )) = c1 q.
0≤i≤n
0≤i≤n
(3.2)
Proposition 2.6 also implies



q X
n+q
q
n
Y X
Y
(1 + ∂(bi ))H(bi )
H(T ) ≤ 
(1 + ∂(ai ))H(ai ) 
j=q+1 i=0
j=1 i=0
where the first factor corresponds to the first q columns of Ms (P (x, y), Q(y)) and the
second factor corresponds to the remaining n columns of Ms (P (x, y), Q(y)). Since the
sums in both factors are independent of j it follows that
!q
!n
q
n
X
X
H(T ) ≤
(1 + ∂(ai ))H(ai )
H(bi ) .
i=0
i=0
Note that the ai and n depend only on η and θ, and that H(bi ) = |bi |∞ ≤ H(Q). Thus
H(T ) ≤ cq2 ((q + 1)H(Q))n
(3.3)
27
where c2 ≥ 1 is another constant depending only on η and θ. From the definition of Λ it
follows from (3.2) and (3.3) that
0
Λ(T ) = 2∂(T ) H(T ) ≤ (2c1 c2 )q (q + 1)n H(Q)n ≤ (2q )c (q + 1)n H(Q)n .
for some constant c0 depending only on θ and η. For q sufficiently large the term (2q )c
0
will be greater than (q + 1)n and thus there exists a constant c ≥ n depending only on θ
and η such that
Λ(T ) ≤ (2q )c H(Q)n ≤ (Λ(Q))c .
Hence
log
1
|T (θ)|
≤
max
Λ(P )≤Λ(Q)c
log
1
|P (θ)|
= O(Λ(Q)c |θ) ≤ O(uc |θ).
Now that we have proved
log
1
|T (θ)|
≤ O(uc |θ)
the next step will be to find a lower bound on log
the matrix

an (θ)
1
|T (θ)|
(3.4)
in terms of O(u|η). Consider

bq


 an−1 (θ) an (θ)
bq−1 bq


..
..
..
..
..

. an (θ)
.
.
.
.


.
..
M =
b0 b1
 a0 (θ) a1 (θ)

.

..
a0 (θ)
b0


..

..
.

.

a0 (θ)
..
. bq
..
.
..
.
.
..
. ..
b0

















and note M = Ms (P (θ, y), Q(y)). By applying elementary row operations construct a new
matrix M 0 as follows. For every 1 ≤ i ≤ n + q multiply the ith row of M by η n+q−i and
add it to the last row. Thus the lower left entry of M 0 becomes
an (θ)η n+q−1 + an−1 (θ)η (n−1)+q−1 + · · · + a0 (θ)η q−1 = η q−1 P (θ, η).
By simplifying the rest of the entries in this manner the bottom row of M 0 is equal to
η q−1 P (θ, η), η q−2 P (θ, η), . . . , P (θ, η), η n−1 Q(η), η n−2 Q(η), . . . , Q(η).
28
Since P (θ, η) = 0 the first q entries of the bottom row of M 0 are 0. Let Mi be the
determinant of the minor obtained by deleting the bottom row and ith column of M 0 .
Then expanding along the last row of M 0 gives
X
n+q+q+i
n−i
0
(−1)
η Q(η)Mq+i | det(M )| = | det(M )| = 1≤i≤n
≤ |Q(η)| max{|η n−1 |, 1} max {|Mq+i |}.
1≤i≤n
Now apply Lemma 3.7 to the |Mq+i |. The bi are integers and must satisfy |bi | ≤ 1 for all
i. Thus when applying Lemma 3.7 we only consider the first q columns. Hence
q
n−1
| det(M )| ≤ |Q(η)| max{|η
|, 1} max {|ai (θ)|} .
0≤i≤n
(3.5)
To simplify notation let κ = max{∂(a0 ), . . . ∂(al )}, and note that κ depends only on η
and θ because the ai depend only on η and θ. The ai (θ) are polynomials with integer
coefficients in θ. Since the p-adic absolute value of any integer is at most 1 it follows
|ai (θ)| ≤ max{|θ|κ , 1} for 0 ≤ i ≤ n. Thus (3.5) becomes
| det(M )| ≤ |Q(η)| max{|η n−1 |, 1} max{|θ|qκ , 1}.
Let c3 and c4 be constants depending only on θ and η such that c3 = max{1, |η|n−1 } and
if |θ| ≥ 1 then |θ|κ = 2c4 and c4 = 1 otherwise. Substituting these constants into the last
inequality gives
|T (θ)| = | det(M )| ≤ c3 |Q(η)|(2q )c4
log |T (θ)| ≤ log c3 + log |Q(η)| + c4 log(2q ).
Having now obtained an upper bound for log |T (θ)| recall that 2q ≤ 2q H(Q) = Λ(Q) ≤ u
and apply (3.4) to obtain
1
1
O(u |θ) ≥ log
≥ log
− log c3 − c4 log u.
|T (θ)|
|Q(η)|
1
Since Q was defined such that log |Q(η)|
= O(u|η) it follows that
c
O(uc |θ) + c4 log u + log c3 ≥ O(u|η).
29
By Proposition 3.3 there exists a positive constant c5 depending on θ such that for all sufficiently large u, O(u|θ) ≥ c5 log u and consequently there exists another positive constant
c6 depending only on θ and η such that
O(u|η) ≤ O(uc |θ) + c4 log u + c3 ≤ c6 O(uc |θ).
From the definition of the partial ordering it thus follows that O(u|θ) O(u|η).
Corollary 3.8. If θ, η ∈ Cp are algebraically dependent then O(u|θ) O(u|η).
Proof. This follows directly from Theorem 3.6 by interchanging θ and η.
Corollary 3.9. If α ∈ Q then O(u|α) log u.
Proof. By Corollary 3.8 it suffices to prove O(u|1) log u. This is because Q(1) = Q and
hence 1 and α are algebraically dependent for any α ∈ Q. From Lemma 3.3 O(u|1) log u
and thus it suffices to prove O(u|1) log u.
Fix u ∈ N and let n be such that pn ≤ u < pn+1 . Let Q(x) = aq xq + · · · + a0 ∈ Z[x]
be a nonzero polynomial of degree q satisfying Λ(Q) = 2q H(Q) ≤ u. Then
|Q(1)|∞ ≤ |aq |∞ + · · · + |a0 |∞ ≤ (q + 1)H(Q) ≤
q + 1 n+1
q+1
u<
p
< pn+1
q
2
2q
and it follows that the largest power of p that can divide Q(1) is pn . Thus for all integer
polynomials Q with Λ(Q) ≤ u and Q(1) 6= 0 we have |Q(1)| ≥ p−n . Moreover, the constant
polynomial defined by Qn (x) = pn satisfies Λ(Qn ) ≤ u and |Qn (1)| = p−n . Hence
1
O(u|1) = log
= n ≤ log u.
|pn |
and thus O(u|1) << log u.
With the exception of Corollary 3.5, which stated that there are transcendental
θ ∈ Cp for which O(u|θ) log u, all of our results in this chapter hold in the complex
case. In particular, the results that every θ ∈ Cp satisfies O(u|θ) log u, algebraically
dependent elements are equivalent under , and every α ∈ Q satisfies O(u|α) log u were
all proved in the complex case in the original paper on order functions by Mahler [22].
30
4
RESULTS ON O ∗
∗ in the complex case. Most results are
Relatively little work has been done with OM
due to Durand [11]. Here we again use Mahler’s order functions as motivation and define
∗ on C . If α ∈ Q recall that we defined the
an analogue to the complex order function OM
p
minimal polynomial of α over Z to be the minimal polynomial of α over Q multiplied by
the least common multiple of the denominators of its coefficients. Also recall that if m is
the minimal polynomial of α over Z then we defined Λ(α) = Λ(m).
Definition 4.1. Let θ ∈ Cp . Define O∗ (∗|θ) : N \ {1} → R by
O∗ (u|θ) = max log
Λ(α)≤u
α6=θ
1
|α − θ|
where α ∈ Q.
As with O the maximum can be used in the definition because given any u ∈ N
there are only finitely many α ∈ Q with minimal polynomial m over Z which satisfy
Λ(α) = Λ(m) ≤ u. From Corollary 2.18, Q is dense in Cp and hence O∗ (u|θ) is a positive
increasing function for sufficiently large u. Thus the partial order and equivalence
relation defined in Definition 3.2 can be used on the set of functions {O∗ (u|θ) : θ ∈ Cp }.
Define the partial order ∗ on Cp by θ ∗ η if O∗ (u|θ) O∗ (u|η) and define the
equivalence relation ∗ on Cp by θ ∗ η if O∗ (u|θ) O∗ (u|η).
Since we will be considering extensions of both Qp and Q in this chapter in order
to avoid confusion when the field is not specified the term “degree” will always refer to
the degree over Q. The main result of Section 4.1 is that if α ∈ Qp then O∗ (u|α) log u.
In Section 4.2 we construct θ ∈ Cp for which O∗ (u|θ) grows slowly and in Theorem 4.6
we construct a lower bound for these θ under certain conditions. A particular case of this
result will then allow us to construct θ ∈ Cp which satisfy O∗ (u|θ) logn u, where logn
denotes the logarithm composed n times and n ≥ 3.
31
4.1
A Lower Bound for O ∗ on Elements of Qp
Given α ∈ Qp Theorem 2.16 (Krasner’s Lemma) will allow us to construct a sequence in Qp converging to α for which the terms in the sequence are roots of polynomials
converging to the minimal polynomial of α. This idea will be used to bound O∗ from
below when considering elements of Qp .
