U.P.B. Sci. Bull., Series A, Vol. 71,Iss. 2, 2009
ISSN 1223-7027
DEPENDENCE OF THE ROOTS ON COEFFICIENTS
Alina NIŢĂ1, Ana NIŢĂ2
În aceasta notă, se dă o demonstraţie nouă pentru dependenţa continuă a
rădăcinilor unui polinom algebric de coeficienţii polinomului şi se discută cazul
polinoamelor cu coeficienţi aleatori [1], demonstrând existenţa unor soluţii
măsurabile, ordonarea acestora şi dependenţa măsurabilă de coeficienţi [2].
In this note, we give a new proof for the continuous dependence of an
algebraic polynomial’s roots on its coefficients and we discuss the case of
polynomials with random coefficients [1], proving the existence of measurable
solutions and also regarding their ordering and their measurable dependence upon
coefficients, by making use of [2].
Keywords : algebraic polynomial, random polynomial, solutions depending on
coefficients.
1. Introduction
We denote by M n the set of all monic polynomials with complex
coefficients of degree n ( n  1), identified with the associated polynomial
functions.
If P :  , P( x)  x n  c1 x n 1  ...  cn is a complex polynomal function,
with the coefficients ck  ak  ibk ,1  k  n , then P is identified with the point
 a1, b1,....., an , bn 
from
2n
. We will consider the lexicographical order in
.
Let I   x1 ,...., xn , y1 ,..., yn  | for each i , xi  xi 1 and if xi  xi 1 , then
yi  yi 1 , be a set in the space
n

This set is obviously closed ( in
2n
, with coordinates zk  xk  iyk ,1  k  n .
2n
).
Example:
In 2  4 , I   z1 , z2  | x1  x2 or x1  x2 , y1  y2  .
1
Lector, Department of Mathematics, University POLITEHNICA of Bucharest, Romania,
e-mail: nita_alina@yahoo.com
2
Reader, Department of Mathematics, University POLITEHNICA of Bucharest, Romania
20
Alina Niţă, Ana Niţă
2. The case of algebraic polynomials
We define the map F : M n  I , that associates to each polynomial
P  M n , the set of its roots z1 , z2 ,..., zn (lexicographically ordered).
Obviously,
F is a bijective
map , and let G  F 1 .
Thus, G( z1 , z2 ,..., zn )  the monic polynomial  X  z1  ....( X  zn )  M n , identified
with the 2n-dimensional system of the real and imaginary parts of the
coefficients.
Lemma 1. The map G is continuous.
Proof:
Using the Viète’s relations, the its coefficients of  X  z1  ....( X  zn ) are
symmetric functions on z1 , z2 ,..., zn , therefore continuous.
Lemma 2. Let X , Y be two metric spaces , A  X and f : A  Y a
continuous bijective map. Assume that for any x  A , there is a neighbourhood U
of f(x), such that f 1 U  is contained in a compact set. Then the map
f 1 : Y  A is continuous.
Proof:
We use the definition of continuity by sequences. For any b Y fixed, let
yn  b in Y , xn  f 1  yn  and a  f 1 (b) . According to our assumption, there
is a neighbourhood V of b and a compact K  A such that f 1 V   K .
Since yn  b ,
xn  f
1
V   K
there is a rank N such that yn  V for n  N .
Then
for n  N and K being compact , we can find a subsequence
xkn   in K. The map f being continuous, then ykn  f ( ) . But yn  b
therefore f    b ; that is ,   a and so, xkn  a .
In order to prove that f 1 is continuous in b, let W be a neighbourhood of
f 1  b   a . We have to prove that there is a neighbourhood U of b such that
 1
f 1 U   W ; otherwise , there exists bn  B  b,  such that f 1  bn  W .
 n
 
But bn  b and we know that there is a subsequence f 1 bkn  a .
 
