IJMMS 29:6 (2002) 349–353
PII. S0161171202007561
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON THE ROOTS OF THE SUBSTITUTION
DICKSON POLYNOMIALS
JAVIER GOMEZ-CALDERON
Received 10 April 2001
We show that under the composition of multivalued functions, the set of the y-radical
roots of the Dickson substitution polynomial gd (x, a)−gd (y, a) is generated by one of the
roots. Hence, we show an expected generalization of the fact that, under the composition
of the functions, the y-radical roots of x d − y d are generated by ζd y.
2000 Mathematics Subject Classification: 11Txx, 11T06.
Let Fq denote the finite field of order q and characteristic p. For f (x) in Fq [x], let
f ∗ (x, y) denote the substitution polynomial f (x) − f (y). The polynomial f ∗ (x, y)
has frequently been used in questions on the set of values f (x), see for example
Wan [8], Dickson [4], Hayes [6], and Gomez-Calderon and Madden [5]. The linear and
quadratic factors of f ∗ (x, y) have been studied by Cohen [2, 3] and also by Acosta
and Gomez-Calderon [1]. A factor of f ∗ (x, y) is said to be a radical factor if it has
the form
c x − R1 (y) x − R2 (y) · · · x − Rm (y) ,
c ∈ Fq ,
(1)
where rj (y), 1 ≤ j ≤ m, denotes a radical expression in y over the algebraic closure of
the field of functions Fq (y). If Ri (y) and Rj (y) are radical roots of f ∗ (x, y), then the
composite multivalued function Ri (Rj (y)) provides a set of radical roots of f ∗ (x, y);
that is, f (Ri (Rj (y))) = f (y) for all values of Ri (Rj (y)). For example, for q odd,
x 3 + x − y 3 − y = x − R0 (y) x − R1 (y) x − R2 (y) ,
(2)
where R0 (y) = y, 2R1 (y) = −y + −3y 2 − 4, and 2R2 (y) = −y − −3y 2 − 4. Thus,
R1 R1 (y) =
y−
−3y 2 − 4 +
2 1/2
2
3y + −3y − 4
4
= R0 (y), R2 (y) .
(3)
Definition 1. Let Fq denote the finite field of order q and characteristic p. For
a ∈ Fq and an integer d ≥ 1 , let
gd (x, a) =
[|d/2|]
t=0
d
d−t
d−t
(−a)t x d−2t
t
denote the Dickson polynomial of degree d over Fq .
(4)
350
JAVIER GOMEZ-CALDERON
Lemma 2. Let d be a positive integer and assume that Fq contains a primitive dth
root of unity ζ. Put
Ak = ζ k + ζ −k ,
Bk = ζ k − ζ −k .
(5)
Then, for each a in Fq ,
(i) if d is odd,
gd (x, a) − gd (y, a) =
(d−1)/2
(x − y) x 2 − Ak xy + y 2 + Bk2 a ;
(6)
2
x − y 2 x 2 − Ak xy + y 2 + Bk2 a .
(7)
i=1
(ii) if d is even,
gd (x, a) − gd (y, a) =
d/2
i=1
Moreover for a = 0, the quadratic factors are different from each other and irreducible
in Fq [x, y].
Proof. See [7, Theorem 3.12].
Theorem 3. If q is odd, 0 = a ∈ Fq , and (d, q) = 1, then
d
(i) gd (x, a) − gd (y, a) = i=1 (x − Ri (y)), where R1 (y), R2 (y), . . . , Rd (y) denote
d-radical expressions in y over the algebraic closure of the field of functions
Fq (y);
(ii) under the composition of multivalued functions, the set of roots R1 (y), R2 (y),
. . . , Rd (y) is generated by one of the roots Ri (y).
Proof. Let ζ be a dth primitive root over the field Fq . With notation as in Lemma 2,
write,
(a) if d is odd,
gd (x, a) − gd (y, a) = (x, y)
(d−1)/2
2
x − Ak xy + y 2 + Bk2 a
i=1
(d−1)/2
x − σk y + x − σk y − ,
= x − σ0 (y)
(8)
i=1
where σ0 (y ± ) = y, 2σk (y + ) = Ak y + Bk y 2 − 4a, and 2σk (y − ) = Ak y −
Bk y 2 − 4a for 1 ≤ k ≤ (d − 1)/2;
(b) if d is even,
d/2
x 2 − y 2 x 2 − Ak xy + y 2 + Bk2 a
gd (x, a) − gd (y, a) = x 2 − y 2
i=1
d/2
= x − σ0 (y) x − σd/2 (y)
x − σk y + x − σk y − ,
(9)
i=1
where σ0 (y) = y, σd/2 (y) = −y, 2σk (y + ) = Ak y +Bk y 2 − 4a, and 2σk (y − ) =
Ak y − Bk y 2 − 4a for 1 ≤ k ≤ d/2.
ROOTS OF DICKSON POLYNOMIALS
351
Now we consider the composite multivalued function σ1 (y + ) ◦ σk (y + )
σ1 y + ◦ σk y +


2 − 4a
y
+
B
y
A
k
k


= σ1 

2
=
=
4
1/2 2
A1 Ak y+A1 Bk y 2 −4a+B1 Ak y 2 + 2yAk Bk y 2 − 4a + Bk2 y 2 −4aBk2 −16a
4
1/2 A1 Ak y + A1 Bk y 2 − 4a + B1 A2k y 2 + 2yAk Bk y 2 − 4a + Bk2 y 2 − 4aA2k
4
A1 Ak y + A1 Bk y 2 − 4a + B1 Bk y + Ak y 2 − a
=
=
=
2
1/2 A1 Ak y + A1 Bk y 2 − 4a + B1 Ak y + Bk y 2 − 4a − 16a
4
A1 Ak y + A1 Bk y 2 − 4a ± B1 Bk y + Ak y 2 − 4a
4



A1 Bk + Ak B1 y 2 − 4a
,
= A1 Ak + B1 Bk y +

4



A1 Bk − Ak B1 y 2 − 4a 
A1 Ak − B1 Bk y +
.