Theorem 4.2. If α ∈ Qp then O∗ (u|α) log u.
Proof. Assume α ∈ Qp and let R(x) ∈ Qp [x] be the minimal polynomial of α over Qp .
Recall that the minimal polynomial of α over Zp = {a ∈ Qp : |a| ≤ 1} is defined by
multiplying R by the reciprocal of the least power of p appearing in the base p expansion
of the coefficients of R. Let Q(x) = an xn + · · · + a0 ∈ Zp [x] be the minimal polynomial
of α over Zp . Given m ∈ N let 0 ≤ ai,m ≤ pm − 1 be the positive integers such that
ai,m ≡ ai mod pm . Define
Qm (x) = (an,m xn + · · · + a0,m ) + pm ∈ Z[x].
Recall that for a polynomial P (x) ∈ Qp [x] we defined kP k to denote the maximum p-adic
absolute value of the coefficients of P . By construction the sequence {Qm } converges to
Q with respect to k∗k and ∂(Qm ) ≤ n for all m ∈ N. Thus Corollary 2.17 implies there
exist M ∈ N and a sequence {βm } ⊂ Qp such that if m ≥ M then Qp (α) = Qp (βm ),
Qm (βm ) = 0, and the inequality |α − βm | ≤ c1 kQ − Qm k1/n is satisfied for some positive
constant c1 depending on α. Since the Qm are integer polynomials we have {βm } ⊂ Q. It
follows by the definition of the Qm that
|α − βm | ≤ c1 kQ − Qm k1/n ≤ c1 |pm |1/n = c1 p−m/n .
(4.1)
Increase M as necessary such that taking the logarithm gives that for all m ≥ M
log
1
m
≥
− log c1 ≥ c2 m
|α − βm |
n
(4.2)
32
where c2 is another positive constant depending only on α.
Since Qp (α) = Qp (βm ) for all m ≥ M the degree of the minimal polynomial of βm
over Qp must be n. Moreover, the field Q is contained in Qp and thus n ≤ ∂(βm ). Since
βm is a root of the integer polynomial Qm and ∂(Qm ) ≤ n it follows that ∂(βm ) = n.
Thus Qm must be a rational integer multiple of the minimal polynomial of βm over Z for
all m ≥ M . From the definition of Qm the coefficients of Qm are between 0 and pm − 1
except for the constant coefficient which must be between pm and pm − 1 + pm = 2pm − 1.
It follows that the height of Qm is the standard absolute value of the constant coefficient
and consequently
pm ≤ |a0,m + pm |∞ = H(Qm ) < 2pm .
Thus for all m ≥ M
2n pm ≤ Λ(Qm ) = 2n H(Qm ) < 2n (2pm ) = 2n+1 pm .
(4.3)
In particular the sequence {Λ(Qm )} is strictly increasing because
Λ(Qm ) < 2n+1 pm ≤ 2n pm+1 ≤ Λ(Qm+1 ).
Hence given u ≥ Λ(QM ) there exists a unique m ≥ M such that Λ(Qm ) ≤ u < Λ(Qm+1 ).
Take the logarithm and apply (4.3) to obtain
log u < log Λ(Qm+1 ) < log(2n+1 pm+1 ) = (n + 1) log 2 + m + 1 ≤ c3 m
where c3 is a positive constant depending only on α. Combining this with (4.2) implies
there exists a positive constant c depending only on α which satisfies
log
1
> c log u.
|α − βm |
Since m ≥ M and Qm is an integer multiple of the minimal polynomial of βm over Z it
follows that Λ(βm ) ≤ Λ(Qm ) ≤ u. Thus the definition of O∗ gives that for all u ≥ Λ(QM ),
O∗ (u|α) ≥ log
1
> c log u.
|α − βm |
33
In the next chapter we will prove O∗ (u|θ) O(u|θ) for all θ ∈ Cp . Corollary 3.9
states O(u|α) log u for all α ∈ Q and thus Theorem 4.2 will imply O∗ (u|α) log u for
all α ∈ Q.
4.2
Elements of Cp for Which O ∗ Grows Slowly
Recall that the fields Qp and Cp are not locally compact. Consequently given any
finite extension field K of Qp there are always elements of Qp that cannot be approximated
to arbitrary precision by elements of K. For example, if θ ∈ Qp with 1 < |θ| < p then
for any nonzero γ ∈ Qp , |γ − θ| = max{|θ|, |γ|} because v(γ) ∈ Z. Applying this idea
will allow us to construct elements of Cp for which O∗ grows slowly. We first prove the
following two lemmas.
Lemma 4.3. Let n ∈ N and define Kn to be the extension field of Qp generated by
adjoining the roots of all polynomials in Qp [x] with degree less than or equal to n. If q ∈ N
is any prime greater than n and f ∈ Z is not divisible by q then there does not exist
α ∈ Kn satisfying |α| = pf /q .
Proof. From Theorem 2.15 there are only finitely many extensions of Qp of a given finite
degree. By Theorem 2.10 there exist α1 , . . . αm ∈ Qp such that Qp (α1 ), . . . , Qp (αm ) is
every extension of Qp of degree less than or equal to n. It follows that
Kn = Qp (α1 , . . . , αm ).
For all 1 ≤ i ≤ m we have
[Qp (α1 , . . . , αi−1 , αi ) : Qp (α1 , . . . , αi−1 )] ≤ [Qp (αi ) : Qp ] ≤ n
and hence
m
Y
[Kn : Qp ] = [Qp (α1 , . . . , αm ) : Qp ] =
[Qp (α1 , . . . , αi ) : Qp (α1 , . . . , αi−1 )],
i=1
(4.4)
34
where we define the i = 1 term to be [Qp (α1 ) : Qp ]. From (4.4) it follows that each of the
terms in the product are less than or equal to n and therefore
n
Y
[Kn : Qp ] =
ibi
i=2
where each bi ≥ 0 is an integer. Since every element of Kn must have degree over Qp
dividing [Kn : Qp ] it follows that if α ∈ Kn then the degree of α over Qp is of the form
d=
n
Y
idi
i=2
where each di ≥ 0 is an integer. In particular, d is not divisible by any prime greater than
n. From the definition of the p-adic absolute value on Qp it follows that
|α| = (pa )1/d = pa/d
for some a ∈ Z. Thus |α| cannot be of the form pf /q where q ∈ N is a prime greater than
n and f ∈ Z is not divisible by q.
Lemma 4.4. If f, q ∈ N with q prime and f not a multiple of q then the minimal polynomial of pf /q over Q is xq − pf .
Proof. Since f and q are relatively prime there exist a, b ∈ Z such that af + bq = 1. In
particular,
af
q
+ b = 1q . Thus
p1/q = (pf /q )a pb ∈ Q(pf /q )
and hence Q(p1/q ) ⊆ Q(pf /q ). Since (p1/q )f = pf /q it follows that Q(p1/q ) = Q(pf /q ) and
thus ∂(p1/q ) = ∂(pf /q ). The polynomial xq − p has p1/q as a root and xq − p satisfies
Theorem 2.8 (Eisenstein’s Criterion). Thus xq − p is irreducible and must be the minimal
polynomial of p1/q over Q. It follows that ∂(p1/q ) = ∂(pf /q ) = q and hence the minimal
polynomial of pf /q over Q is xq − pf .
Let M0 ∈ R+ . Suppose g : [M0 , ∞) → R+ is an unbounded increasing function for
which there exist positive integers s, m, and M ≥ M0 such that
g(sk + s) − g(sk) ≥
3
mk
(4.5)
35
for all positive integers k ≥ M . Without loss of generality assume m ≥ 3. Theorem 2.11
(Bertrand’s Postulate) states that for all k ≥ 1 there exists a prime number qk which
satisfies
mk < qk < 2mk < mk+1 .
Since g(sk + s) − g(sk) ≥
3
mk
>
3
qk
(4.6)
there exist three consecutive positive integers fk , fk + 1,
and fk + 2 which satisfy
g(sk) ≤
fk
fk + 1
fk + 2
<
<
≤ g(sk + s).
qk
qk
qk
(4.7)
Let Ak be the intersection of {fk , fk + 1, fk + 2} with the set of integers relatively prime
to qk and note that Ak must contain at least two elements since qk is a prime greater than
or equal to 5. Now let B be an infinite binary sequence and define fk∗ as follows. If the
kth term of B is 0 let fk∗ be the smallest element of Ak and otherwise define fk∗ to be the
largest element of Ak . Let hk =
fk∗
qk
and define
θg,B =
∞
X
phk
(4.8)
k=M
where M is given in the definition of g. Since hk ≥ g(sk) the sum defining θg,B converges
with respect to the p-adic absolute value because g(sk) is increasing and unbounded.
The conditions on g are very general since a function which satisfies equality in (4.5)
for all k must be a geometric series and thus bounded. In fact we will prove later that
for any finite number of compositions of logarithms the function log log · · · log(x) satisfies
the conditions for g. Our first main result in this section is O∗ (u|θg,B ) g(c log log u) for
some constant c depending only on g. Define
θl =
l
X
phk
(4.9)
k=M
where l ≥ M and M is from the definition of g. We are now prepared to prove our first
result on θg,B .
Theorem 4.5. Let g be an unbounded positive increasing function satisfying (4.5) and
let B be an infinite binary sequence. If θg,B ∈ Cp is defined as in (4.8) then there exists
a constant c depending only on g such that O∗ (u|θg,B ) g(c log log u).