But f 1 bk n W and so get a contradiction.
Lemma 3. Let f : A 
n
is a closed subset.
A
m
be a continuous
bijective map, where
Dependence of the roots on coefficients
21
a) If A is a compact set, then f 1 is continuous ;
b) Moreover , if f 1 is a bounded map ( i.e the image of any bounded set is
bounded ) , then f 1 is also continuous .
Proof:
a) The condition from Lemma 2 is fulfilled.
b) For any x  A fixed, take a bounded neighbourhood U of f(x). Then the
set f 1 (U ) is bounded and its closure is compact (being closed and bounded in
n
and by applying the Borel-Lebesgue theorem).Using Lemma 2, it follows that
1
f is also continuous.
Proposition 1. The map F  G 1 is continuous.
Proof:
Using Lemma 1 and Lemma 3,b) for f  G : I  M n , A  I , it is enough
2n
to show that the map F is bounded. For this, let B  M n
be a bounded
subset, then there is M  0 , such that B is contained in the ball B(0, M ) .
Let Q  B, Q  X n  c1 X n 1  ....  cn , hence ci  M for any i.
For each not-null root z of Q , we have z n  c1 z n 1  ...  cn  0 ,
and so dividing by z n 1 , z  c1 
c
c
c
c2
 ...  nn1 and z  c1  2  ...  nn1 .
z
z
z
z
Then we have either z  1 or z  1 and so it follows that

1
1 
1
z  M 1   ...  n 1   M 

1
z
z 
1

z

1
hence z  1    M .
z

Therefore z  max 1, M  1 ; it results that the set F(B) is bounded.
Corollary. The roots of any algebraic polynomial depend continuously on
the coefficients of that polynomial .
We add following results :
Proposition 2. If the polynomial P  X n  c1 X n 1  c2 X n  2  ...  (1) n cn
D (c1 , c2 ,..., cn )
has the complex roots x1 , x2 ,..., xn , then jacobian
equals the
D ( x1 , x2 ,..., xn )
Vandermonde determinant
  x  x .
1 j i  n
i
j
22
Alina Niţă, Ana Niţă
Proof:
We prove this by induction ; for n=2 it is clear and suppose the assertion
true for n 1 . By the Viète’s relations,
ck  Sk  x1 ,..., xn  ; c0  1  S0  x1 ,..., xn  ( the symmetrical polynomials).

S k  x1 ,..., xn   S k 1  x1 ,..., xi 1 , xi 1 ,..., xn  . In the above considered
xi
jacobian , substract the first colum from each of the next columms and use the
relation
Sk 1  x2 ,..., xn   Sk 1  x1 ,..., x j 1 , x j 1 ,..., xn   ( x j  x1 ) S k  2  x2 ,..., x j 1 , x j 1 ,..., xn  ,
But
for 2  j , k  n.
By developping the obtained determined by the first row and by factoring
,..., d n 1 ) n
  x1  x j ,
 x1  x j , one gets that jacobian is just DD((dx1 ,,dx2 ,...,

xn 1 ) j 2
2 j  n
2
3
where d i  Si  x2 ,..., xn  . The proposition follows from the induction hypothesis.
Corollary.
If
the roots x1 , x2 ,..., xn are district and the coefficients
c1 , c2 ,..., cn as C 1 functions , then locally
n ( n 1)
D ( x1 , x2 ,..., xn )
=  1 2 
D (cn , cn 1 ,..., c1 )
1
 x  x 
1 j i  n
i
.
j
It is enough to apply the theorem of the local inversion theorem.
3. The case the random polynomials
Let us now consider a fixed probability field
f k : (a, b) 
 , K , P  .
Let
, be functions of C 1 class on the open interval (a, b) and ck
p
random variables , ck :   , 1  k  p . Denote F ( x,  )   ck  . f k  x  .
k 1
The equations F  x,    0 generalize both the random algebraic equations
(considering
f k as
f k  x   xk 1 ,1  k  p ) and the random trigonometric
equations ( f1  x   1, f 2  x   cos x, f3  x   sin x, f 4  x   cos 2x, etc .).
A solution of the equation F ( x,  )  0 , is any function   ,
  z   such that F ( z ( ),  )  0 a.e. In most cases, there are no measurable
(random) solutions. However, we will specify some conditions when this happens.
23
Dependence of the roots on coefficients
Let us consider a deterministic polynomial
P  M n , P( x)  x n  c1 x n 1  ....  cn , ck 
.
For each integer q  1, there are complex numbers  c pq  , 0  p  n, q  1 ,
such that the identity