4

(10)
Thus,




σk+1 y + , σk−1 y + ,





−
+


+
+ σ(d−1)/2 y , σ(d−3)/2 y ,
σ1 y ◦ σk y =

σ y + , σ y + ,

k−1

 k+1



−

σ

y+ , σ
y ,
d/2−1
Similarly, we get
σ1 y + ◦ σk y + =
d/2−1




σk+1 y − , σk−1 y − ,









σ(d−1)/2 y + , σ(d−3)/2 y − ,

−



y , σk−1 y − ,
σ

 k+1






σd/2−1 y + , σd/2−1 y − ,
d−3
, d is odd,
2
d−1
if k =
, d is odd,
2
d
if 1 ≤ k ≤ − 1, d is even,
2
d
if k = , d is even.
2
(11)
(d − 3)
, d is odd,
2
d−1
if k =
, d is odd,
2
d
if 1 ≤ k ≤ − 1, d is even,
2
d
if k = , d is even.
2
(12)
if 1 ≤ k ≤
if 1 ≤ k ≤
Therefore, σ1 (y + ) generates the set of radical roots σi (y + ), σi (y − ), for all values i.
352
JAVIER GOMEZ-CALDERON
The set of the y-radical roots of a substitution polynomial may require more than
one generator as we illustrate in the following theorem.
Theorem 4. For 0 ≠ b ∈ Fq and (mn, q) = 1, let fm,n (x, b) denote the polynomial
(x m + b)n . Then,
mn
(i) fm,n (x, b) − fm,n (y, b) = i=1 (x − Ri (y)), where R1 (y), R2 (y), . . . , Rmn (y) denote radical expressions in y over algebraic closure of the field of functions Fq (y).
(ii) Under the composition of multivalued functions, the set of roots R1 (y), R2 (y),
. . . , Rmn y is generated by at least m of the roots Ri (y).
Proof. Let ζ and ξ be primitive roots of unity of order n and m, respectively, over
the field Fq . Then
fm,n (x, b) − fm,n (y, b) =
n
#
" m
x + b − ζk ym + b
k=1
=
m n 1/m x − ξ i ζ k y m + b(1 − ζ)
(13)
k=1 i=1
=
m
n x − σik (y) .
k=1 i=1
Now we consider the composite multivalued function σj1 (y) ◦ σik (y).
σji (y) ◦ σik (y) = ξ j
= ξj
i k m
1/m m
1/m
ζ ξ ζ y + b 1 − ζk
+ b(1 − ζ)
k m
1/m
ζ ζ y + b 1 − ζ k + b(1 − ζ)
1/m
(14)
= ξ j ζ k+1 y m + b 1 − ζ k+1


σjk+1 (y),
if 1 ≤ k ≤ n − 2, 1 ≤ j ≤ m,
= 
 σ10 (y), σ20 (y), . . . , σm0 (y) , if k = n − 1, 1 ≤ j ≤ m.
Therefore, σ11 (y), σ21 (y), . . . , σm1 (y) generate the set of roots {σjk (y) : 1 ≤ j ≤ m,
1 ≤ k ≤ n}.
References
[1]
[2]
[3]
[4]
[5]
[6]
M. T. Acosta and J. Gomez-Calderon, The second-order factorable core of polynomials over
finite fields, Rocky Mountain J. Math. 29 (1999), no. 1, 1–12.
S. D. Cohen, Exceptional polynomials and the reducibility of substitution polynomials, Enseign. Math. (2) 36 (1990), no. 1-2, 53–65.
, The factorable core of polynomials over finite fields, J. Austral. Math. Soc. Ser. A 49
(1990), no. 2, 309–318.
L. E. Dickson, The analytic representation of substitutions on a prime power of letters with
a discussion of the linear group, Ann. of Math. 11 (1897), 65–120, 161–183.
J. Gomez-Calderon and D. J. Madden, Polynomials with small value set over finite fields, J.
Number Theory 28 (1988), no. 2, 167–188.
D. R. Hayes, A geometric approach to permutation polynomials over a finite field, Duke
Math. J. 34 (1967), 293–305.
ROOTS OF DICKSON POLYNOMIALS
[7]
[8]
353
R. Lidl, G. L. Mullen, and G. Turnwald, Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 65, Longman Scientific and Technical,
Harlow, 1993.
D. Q. Wan, On a conjecture of Carlitz, J. Austral. Math. Soc. Ser. A 43 (1987), no. 3, 375–384.
Javier Gomez-Calderon: Department of Mathematics, The Pennsylvania State University, New Kensington Campus, New Kensington, PA 15068, USA
E-mail address: jxg11@psu.edu