36
Proof. Let g be a function as in the statement of the theorem and let B be an infinite
binary sequence. To simplify notation we fix g and B and write θ = θg,B . Recall that
θ = θg,B =
∞
X
phk
k=M
where M is from the definition of g and hk =
the integers fk∗ were defined such that
fk∗
qk
fk∗
qk
, with fk∗ depending on B. Also recall that
is in lowest terms and g(sk) ≤
fk∗
qk
≤ g(sk + s).
1
Let u ∈ N, u ≥ 2 and let γ ∈ Q be such that O∗ (u|θ) = log |θ−γ|
and Λ(γ) ≤ u. Now let
n be the unique positive integer satisfying
n−1
2m
n
≤ Λ(γ) < 2m ,
(4.10)
where m is from the definition of g in (4.5). Since Λ(γ) increases as u increases fix u ∈ N
sufficiently large so that n > M . From (4.10) and the definition of Λ it follows that the
degree of the minimal polynomial of γ over Z is at most mn − 1. Define Kmn again to be
the field obtained by adjoining the roots of all polynomials of degree less than or equal to
mn over Qp . Recall from Lemma 4.3 that it is impossible for an element of Kmn to have
∗
p-adic absolute value p−hn = p−fn /qn because qn is prime and qn > mn by (4.6). We now
use this to prove |θ − γ| ≥ p−hn by way of contradiction. Assume |θ − γ| < p−hn and recall
that for every l ≥ M we defined θl by
θl =
l
X
phk
k=M
where M is from the definition of g. From Lemma 4.4 and (4.6) it follows that if k < n
then ∂(phk ) = qk < 2mk < mn . Since the degree of phk over Qp must be less than or equal
to ∂(phk ) we have phk ∈ Kmn for M ≤ k ≤ n − 1. In particular this implies θn−1 ∈ Kmn .
Since the degree of γ over Qp is less than or equal to ∂(γ) < mn it also follows that
γ ∈ Kmn and hence γ − θn−1 ∈ Kmn . Moreover, note that from the definition of θl we
have |θ − θn−1 | = p−hn > |θ − γ| and it follows that
∗
|γ − θn−1 | = max{|γ − θ|, |θ − θn−1 |} = |θ − θn−1 | = p−hn = p−fn /qn .
This is a contradiction to Lemma 4.3 because γ − θn−1 ∈ Kmn and qn > mn is prime.
37
Since we just proved |θ − γ| ≥ p−hn it follows that phn ≥
1
|γ−θ| .
Recall γ was defined
1
= O∗ (u|θ). It thus follows that
such that log |γ−θ|
O∗ (u|θ) ≤ hn .
(4.11)
Our goal is to prove hn is bounded above by g(c log log u) for some constant c depending
only on θ. Take the logarithm of the left side of (4.10) twice to obtain
(n − 1) log m + log log 2 ≤ log log Λ(γ) ≤ log log u
(4.12)
since Λ(γ) ≤ u. Recall s from the definition of g in (4.5). Since s and m are constants
depending only on g and g is independent of B, by (4.12) there exists a positive constant
c depending only on g such that
sn + s ≤ c log log u.
Since g is increasing, (4.7) and (4.11) thus imply
O∗ (u|θ) ≤ hn =
fn∗
≤ g(sn + s) ≤ g(c log log u).
qn
(4.13)
Recall that θ = θg,B where B is an infinite binary sequence. Since (4.13) is independent of
the choice of B it follows that O∗ (u|θ) g(c log log u) for uncountably many θ ∈ Cp .
2
In the next result we obtain a lower bound for O∗ (u|θg,B ) when g(x) ≤ pcx for some
positive constant c depending only on g.
Theorem 4.6. Suppose g satisfies (4.5) and assume there exists a positive constant c
2
depending only on g such that g(x) ≤ pcx for all sufficiently large x ∈ R+ . If θ = θg,B ∈ Cp
is defined as in (4.8) then there exists a positive constant c0 depending only on g such that
O∗ (u|θ) g c0 (log log u)1/2 .
Proof. Let g(x) be defined as in (4.5) and assume there exists positive constants c and x0
2
depending only on g such that g(x) ≤ pcx for all x ≥ x0 . Recall that we defined
θ=
∞
X
k=M
phk
38
and
l
X
θl =
phk
k=M
where hk =
fk∗
qk
is such that
g(sk) ≤ hk =
fk∗
≤ g(sk + s).
qk
(4.14)
Moreover, qk satisfies mk < qk < 2mk < mk+1 where m is also defined in (4.5).
To simplify notation define an = 2n mn(n+1)/2 for n ∈ N. Let u ∈ N be sufficiently
large such that there exists a unique n ∈ N, n ≥ max{M, xs0 − 2} satisfying
nan
n
2an (1 + 2mn )p2m g(sn+s)
≤u
(4.15)
(n+1)an+1
n+1
u < 2an+1 (1 + 2mn+1 )p2m g(sn+2s)
.
(4.16)
and
Recall from Lemma 4.4 that ∂(phk ) = qk . Since θn ∈ Q(phM , phM +1 . . . , phn ) it follows
that ∂(θn ) ≤ qM qM +1 . . . qn and hence
∂(θn ) ≤
n
Y
qk <
k=M
n
Y
2mk = 2n m(n+1)n/2 = an .
(4.17)
k=1
In order to bound the height of θn apply Proposition 2.7 to obtain
H(θn ) ≤
n
Y
!∂(θn )
(1 + ∂(phk ))H(phk )
.
(4.18)
k=M
∗
∗
By Lemma 4.4 the minimal polynomial of phk = pfk /qk over Q is xqk − pfk . Hence
∗
H(phk ) = pfk . Multiplying by qk in (4.14) implies fk∗ ≤ qk g(sk + s) and thus
∗
H(phk ) = pfk ≤ pqk g(sk+s) .
The inequality (4.18) then becomes
H(θn ) <
n
Y
k=M
!an
(1 + qk )pqk g(sk+s)
nan
≤ (1 + qn )pqn g(sn+s)
39
because ∂(phk ) = qk and ∂(θn ) ≤ an by (4.17). Now that the height is bounded it is
possible to bound Λ(θn )
nan
Λ(θn ) = 2∂(θn ) H(θn ) < 2an (1 + qn )pqn g(sn+s)
.
(4.19)
Recall from (4.5) that we defined the qn to be such that qn < 2mn . Thus (4.19) and (4.15)
imply
nan
n
≤ u.
Λ(θn ) < 2an (1 + 2mn )p2m g(sn+s)
(4.20)
1
1
Hence O∗ (u|θ) = maxΛ(α)≤u log |θ−α|
≥ log |θ−θ
. From the definitions of θ and θn it
n|
follows that
∞
n
∞
X
X
X
|θ − θn | = phk −
phk = phk = p−hn+1
k=1
k=1
k=n+1
and thus by (4.14) and (4.20)
O∗ (u|θ) ≥ log
1
= log phn+1 = hn+1 ≥ g(sn + s) > g(sn).
|θ − θn |
(4.21)
Now take the logarithm of (4.16) to obtain
log u < an+1 log 2 + (n + 1)an+1 log(1 + 2mn+1 ) + 2mn+1 g(sn + 2s) .
(4.22)
Recall that we defined an+1 = 2n+1 m(n+1)(n+2)/2 and note that 12 (n2 + 3n + 2) ≤ 3n2 for
all n ∈ N. Thus (4.22) becomes
2
log u < 2n+1 m3n log 2 + (n + 1)2n+1 m3n
2
log(1 + 2mn+1 ) + 2mn+1 g(sn + 2s)
2
Also recall that g(x) ≤ pcx for all x ≥ x0 , where c is a constant depending only on g.
Since we assumed n ≥
x0
s
− 2 it follows that sn + 2s ≥ x0 . Thus g(sn + 2s) ≤ pc(sn+2s)
2
and hence
2
2
log u < 2n+1 m3n log 2 + (n + 1)2n+1 m3n
≤ c1 n2n+2 m3n
2 +n+1
2
log(1 + 2mn+1 ) + 2mn+1 pc(sn+2s)
2
pc(sn+2s)
where c1 is some positive constant independent of n. Taking the logarithm again gives
log log u < log c1 + log n + (n + 2) log 2 + (3n2 + n + 1) log m + c(sn + 2s)2 < c2 n2 (4.23)
40
where c2 is a positive constant independent of n. It follows that n > ( c12 log log u)1/2 .
Recall that (4.15) gave a lower bound for u in terms of n and allowed us to obtain (4.21).
Since g is strictly increasing applying (4.21) to (4.23) implies
s
1/2
∗
O (u|θ) ≥ g(sn) > g √ (log log u)
c2
and the result follows.
Now that we have an upper bound and lower bound on O∗ (u|θg,B ) when g(x) ≤ pcx
2
for a positive constant c it is possible to obtain equality is some cases.
Corollary 4.7. For n ∈ N let logn (x) denote the function log log · · · log(x) where the
logarithm is composed n times. If n ≥ 3 then there exist uncountably many θ ∈ Cp which
satisfy O∗ (u|θ) logn (u).
Proof. Fix n ∈ N. Our goal is to prove that logn satisfies the conditions on g in Theorem
4.5 and Theorem 4.6. The function logn (x) is positive for sufficiently large x, increasing,
and unbounded. In order to prove logn satisfies (4.5) first note that induction on n implies
Z k+1
dx
= logn (k + 1) − logn k.