i


cn  p  z  e q


p 0 


n
p


p
  c pq z  0 takes place ;



in particular, cnq  1 for any q  1 . Let K be the unit cube from
(1)
n
, K   0,1 and
n
we fix x   x1 , x2 ,..., xn   K . For each integer q  1, k  1,
, for 1  k  n
 xk
 n
put
q (k )  
 c pq .q  k  p  n  , for k  n  1
 p 0
for q  k   0

q  k  1 q  k  ,
and
.
(2)
 q (k )  
0
,
for

k

0



q

We thus obtain two double sequences of complex numbers that depend on x.
The following fact is known: for any q  1 , there is  q  lim q  k  and
k 

i 

furthermore, the limit lim   q  e q  dx1....dxn exists and is equal with one of the

q  K 


roots of the polynomial P. ( [ 1 ] )
We draw attention to an ordering of a finite number of complex numbers,
which will be now taken into consideration.
If m  1 is a fixed integer, then there is a function h : m   such that
for any
m distinct complex numbers z1 ,..., zm and for any q  h  z1 , z2 ,..., zm  ,
i

we have z1    z2    ...  zm    0, where   e q . It’s enough to notice
that for each distinct z1 ,..., zm numbers from , there is h   depending on
0    h ,the
z1 ,..., zm
such
that
for
each
β
with
relation
z1  ei  z2  ei  ...  zm  ei  0 takes place.
24
Alina Niţă, Ana Niţă
Proposition 3. Let P( x, )  xn  c1 () xn1  ....  cn   ;  , x  be a
monic polynomial with coefficients random variables defined on  . Then there
is a measurable function ( random variable)  :   such that P    ,    0
a.e.
Proof:
The coefficients c pq which verify (1) can be chosen as random variables
depending on c1 , c2 ,..., cn . Then we can build the sequences of random variables
q  k ,   and  q  k ,   defined by (2). Expanding the above mentioned result, it
results that

i 

z1 ( )  lim   q    e q  dx1...dxn

q  K 


is a random variable and furthermore , P  z1   ,    0 .We then define the
function  as being z1 .
Corollary . Given a random polynom
P( x, )  xn  c1 () xn1  ....  cn   , there is an ordering of the solutions of that
polynom such that these are random variables (measurable solutions)  
depending on the coefficients ck .
Proof:
Using Proposition 3 there is a first solution z1 ( ) . Using induction and
applying Horner’s scheme , by considering the random polynomial
Q( x, )  xn1  b1 () xn2  ....  bn1   where
bj    c j    z1   bj 1   ,1  j  n we obtain bn  0 .
By the induction hypothesis, let z2   , z3   ,..., zn   be the ordered
roots of Q, which are measurable functions of bi . Then the roots of P will be
z1 , z2 ,..., zn which are measurable functions of ck , because bi depends measurably
on ck .
REFFERENCE
[ 1 ] A. Bharucha – Reid , Random algebraic equations , Academic Press , 1970.
[ 2 ] J. Hammeyley, The zeros of random polynomial , Proc . 3 d Berkeley Sympos.Statistics, 2,
89-111 , 1956.
[ 3 ] S. Lang, Algebra, Ed.Springer 1997.
[ 4 ] N. Radu, D. Ion Algebra , Editura Didactica si Pedagogica, Bucuresti 1991.