2
x(log x)(log x) . . . (logn−1 x)
k
Moreover, for k sufficiently large
Z k+1
−1
dx
≥ (k + 1)(log(k + 1)) . . . (logn−1 (k + 1))
2
n−1
x(log x)(log x) . . . (log
x)
k
1
≥
(k + 1)n
3
≥ k.
3
2
Hence (4.5) is satisfied for all n ∈ N with s = 1. Since logn (x) is bounded above by px
the function g(x) = logn (x) satisfies the conditions for Theorem 4.5 and Theorem 4.6.
Thus for all n ∈ N there exist uncountably many θ ∈ Cp and positive constants c and c0
depending only on on n such that for all sufficiently large u
1
∗
n
0
1/2
n−1
0
O (u|θ) > log c (log log u)
= log
log c + log log log u > c1 logn+2 u
2
41
and
O∗ (u|θ) ≤ logn (c log(log u)) ≤ c2 logn+2 u,
where c1 and c2 are constants independent of u. It thus follows that for a fixed n ∈ N
there exist uncountably many θ ∈ Cp satisfying O∗ (u|θ) logn+2 u.
∗ in the complex case the properties
Although not much work has been done with OM
of Cp allowed us to prove several results on O∗ . In this chapter we proved that if α ∈ Qp
then O∗ (u|α) log u. In Theorem 4.5 we constructed elements of Cp for which O∗ grows
slowly and in Theorem 4.6 we gave a lower bound for these elements provided the function
2
g(x) is bounded by pcx for some positive constant c. Finally, in Corollary 4.7 we used
these results to construct θ ∈ Cp which satisfy O∗ (u|θ) logn u for all integers n ≥ 3.
42
5
TRANSCENDENCE TYPE
In the complex case the main tool Durand [11] used to prove his results on OM and
∗ was transcendence type. Our definition of p-adic transcendence type is as follows.
OM
Definition 5.1. For θ ∈ Cp the transcendence type of θ is defined by τ (θ) = sup T (θ)
where
T (θ) =
O(u|θ)
τ ≥ 0 : lim sup
=∞ .
τ
u→∞ (log u)
Because of the lim sup in the definition the set T (θ) considers u ∈ N for which
O(u|θ) is large. Thus the function τ (θ) provides us with a method of measuring the rate
of the best polynomial approximations of θ relative to a power of log u. It is also possible
to construct an analogue of transcendence type corresponding to the function O∗ . Define
τ ∗ (θ) = sup T ∗ (θ) where
∗
T (θ) =
O∗ (u|θ)
=∞ .
τ ≥ 0 : lim sup
τ
u→∞ (log u)
In Section 5.1 we will demonstrate that if τ ∗ (θ) ≥ 2 or τ (θ) > 2 then τ ∗ (θ) = τ (θ).
In doing this we will also prove that for all θ ∈ Cp , O∗ (u|θ) O(u|θ). Given any real
√
number τ ≥ (3 + 5)/2 in Section 5.2 we construct θ ∈ Qp that satisfy τ (θ) = τ ∗ (θ) = τ ,
which will then imply there are uncountably many equivalence classes in Cp under the
equivalence relations and ∗ .
Before proceeding we first give an equivalent definition of the set T (θ) in the next
lemma, which will be used later to prove Theorem 5.11.
Lemma 5.2. Let θ ∈ Cp and let T 0 (θ) be the set of all τ ≥ 0 for which there exists a
sequence of polynomials {Pn } ⊂ Z[x] such that Pn (θ) 6= 0, Λ(Pn+1 ) > Λ(Pn ), and
1
log
≥ n(log Λ(Pn ))τ
|Pn (θ)|
for all n ∈ N. Then T (θ) = T 0 (θ), and hence τ (θ) = sup T 0 (θ).
43
Proof. Suppose τ ∈ T (θ). Then there is an increasing sequence of integers {un } such that
O(un |θ)
= ∞.
n→∞ (log un )τ
lim
Deleting elements as necessary allows us to assume that for all n ∈ N,
O(un |θ)
≥ n.
(log un )τ
For each n define Pn (x) ∈ Z[x] to be a polynomial satisfying log
1
|Pn (θ)|
= O(un |θ) and
Λ(Pn ) ≤ un . If necessary redefine the un such that Λ(Pn ) = un for each n. It follows that
1
= O(Λ(Pn )|θ) ≥ n(log Λ(Pn ))τ
log
|Pn (θ)|
and τ ∈ T 0 (θ).
Now assume τ ∈ T 0 (θ) and let un = Λ(Pn ) where {Pn } is the sequence of polynomials
from the definition of T 0 (θ). By the definitions of O and T 0 (θ) it follows that
1
log
|Pn (θ)|
O(un |θ)
≥
≥ n.
τ
(log un )
(log Λ(Pn ))τ
Since this is true for every n ∈ N,
lim sup
u→∞
O(u|θ)
O(un |θ)
≥ lim
=∞
τ
n→∞ (log un )τ
(log u)
and τ ∈ T (θ).
5.1
Comparing τ and τ ∗
Our goal for this section will be to prove Theorem 5.9, which states that if τ ∗ (θ) ≥ 2
then τ (θ) = τ ∗ (θ). This was first done in the complex case by Durand [11] using results
proved by Fel’dman [13]. If F is a field, a polynomial with coefficients in F is separable
if it has no repeated roots in the algebraic closure of F . Given θ ∈ Cp and a polynomial
P (x) ∈ Z[x] in Proposition 5.4 we obtain an upper bound for |P (θ)| in terms of the roots
44
of P, and in Proposition 5.7 we obtain a corresponding lower bound when P is separable.
These bounds are due to Fel’dman in the complex case. We then take the lim sup of
these bounds to obtain the desired result. Although our bounds are different from those
obtained by Fel’dman, our proof in the p-adic case will follow the same outline as Durand
and Fel’dman in the complex case. The following lemma provides us with an upper bound
on the product of roots of P .
Lemma 5.3. Let P (x) ∈ Z[x] be a polynomial of degree n. Let α1 , . . . , αn ∈ Q be the roots
of P, including repetitions due to multiplicity. Then for any set {i1 , i2 , . . . , ik } contained
in {1, 2, . . . , n} we have |αi1 αi2 . . . αik | ≤ H(P ).
Proof. Suppose P (x) = an xn + · · · + a0 ∈ Z[x] and let α1 , . . . , αn ∈ Q be the roots of P,
including repetitions due to multiplicity. If the roots of P all have p-adic absolute value
less than 1 then the result follows. Without loss of generality we can order the roots of
P such that |αi | ≤ |αj | if i ≤ j. If |α1 | < 1 define m such that |αm | < 1 ≤ |αm+1 | and if
|α1 | ≥ 1 let m = 0. By Proposition 2.14 there exists x0 ∈ Cp with |x0 | = 1 such that
sup |P (x)| = |P (x0 )| = |an xn0 + · · · + a0 | ≤ max {|ai xi0 |} ≤ 1.
0≤i≤n
|x|=1
Let
Q(x) =
n
Y
(5.1)
(x − αi )
i=m+1
and note Q(0) = ±αm+1 αm+2 . . . αn . Moreover, applying the p-adic maximum modulus
theorem (Proposition 2.14) again gives there exists x1 ∈ Cp satisfying |x1 | = 1 and
m
|Q(0)| ≤ sup |Q(x)| = sup |Q(x)| = |Q(x1 )| =
|x|≤1
|x|=1
Y
1
|P (x1 )|
|x1 − αi |−1
|an |
(5.2)
i=1
where if m = 0 the empty product is assumed to equal 1. The inequality (5.1) implies
|P (x1 )| ≤ 1. If i ≤ m then |αi | < 1 and from part (ii) of Proposition 2.13 it follows that
|x1 − αi | = |x1 | = 1. Thus (5.2) becomes |Q(0)| ≤
1
|an |
1
|an | .
Part (i) of Proposition 2.13 implies
≤ |an |∞ ≤ H(P ) and since the largest product of roots of P is
|αm+1 αm+2 . . . αn | = |Q(0)|
the result follows.
45
Proposition 5.4. Let P (x) = an xn + · · · + a0 ∈ Z[x] be a polynomial of degree n with
roots α1 , . . . , αn . Then for any θ ∈ Cp
|P (θ)| ≤ min |θ − αi | max{1, |θ|n−1 } H(P ).
1≤i≤n
Proof. The result is trivial if P (θ) = 0. Thus we assume P (θ) 6= 0. First note that
Q
|P (θ)| = |an | ni=1 |θ − αi | and let
−1
Q(x) = an min |θ − αi |
P (x).
1≤i≤n
(5.3)
Then Q(θ) = βn−1 θn−1 + · · · + β0 ∈ Cp where each βi is a sum of products of the form
α1,i α2,i . . . αn−i,i where the subscripts are distinct. The p-adic triangle inequality and
Lemma 5.3 thus imply |βi | ≤ H(P ) for 0 ≤ i ≤ n − 1. Apply this to equation (5.3) to
obtain
|Q(θ)| = |βn−1 θn−1 + · · · + β0 |
≤ max{|βn−1 |, . . . , |β0 |} max{|θ|n−1 , 1}
≤ H(P ) max{|θ|n−1 , 1}.
Hence
|P (θ)| = |an | min |θ − αi | |Q(θ)| ≤ min |θ − αi | H(P ) max{1, |θ|n−1 }.
1≤i≤n
1≤i≤n
Corollary 5.5. Let θ ∈ Cp . Then O∗ (u|θ) O(u|θ).
1
Proof. Given u ∈ N and θ ∈ Cp let α be such that O∗ (u|θ) = log |θ−α|
and Λ(α) ≤ u. Let
P (x) ∈ Z[x] be the minimal polynomial of α over Z and let α1 , . . . , αn ∈ Q be the roots
of P . By Proposition 5.4
1
min1≤i≤n |θ − αi |
1
+ log H(P ) + log(max{|θ|n−1 , 1})
≤ log
|P (θ)|
O∗ (u|θ) = log
≤ O(u|θ) + log Λ(P ) + max{0, (n − 1) log |θ|}.
46
Note that log Λ(α) = log Λ(P ) = n log 2 + log H(P ) ≤ log u and n ≤
1
log 2
log u. Thus since
Theorem 3.3 gives log u O(u|θ) there exists a positive constant c depending only on θ
which satisfies O∗ (u|θ) ≤ c O(u|θ).
Corollary 5.6. If α ∈ Q then O∗ (u|α) log u.
Proof. Suppose α ∈ Q. Corollary 3.9 implies O(u|α) log u and Theorem 4.2 gives
O∗ (u|α) log u. Apply Corollary 5.5 to obtain log u O∗ (u|α) O(u|α) log u.
Proposition 5.4 gives an upper bound on the size of |P (θ)|. Obtaining a lower bound
on |P (θ)| in the next proposition will allow us to relate τ and τ ∗ .
Proposition 5.7. Let P (x) = an xn + · · · + a0 ∈ Z[x] be a separable polynomial of degree
n with height H and distinct roots α1 , . . . , αn ∈ Q. Then for all θ ∈ Cp ,
|P (θ)| ≥ min |θ − αi ||an |−n+2 H −2n n−n/2 ((2n − 1)!)−1/2 .
1≤i≤n
Proof. Let P be as given. Without loss of generality let α1 be such that |θ − α1 | ≤ |θ − αi |
for 2 ≤ i ≤ n. Relabel the other αi such that |αi − α1 | ≤ |αj − α1 | for all 2 ≤ i ≤ j and to
simplify notation define δ = |θ − α1 |. The theorem follows directly when n = 1 because
then |P (θ)| = |a1 θ + a0 | = |a1 |δ. We thus assume n ≥ 2.
Let m be such that |αi − α1 | ≤ δ for 1 ≤ i ≤ m and |αi − α1 | > δ for m + 1 ≤ i ≤ n.
Then
|P (θ)| = |an |
n
Y
i=1
|θ − αi | ≥ |an |δ m
n
Y
|(θ − α1 ) − (αi − α1 )|.
(5.4)
i=m+1
Since |αi − α1 | > |θ − α1 | = δ for all m + 1 ≤ i ≤ n part (ii) of Proposition 2.13 implies
|(θ − α1 ) − (αi − α1 )| = max{|αi − α1 |, |θ − α1 |} = |αi − α1 |.
47
Thus (5.4) becomes
n
Y
|P (θ)| ≥ |an |δδ m−1
|αi − α1 |
i=m+1
= |an |δ
≥ |an |δ
m
Y
i=2
m
Y
!
δ
n
Y
!
|αi − α1 |
i=m+1
|αi − α1 |
i=2
Defining D =
Qn
j=2 |α1
n
Y
!
!
|αi − α1 | .
i=m+1
− αj | then gives
|P (θ)| ≥ |an |δD.
(5.5)
In order to estimate the right hand side of (5.5) in terms of n and H first recall
from (2.1) that the discriminant of P is equal to
∆ = disc(P ) =
an2n−2
n j−1
Y
Y
(αk − αj )2 .
j=2 k=1
Since P has distinct roots, ∆ is a nonzero integer. Let Γ = (2n−1)!H 2n−1 nn and applying
Proposition 2.3 gives |∆|∞ ≤ Γ. Combining this with part (i) of Proposition 2.13 then
implies that
1
Γ
≤
1
|∆|∞
≤ |∆| and consequently
n j−1
n j−1
j=2 k=1
j=3 k=2
YY
YY
1
≤ |∆| = |an |2n−2
|αk − αj |2 = |an |2n−2 D2
|αk − αj |2 .
Γ
Thus we obtain the following upper bound for
1
D2
n j−1
YY
1
2n−2
≤
Γ|a
|
|αk − αj |2 .
n
D2
(5.6)
j=3 k=2
Keeping α1 fixed relabel the other αi such that |α2 | ≤ · · · ≤ |αn |. Note (5.6) is still
satisfied with the relabeled αi . Moreover, |αj − αi | ≤ |αi | for all 2 ≤ j ≤ i and applying
this to (5.6) implies
n j−1
n
j=3 k=2
j=3
YY
Y
1
2n−2
2
2n−2
≤
Γ|a
|
|α
−
α
|
≤
Γ|a
|
|αj |2n .
n
j
n
k
D2
48
By Lemma 5.3 the p-adic absolute value of the product of distinct roots must be less than
the height. Thus
1
≤ Γ|an |2n−2 H 2n
D2
and hence
−1/2
D ≥ |an |2n−2 H 2n Γ
= |an |−n+1 H −n Γ−1/2 .
Finally use this in (5.5) and apply the definition of Γ to obtain
|P (θ)| ≥ δ|an |−n+2 H −n Γ−1/2
≥ δ|an |−n+2 H −n ((2n − 1)!H 2n−1 nn )−1/2
≥ min |θ − αi ||an |−n+2 H −2n n−n/2 ((2n − 1)!)−1/2 .
1≤i≤n
One more lemma is still required before using Propositions 5.4 and 5.7 to prove
Theorem 5.9. In the complex case this result can be found in Chapter 6 of Lang [18]. The
proof of Lang transfers to the p-adic case.
Lemma 5.8. Let θ ∈ Cp . Suppose τ ≥ 1 is a constant and P (x) ∈ Z[x] is a polynomial
with P (θ) 6= 0 satisfying
log
1
|P (θ)|
≥ c (log Λ(P ))τ
for some positive constant c. Then there exists an irreducible polynomial Q(x) ∈ Z[x]
dividing P such that
log
1
|Q(θ)|
≥
c
(log Λ(Q))τ .
2τ
Proof. The proof follows by contraposition. Let θ ∈ Cp . Suppose P (x) ∈ Z[x] is such
that P = P1 P2 . . . Pl where P1 (x), . . . , Pl ∈ Z[x] are (not necessarily distinct) irreducible
polynomials and P (θ) 6= 0. Assume there exists a positive constant c such that
1
c
log
< τ (log Λ(Pi ))τ
|Pi (θ)|
2
49
for every 1 ≤ i ≤ l. Then since τ ≥ 1
log
1
|P (θ)|
=
l
X
log
i=1
<
1
|Pi (θ)|
l
c X
(log Λ(Pi ))τ
2τ
i=1
c
≤ τ
2
=
l
X
!τ
log(Λ(Pi ))
i=1
c
(log(Λ(P1 ) . . . Λ(Pl )))τ .
2τ
Now apply Proposition 2.4 to obtain
τ
τ
1
c c log
< τ log 2∂(P ) Λ(P )
≤ τ log (Λ (P ))2 = c (log Λ(P ))τ .
|P (θ)|
2
2
This completes the proof.
We can now prove our main result comparing τ and τ ∗ .
Theorem 5.9. Suppose θ ∈ Cp . If τ ∗ (θ) ≥ 2 or τ (θ) > 2 then τ (θ) = τ ∗ (θ).
Proof. Suppose τ ≥ 0. Given θ ∈ Cp from Corollary 5.5 we have O∗ (u|θ) O(u|θ), and
there exist constants c and u0 such that O∗ (u|θ) ≤ c O(u|θ) for all u ≥ u0 . Dividing by
(log u)τ implies
lim sup
u→∞
O∗ (u|θ)
c O(u|θ)
≤ lim sup
τ
(log u)τ
u→∞ (log u)
and by the definition of transcendence type it follows that τ ∗ (θ) ≤ τ (θ).
It now suffices to prove that if τ (θ) > 2 then τ (θ) ≤ τ ∗ (θ). This is because if
τ ∗ (θ) > 2 then τ (θ) > 2 and if τ ∗ (θ) = 2 then either τ (θ) = 2, in which case we are
done, or τ (θ) > 2, which would then lead to a contradiction. In order to demonstrate
τ (θ) ≤ τ ∗ (θ) when τ (θ) > 2 it suffices to prove for fixed τ ≥ 2, if
lim sup
u→∞
O(u|θ)
=∞
(log u)τ
(5.7)
50
then
lim sup
u→∞
O∗ (u|θ)
= ∞.
(log u)τ
(5.8)
Assume equation (5.7) is satisfied for some fixed τ ≥ 2. Then there exists a sequence of
polynomials {Pl (x)} ⊂ Z[x] such that for every l ∈ N
log
1
O(Λ(Pl )|θ)
(log Λ(Pl ))−τ =
≥ l.
|Pl (θ)|
(log Λ(Pl ))τ
By Lemma 5.8 for every l there exists an irreducible Ql (x) ∈ Z[x] dividing Pl (x) such that
1
l
log
≥ τ (log Λ(Ql ))τ .
(5.9)
|Ql (θ)|
2
Although it is possible the sequence {Ql } contains repetitions there must be an infinite
number of distinct Ql because
l
2τ
can be arbitrarily large. Since only a finite number of
polynomials can have a given size it is thus possible to pick a subsequence and relabel the
Ql such that Λ(Ql ) < Λ(Ql+1 ) for all l.
For a fixed l, let n = ∂(Ql ), H = H(Ql ), and an be the leading coefficient of Ql . It
can be proved that all irreducible polynomials with coefficients in Q are separable with
roots in Q (see Chapter 13 of Dummit and Foote [9]), and since Ql is irreducible, Ql
must be separable. Define α1 , . . . , αn ∈ Q to be the distinct roots of Ql . Without loss of
generality let α = α1 be such that |θ − α| = min1≤i≤n |θ − αi |. From Proposition 5.7
1
log
≤ log |θ − α|−1 |an |n−2 H 2n nn/2 ((2n − 1)!)1/2 ,
|Ql (θ)|
where the logarithm on the right hand side can be written as the sum
log
1
1
1
+ (n − 2) log |an | + 2n log H + n log n + log((2n − 1)!).
|θ − α|
2
2
(5.10)
1
First note that log |θ−α|
≤ O∗ (Λ(Ql )|θ) because Λ(α) = Λ(Ql ). We claim the
remaining terms of (5.10) are less than or equal to a constant multiple of (log Λ(Ql ))2 . If
n ≥ 2 in the second term then (n − 2) log |an | is not positive because |an | ≤ 1. If n = 1
then by part (ii) of Proposition 2.13, |an |−1 ≤ |an |∞ ≤ H(Ql ) and consequently
(n − 2) log |an | = log |an |−1 ≤ log H(Ql ) ≤ log Λ(Ql ).
51
To simplify notation let c = (log 2)−1 and recall that from the definition of Λ
log Λ(Ql ) = nc−1 + log H.
In particular, log H ≤ log Λ(Ql ) and
n ≤ c log Λ(Ql ) − c log H ≤ c log Λ(Ql ).
(5.11)
Thus the third term of (5.10) satisfies
2n log H ≤ 2c(log Λ(Ql ))2
and similarly the fourth term of the sum is bounded above by
1
1
n log n ≤ c(log Λ(Ql ))(log(c log Λ(Ql )).
2
2
For the final term of (5.10)
1
1
log((2n − 1)!) ≤ log (2n − 1)2n−1 =
2
2
1
n−
log(2n − 1).
2
Now apply (5.11) to obtain
1
log((2n − 1)!) ≤ c(log Λ(Ql ))(log(2c log Λ(Ql )).
2
Thus we have proved the first term in (5.10) is less than or equal to O∗ (Λ(Ql )|θ) and
the sum of the remaining terms is less than or equal to c1 (log Λ(Ql ))2 for some constant
c1 . From (5.9) it thus follows that
∗
O (Λ(Ql )|θ)
(log Λ(Ql ))2
1
τ
τ
−τ
l ≤ 2 (log Λ(Ql )) log
≤2
+ c1
.
|Ql (θ)|
(log Λ(Ql ))τ
(log Λ(Ql ))τ
Since it was assumed that τ ≥ 2, then
0 ≤ lim c1
l→∞
(log Λ(Ql ))2
≤ c1
(log Λ(Ql ))τ
and thus (5.12) implies
O∗ (Λ(Ql )|θ)
(log Λ(Ql ))2
+
c
1
(log Λ(Ql ))τ
(log Λ(Ql ))τ
O∗ (Λ(Ql )|θ)
= lim
l→∞ (log Λ(Ql ))τ
O∗ (u|θ)
≤ lim sup
.
τ
u→∞ (log u)
l
lim τ = lim
l→∞
l→∞ 2
Hence we have demonstrated (5.7) implies (5.8) and the result follows.
(5.12)
52
Under the given conditions this result relates O and O∗ when considering the best
approximations relative to a power of log u. In particular, if θ ∈ Cp is such that the
largest values of O(u|θ) or O∗ (u|θ) grow faster than (log u)2 then from this perspective
approximation by algebraic numbers is the same as approximation by polynomials.
5.2
Constructing Numbers With a Given Transcendence Type
Theorem 5.11 will allow us to construct elements of Qp with transcendence type τ
for any real number τ ≥
√
3+ 5
2
≈ 2.618. In the complex case Durand [11] constructed
numbers with transcendence type τ for all τ ≥ 3. This result was then improved by
Amoroso [2] when he constructed real numbers with transcendence type τ for all τ ≥ 2.
Our construction is similar to that of Durand and Amoroso. The following lemma gives a
lower bound on the p-adic absolute value of a polynomial evaluated at a positive integer
and will be used in our construction in Theorem 5.11.
Lemma 5.10. Let α ∈ N and let α = α0 + α1 p + · · · + αn pn be the base p expansion of α
where αn 6= 0. Let Q(x) ∈ Z[x] be a polynomial of degree d. If Q(α) 6= 0 then
|Q(α)| > H(Q)−1 2−(d+1) αn−d p−nd .
Proof. First consider the case when α = 1. It follows that |Q(1)|∞ ≤ H(Q)(d + 1) and
then part (i) of Proposition 2.13 implies
|Q(1)| ≥
1
≥ H(Q)−1 (d + 1)−1 .
|Q(1)|∞
Thus the result follows because d + 1 ≤ 2d+1 pnd for all d ∈ N. Now suppose α 6= 1. It
therefore follows from the geometric series formula that
|Q(α)|∞ ≤ H(Q)
d
X
i=0
αd − 1
d
α = H(Q) α +
≤ H(Q)2αd
α−1
i
53
and from the definition of α we have
|Q(α)|∞ ≤ 2H(Q)(α0 + · · · + αn pn )d < 2H(Q)(2αn pn )d = H(Q)2d+1 αnd pnd .
By part (i) of Proposition 2.13, 1 ≤ |Q(α)||Q(α)|∞ . Thus 1 < H(Q)2d+1 αnd pnd |Q(α)| and
hence
|Q(α)| > H(Q)−1 2−(d+1) αn−d p−nd .
Given τ ≥
√
3+ 5
2
we are now prepared to construct α ∈ Qp satisfying τ (α) = τ .
From Theorem 5.9 this also gives that τ ∗ (α) = τ . The notation b∗c will be used to denote
the greatest integer function.
Theorem 5.11. Let f : N → {0, 1} be an arbitrary function and let a ≥ 2. For any real
number τ ≥
√
3+ 5
2
let
αn = 1 +
n
X
τ i c−f (i)
pba
(5.13)
i=1
and α = limn→∞ αn ∈ Qp . Then the transcendence type of α is τ .
Proof. For any a ≥ 2 and τ ≥
√
3+ 5
2
let {αn } and α be defined as in (5.13). Then α ∈ Qp
τi
because α = limn→∞ αn since ba c − f (i) monotonically increases without bound as i
approaches infinity and Qp is complete with respect to | ∗ |. Our proof that τ (α) = τ will
consist of two parts. Recall Lemma 5.2 states that if t > 0 and there exists a sequence of
integer polynomials {Qn } with monotone increasing Λ(Qn ) which satisfies
1
≥ n(log Λ(Qn ))t
log
|Qn (α)|
(5.14)
for every integer n ≥ 1 then t ∈ T (α). Given 0 < < τ our first step will be to construct
a sequence of integer polynomials {Qn } which satisfies (5.14) for t = τ − . Since this
implies τ − ∈ T (α) for all 0 < < τ it will then follow that τ ≤ sup T (α) = τ (α). The
second step is to prove τ ≥ τ (α). To do this we prove there exists a constant c0 depending
only on a and τ such that every polynomial Q(x) ∈ Z[x] satisfies
1
log
≤ c0 (log Λ(Q))τ .
|Q(α)|
(5.15)
54
If this holds then for any u ∈ N define the polynomial Q(x) ∈ Z[x] to be such that
1
O(u|α) = log |Q(α)| and Λ(Q) = u. Then (5.15) implies
1
log( |Q(α)|
)
O(u|α)
=
≤ c0
τ
τ
(log u)
(log u)
and by Definition 5.1, τ ≥ τ (α).
Before proving the first step we first note that since f (n) ∈ {0, 1}, (5.13) implies
τ n c−f (n)
αn−1 ≤ pba
τn
≤ pa
and
τn
p−2 pa
τ n c−f (n)
≤ pba
τ n c−f (n)
≤ αn = pba
τn
+ αn−1 ≤ 2pa .
For Pn (x) = x − αn ∈ Z[x], we obtain the following rough upper and lower bounds on
Λ(Pn ) in terms of a, τ and n
τn
2p−2 pa
τn
≤ 2αn = 2H(Pn ) = Λ(Pn ) ≤ 4pa .
(5.16)
The proof of the first step follows by first taking the logarithm of the right inequality
in (5.16)
n
(log Λ(Pn ))τ ≤ (log 4 + aτ )τ .
Then there exists a constant c1 > 0 depending only on τ and a such that
n
(log 4 + aτ )τ
≤ 2τ aτ
n+1
≤ 2τ ((baτ
≤ 2τ (baτ
≤ c1 (baτ
n+1
n+1
n+1
c + 1) + (1 − f (n + 1)))
c − f (n + 1) + 2)
c − f (n + 1))
and hence
(log Λ(Pn ))τ ≤ c1 (baτ
n+1
c − f (n + 1)).
(5.17)
55
Part (ii) of Proposition 2.13 implies
∞
X
i
τ
|α − αn | = pba c−f (i) i=n+1
)
(
∞
baτ n+1 c−f (n+1) X
τ i c−f (i) ba
,
= max p
p
i=n+2
τ n+1 c−f (n+1) = pba
and since Pn (α) = α − αn it follows that
1
n+1
= baτ c − f (n + 1).
log
|Pn (α)|
(5.18)
Up to the positive factor of c1 this is equal to the right hand side of (5.17) and hence
1
τ
.
(log Λ(Pn )) ≤ c1 log
|Pn (α)|
Therefore, for any > 0
1
(log Λ(Pn )) (log Λ(Pn ))τ − ≤ log
c1
1
|Pn (α)|
.
The sequence {Λ(Pn )} increases monotonically without bound by (5.16) and consequently
for fixed > 0 the sequence {(log Λ(Pn )) } must also be increasing and unbounded. Thus
there exists a subsequence {Qn } of {Pn } such that for all n ∈ N, n ≤
1
c1 (log Λ(Qn ))
and
hence
τ −
n(log Λ(Qn ))
≤ log
1
|Qn (α)|
.
Thus for every > 0 we have found a sequence of integer polynomials with monotone
increasing size satisfying (5.14) with t = τ − and t ∈ T (α). Hence τ ≤ τ (α) and the
proof of the first step is complete.
We now prove the second step by demonstrating (5.15). We proceed by proving
n
(5.15) first for Pn (x) = x − αn . From (5.18), log |Pn1(α)| ≤ (aτ )τ . Take the logarithm
of the left side of (5.16) to obtain
2
τ
p
τn τ
(a ) ≤ log
Λ(Pn )
≤ c2 (log Λ(Pn ))τ
2
56
where c2 is a constant depending only on τ . Hence
1
log
≤ c2 (log Λ(Pn ))τ
|Pn (α)|
(5.19)
which proves (5.15) when Q(x) = Pn (x) = x − αn for all n ∈ N.
Now suppose Q(x) ∈ Z[x] is arbitrary. To simplify notation define exp(x) to denote
px . Let N0 ∈ N be such that if Λ(Q) ≥ N0 then there exists a positive integer n such that
n
n−1
n+1
n
≤ Λ(Q) < exp (c + 1)−1 aτ −τ .
exp (c + 1)−1 aτ −τ
where c =
(5.20)
1
log 2 .
Now assume Λ(Q) ≥ N0 and Q(αn ) 6= 0. We will consider the other cases at the
τ n c−f (n)
end of the proof. Let d = ∂(Q). The last term in the base p expansion of αn is pba
with a coefficient of 1. Since αn ∈ N, Lemma 5.10 implies
τ n c+df (n)
(H(Q))−1 2−(d+1) p−dba
τ n c−f (n)
Define α∗ = pba
≤ |Q(αn )| ≤ max{|Q(α) − Q(αn )|, |Q(α)|}.
to simplify notation. To proceed we show
max{|Q(α) − Q(αn )|, |Q(α)|} = |Q(α)|
(5.21)
|Q(α) − Q(αn )| < (H(Q))−1 2−(d+1) (α∗ )−d .
(5.22)
by demonstrating
The ultrametric triangle inequality implies
d
X
i
i ai (α − αn )
|Q(α) − Q(αn )| = i=1
≤
≤
max {|ai |} max {|αi − αni |}
1≤i≤d
max {|α
1≤i≤d
i
1≤i≤d
− αni |}.
Note that αi = (αn + (α − αn ))i = αni + (α − αn )ρ where ρ is simply the remaining terms
obtained from expanding. Moreover, |ρ| ≤ 1 and hence
|αi − αni | = |α − αn ||ρ| ≤ |α − αn |.
57
It follows that
τ n+1 c+f (n+1)
max {|αi − αni |} ≤ |α − αn | = p−ba
1≤i≤d
τ n+1 +2
≤ p−a
and consequently
τ n+1 +2
|Q(α) − Q(αn )| ≤ p−a
1
log 2
Recall we defined c =
.
(5.23)
and by the change of base formula for logarithms we have
1
∗
(H(Q))−1 2−d 2−cd log α
2
1
τn
(Λ(Q))−1 2−cd(ba c−f (n))
2
1
τn
(Λ(Q))−1 (2d )−ca
2
1
τn
(Λ(Q))−1 (2d H(Q))−ca
2
1
τn
(Λ(Q))−ca −1 .
2
(H(Q))−1 2−(d+1) (α∗ )−d =
=
≥
≥
=
Thus
1
τn
(H(Q))−1 2−(d+1) (α∗ )−d ≥ (Λ(Q))−ca −1 .
2
(5.24)
Recall that exp(x) denotes px . Now apply (5.20) to obtain
n
1
1 τ n+1 −τ n
1
−caτ −1
τn
(Λ(Q))
exp
a
−ca − 1
≥
2
2
c+1
1
−c τ n+1
1 τ n+1 −τ n
=
exp
a
−
a
2
c+1
c+1
and thus (5.24) implies
−1 −(d+1)
(H(Q))
2
∗ −d
(α )
1
> exp
2
−c τ n+1
1 τ n+1 −τ n
a
a
−
.
c+1
c+1
(5.25)
If we assume by way of contradiction to (5.22) that
(H(Q))−1 2−(d+1) (α∗ )−d ≤ |Q(α) − Q(αn )|
then taking the logarithm of (5.23) and (5.25) gives
log
1
c τ n+1
1 τ n+1 −τ n
n+1
−
a
−
a
≤ −aτ
+ 2.
2 c+1
c+1
(5.26)
58
Now multiply by −a−τ
n+1
− a−τ
< 0 to obtain
n+1
log
1
c
1 −τ n
n+1
+
+
a
≥ 1 − 2a−τ .
2 c+1 c+1
(5.27)
Since n, as defined in (5.20), is an unbounded increasing function of Λ(Q), and Λ(Q) can
be arbitrarily large, (5.27) must hold for arbitrarily large positive integers n. However, if
we let n increase without bound taking the limit gives
c
c+1
≥ 1, which is impossible. Thus
(5.26) can only hold for finitely many n and there exists a positive integer N1 ≥ N0 for
which all Q(x) ∈ Z[x] with Λ(Q) ≥ N1 and Q(αn ) 6= 0 cannot satisfy (5.27) and therefore
satisfy (5.21). Consequently (H(Q))−1 2−(d+1) (α∗ )−d ≤ |Q(α)| and hence
log
1
≤ log H(Q)2d+1 (α∗ )d .
|Q(α)|
(5.28)
We can now use our previous results to bound the right hand side of (5.28) in terms of
Λ(Q). The inequality (5.24) gives
τ n +1
≤ log 2Λ(Q)ca
log H(Q)2d+1 (α∗ )d
n
= log 2 + (caτ + 1) log Λ(Q)
n
≤ 2caτ log Λ(Q)
and thus
log
1
n
≤ 2caτ log Λ(Q).
|Q(α)|
τ
Now note (τ n − τ n−1 ) τ −1
= τ n . To simplify notation let γ =
(5.29)
τ
τ −1 .
that
n n−1 γ
n
(c + 1)γ (log Λ(Q))γ ≥ a(τ −τ ) = aτ .
Using this in (5.29) and defining c3 = 2c(1 + c)γ gives
log
Recall that we assumed τ ≥
1
≤ c3 (log Λ(Q))1+γ .
|Q(α)|
√
3+ 5
2 .
Thus it follows that
√
√
τ
3+ 5
3+ 5
√ =
1+γ =1+
≤1+
≤τ
τ −1
2
1+ 5
By (5.20) it follows
59
and hence
log
1
≤ c3 (log Λ(Q))τ .
|Q(α)|
(5.30)
This is true for all integer polynomials Q such that Q(αn ) 6= 0 and Λ(Q) ≥ N1 . Moreover,
there are only finitely many polynomials Q with Λ(Q) < N1 and thus there exists a
constant c4 such that if Λ(Q) < N1 then
1
log
≤ c4 (log Λ(Q))τ .
|Q(α)|
(5.31)
It remains to consider when αn is a root of Q (where n is still defined as it was in
(5.20)). Recall that Pn (x) = x − αn is the minimal polynomial of αn over Q and Pn must
divide any integer polynomial that has αn as a root. Thus if Q(αn ) = 0 then Q = Pnl R
for some l ∈ N and R(x) ∈ Q[x] with R(αn ) 6= 0. By Theorem 2.9 (Gauss’s Lemma)
R(x) ∈ Z[x]. From Theorem 2.4, and in particular (2.3) it follows that (Λ(Pn ))l ≤ (Λ(Q))2
and Λ(R) ≤ (Λ(Q))2 . Recall by (5.19)
1
log
≤ c2 (log Λ(Pn ))τ
|Pn (α)|
where c2 depends only on α and τ . Since R(αn ) 6= 0, (5.30) and (5.31) imply
1
log
≤ c5 (log Λ(R))τ .
|R(α)|
where c5 = max{c3 , c4 } depends only on a and τ . Applying these inequalities gives
1
1
1
log
= l log
+ log
|Q(α)|
|Pn (α)|
|R(α)|
τ
τ
< c2 l (log Λ(Pn )) + c5 (log Λ(R))τ
≤ c2 (log(Λ(Pn ))l )τ + c5 (log Λ(R))τ
≤ c2 (log(Λ(Q))2 )τ + c5 (log(Λ(Q)2 ))τ
= (2τ c2 + 2τ c5 )(log Λ(Q))τ .
Hence when Q(αn ) = 0
log
1
|Q(α)|
≤ (2τ c2 + 2τ c5 )(log Λ(Q))τ .
(5.32)
60
If we let c0 = 2τ c2 + 2τ c5 and note c0 ≥ max{c2 , c5 } is a constant depending only on
α and τ it thus follows from (5.19), (5.30), (5.31), and (5.32) that
1
log
≤ c0 (log Λ(Q))τ
|Q(α)|
for all polynomials Q. Hence the second step of the proof is complete and τ ≥ τ (α). Thus
the transcendence type of α must equal τ .
Corollary 5.12. There are uncountably many different equivalence classes on Cp under
the equivalence relation defined by θ η if O(u|θ) O(u|η). The same holds for
the equivalence classes on Cp under the equivalence relation ∗ defined by θ ∗ η if
O∗ (u|θ) O∗ (u|η).
Proof. Given any τ ≥
√
3+ 5
2
Theorem 5.11 gives that there exists θ ∈ Cp with transcen-
dence type τ . Thus the result on will follow if we can prove that if θ, η ∈ Cp are such
that τ (θ) 6= τ (η) then O(u|θ) and O(u|η) are not equivalent.
Let θ, η ∈ Cp and suppose O(u|η) O(u|θ). Then there exist u0 , c ∈ N and positive
γ ∈ R such that O(uc |η) ≥ γO(u|θ) for all u ≥ u0 . It follows that for all τ > 0
lim sup
u→∞
γO(u|θ)
O(uc |η)
O(uc |η)
O(u|η)
τ
≤
lim
sup
=
c
lim
sup
≤ cτ lim sup
τ
τ
c
τ
τ
(log u)
u→∞ (log u)
u→∞ (log u )
u→∞ (log u)
and thus τ (θ) ≤ τ (η). In particular, if τ (θ) > τ (η) then O(u|η) O(u|θ) cannot hold
and hence θ and η do not satisfy O(u|θ) O(u|η). Thus it follows that any two elements
of Cp with different transcendece types must be in different equivalence classes under
the equivalence relation . Since Theorem 5.11 gives that there are uncountably many
possible values for the transcendence type of elements of Cp the result follows. The proof
for ∗ is identical since if τ ≥
√
3+ 5
2
> 2 then Theorem 5.9 implies that τ ∗ (θ) = τ .
All of our results in this chapter are also true in the complex case. The proof that
O∗ (u|θ) O(u|θ) and the equality of τ (θ) and τ ∗ (θ) when τ ∗ (θ) ≥ 2 or τ (θ) > 2 are
√
analogous to the results of Fel’dman [13] and Durand [11]. Given τ ≥ (3 + 5)/2 the
construction of θ ∈ Qp with τ (θ) = τ is similar to results of Durand [11] and Amoroso [2].
61
6
CONCLUSION
Our initial goal was to consider the results on complex order functions due to Mahler
[22] and Durand [11] in the p-adics. Several of our results are analogous to those in the
complex case. Theorem 3.6 established that the order functions O(u|θ) and O(u|η) are
equivalent when θ, η ∈ Cp are algebraically dependent. Given τ ≥
√
3+ 5
2 ,
Theorem 5.11
constructs θ ∈ Qp such that τ (θ) = τ and Theorem 5.9 demonstrates if θ ∈ Cp is such
that τ ∗ (θ) ≥ 2 or τ (θ) > 2 then τ ∗ (θ) = τ (θ). Further study revealed that the properties
of the p-adics could be used to obtain results different from those proved in the complex
case. For instance, Corollary 3.5 is a direct result of work done by Escassut [12] and states
that there are transcendental θ ∈ Cp for which O(u|θ) log u. Also, Theorem 4.2 gives
that if α ∈ Qp then O∗ (u|α) log u. Theorem 4.5 gave θ ∈ Cp for which O∗ (u|θ) grows
slowly, and under certain conditions a lower bound for O∗ (u|θ) then could be obtained
from Theorem 4.6. Corollary 4.7 then used these results to demonstrate that if n ≥ 3
then there exist uncountably many θ ∈ Cp which satisfy logn u O∗ (u|θ).
There are many open questions concerning both the complex and p-adic order
functions. A likely subject for further study would be a proof or counterexample that
O∗ (u|θ) O∗ (u|η) when θ and η are algebraically dependent elements of Cp . Given τ ≥ 2
Amoroso [2] was able to construct θ ∈ C such that τ (θ) = τ and thus it seems likely
that Theorem 5.11 could be extended in this manner. A more difficult question would
be to consider the existence of θ ∈ Cp for which 1 < τ (θ) < 2. This is impossible in the
complex case because O(u|θ) (log u)2 for all transcendental θ ∈ C. However, Corollary
3.5 implies there exist transcendental θ ∈ Cp for which τ (θ) = 1 and therefore it is certainly possible there are θ ∈ Cp with τ (θ) between 1 and 2. Another open problem is to
determine for which functions a : R+ → R+ there exist θ ∈ Cp for which O(u|θ) a(u) or
O∗ (u|θ) a(u). In the complex case this problem was posed by Mahler [22] and remains
open.
62
BIBLIOGRAPHY
1. W. Adams. Transcendental numbers in the p-adic domain. Amer. J. Math., 88:279–
308, 1966.
2. F. Amoroso. On the distribution of complex numbers according to their transcendence types. Ann. Mat. Pura Appl. (4), 151:359–368, 1988.
3. V. Beresnevich, V. Bernik, and E. Kovalevskaya. On approximation of p-adic numbers
by p-adic algebraic numbers. J. Number Theory, 111:33–56, 2005.
4. V. Bernik and M. Dodson. Metric Diophantine Approximation on Manifolds. Cambridge University Press, Cambridge, 1999.
5. N. Bourbaki. Algebra. II. Chapters 4–7. Springer-Verlag, Berlin, 1990. Translated
from the French by P. M. Cohn and J. Howie.
6. Y. Bugeaud. Approximation by Algebraic Numbers. Cambridge University Press,
Cambridge, 2004.
7. P. Cijsouw. Transcendence Measures. PhD thesis, Amsterdam University, 1972.
8. A. Dubickas and C. Smyth. Length of the sum and product of algebraic numbers.
Math. Notes, 77:854–860, 2004.
9. D. Dummit and R. Foote. Abstract Algebra. John Wiley and Sons, Hoboken, NJ,
third edition, 2004.
10. A. Durand. Une nouvelle classification des nombres complexes, selon K. Mahler.
In Séminaire Delange-Pisot-Poitou (Groupe d’étude de Théorie des nombres), pages
G17.01–G17.06. 1973/74.
11. A. Durand. Quatre problèmes de Mahler sur la fonction ordre d’un nombre transcendant. Bull. Soc. Math. France, 102:365–377, 1974.
12. A. Escassut. Transcendence order over Qp in Cp . J. Number Theory, 16:395–402,
1983. Correction in J. Number Theory, 19:451, 1984.
13. N. Fel’dman. The approximation of certain transcendental numbers. I. Approximation of logarithms of algebraic numbers. Izvestiya Akad. Nauk SSSR. Ser. Mat,
15:53–74, 1951. (in Russian). English transl. in Amer. Math. Soc. Transl. Series 2,
59:224-245, 1966.
14. F. Gouvêa. p-adic Numbers: An Introduction. Springer-Verlag, New York, second
edition, 1997.
63
15. R. Güting. Approximation of algebraic numbers by algebraic numbers. Michigan
Math. J., 8:149–159, 1961.
16. K. Hensel. Zahlentheorie. Göschen, Berlin, 1913.
17. N. Koblitz. p-adic Numbers, p-adic Analysis, and Zeta Functions. Springer-Verlag,
New York, second edition, 1984.
18. S. Lang. Introduction to Transcendental Numbers. Addison-Wesley Pub. Co., Reading, MA, 1966.
19. É. Lutz. Sur les Approximations Diophantiennes Linéaires p-adiques. Hermann,
Paris, 1955.
20. K. Mahler. Über Approximation der Exponentialfunktionen und des Logarithmus,
I,II. J. Reine Angew. Math., 166:118–150, 1932.
21. K. Mahler. Über eine Klassen-Einteilung der p-adischen Zahlen. Compositio Math.,
2:259–275, 1935.
22. K. Mahler. On the order function of a transcendental number. Acta Arith., 18:63–76,
1971.
23. M. Mignotte and D. Ştefănescu. Polynomials: An Algorithmic Approach. SpringerVerlag, Singapore, 1999.
24. J. Morrison. Approximation of p-adic numbers by algebraic numbers of bounded
degree. J. Number Theory, 10:334–350, 1978.
25. Y. Nesterenko. Diophantine approximation in the field of p-adic numbers. Mat.
Zametki, 35:653–662, 1984. (in Russian). English transl. in Math. Notes, 35:342-347,
1984.
26. Y. Nesterenko. Measure of algebraic independence for almost all pairs of p-adic
numbers. Mat. Zametki, 36:295–304, 1984. (in Russian). English transl. in Math.
Notes, 36:642-647, 1984.
27. I. Niven, H. Zuckerman, and H. Montgomery. An Introduction to the Theory of
Numbers. John Wiley and Sons, New York, fifth edition, 1991.
28. P. Philippon. Classification de Mahler et distances locales. Bull. Austral. Math. Soc.,
49:219–238, 1994.
29. A. Robert. A Course in p-adic Analysis. Springer-Verlag, New York, 2000.
30. T. Schneider. Einführung in die Transzendenten Zahlen. Springer-Verlag, Berlin,
1957.
64
31. J. Serre. Local Fields. Springer-Verlag, New York, 1979.
32. O. Teulié. Approximation d’un nombre p-adique par des nombres algébriques. Acta
Arith., 102:137–155, 2002.
33. M. Waldschmidt. Nombres Transcendants. Springer-Verlag, New York, 1973.
34. T. Wang. p-adic transcendence and p-adic transcendence measures for the values of
Mahler type functions. Acta Math. Sin. (Engl. Ser.), 22:187–194, 2006.
35. D. Zelo. Simultaneous Approximation to Real and p-adic Numbers. PhD thesis,
University of Ottawa, 2008.