On a decomposition of polynomials in several variables

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

A

NDRZEJ

S

CHINZEL

On a decomposition of polynomials in several variables

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{2 (2002),}

p. 647-666

<http://www.numdam.org/item?id=JTNB_2002__14_2_647_0>

© Université Bordeaux 1, 2002, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

On

a

decomposition

of

polynomials

in several variables

par ANDRZEJ SCHINZEL

Dedscated to Michel France

RÉSUMÉ. On considère la

_{représentation}

d’un

_polynôme

a

plu-sieurs variables comme une somme de

polynômes

à une variable en combinaisons linéaires des variables.

ABSTRACT. One considers

representation

of a

polynomial

in

sev-eral variables as the sum of values of univariate

_polynomials

taken

at linear combinations of the variables.

K. Oskolkov has called my attention to the

following

theorem used in

the

_theory

of

_polynomial

_{approximation}

_(see

_[6],

Lemma 1 and

_below,

Lemma

_4):

for _{every sequence} of d + 1

_pairwise

_{linearly independent}

vec-tors

_{[a,., , a,,2]}

_(l~/~~+l)

and _every

_polynomial

F ~

_{C [Xl, X2]}

of

degree

d there exist

_polynomials

_fILE

_{C [z]}

₍₁

_It d +

₁₎

such that

He has asked for a

_{generalization}

and a refinement of this result. The

following

theorem is a

_step

in this direction.

Theorem 1. Let

_{n, d}

be

_positive

_integers

and K a

field

with char K = 0

or char K &#x3E; d. For _{every sequence}

_S"

_{(2 v}

_n)

_of

subsets

_of

K each

_{of cardinality}

at least d -f- 1 there exist M =

Cn n 1 1 J

_vectors

( n-1 )

[a,1 ,

2? - " ? E

_{I}

x

S2

x... x

8n

with the

_following

_property.

For every

polynomial

F E

_{K~~1, ... ,}

_{of degree}

at most d there exist

polyno-mials

_fp

E

_{K[z] (1}

_{S J-t S}

_M)

such that

It is not true that

_polynomials

_fp.

_satisfying

₍₁₎

exist for every sequence of vectors

_[a,,l,

... ,

(1

/-t

M)

such that each n of them are

linearly

independent.

See the

_example

at the end of the _paper.

Let

_P(n,

_d,

_K)

be the set of all

_polynomials

F E

xnJ

of

_degree

d. Let

_{M(n, d, K)}

be the least number M such that for every F E

_P(n,

_d,

_K)

(1)

holds for some _sequence of vectors

_{~a~l, ... ,}

E K’~ and some se-quence of

polynomials

f ~

E

_K[z]

_{( 1 ~a M)}

if such _sequences exist and

00 otherwise. For an infinite field

K,

let

m(n,

d,

K)

be the least number M

such that for a Zariski _open subset S of

P(n, d, K)

and for _every F E

S,

(1)

holds for some _sequences of vectors and

_polynomials

as

before,

if such sequences exist and oo, otherwise. Theorem 1

implies

Corollary.

oo

zf

and

only if

either n = 1 or char K = 0 _or

char K &#x3E; d.

_{If K is infinite}

the same

equivalence

holds

for

m (rt,

d,

K) .

The

_problem

of determination of

_m(n,

d,

K)

is related to the

_problem,

much studied in the XIX th

_century

_(see

_[5],

for a modern

account),

of

representation

of a

_general

_n-ary form of

degree

d as the sum of _powers of

linear forms. The two

_problems

are not

_equivalent

even for K

algebraically

closed,

since in our case neither F nor

fp

are

supposed homogeneous.

Theorem 2. For _every

_{infinite field}

K such that char K = 0 _or char K _&#x3E; d

we have

For K =

Fq,

char K

&#x3E; d,

_we have

In

_particular,

_{every n-ary} form of

_degree

d over a field K of characteristic

0 is

_{representable}

as a linear combination of

(’~~ d i l)

d-th _powers of linear

forms over K. This has been first

proved,

but not

explicitly

stated

by

Ellison

_[3].

Clearly

M(I,

d,

K)

=

M(n,1, K)

=1 and one

_easily

_proves

Theorem 3.

_2,

then

_{M(n, 2, K)}

= n

_{and if,}

in

_{addition, K is}

infinite,

then

_m(n,

_2,

_K)

= _n.

Diaconis and Shahshahani asserted without a formal

_proof

that

We shall show .

Theorem 4. For _every

_field

K such that either char K =

_0,

_or char K _&#x3E; d

and card K &#x3E; 2d - 2 we have

In

_particular,

_every

_binary

form F of

_degree

d over a field K of

charac-teristic 0 is

_{representable}

as a linear combination of d d-th _powers of linear

forms over

_K,

which

_slightly

_improves

Theorem A of

[4].

For K = C this

was

_{proved by}

Reznick

_[8].

The

_following

theorem shows that the condition card K &#x3E; 2d - 2 in

Theorem _{4 may} be

_superfluous.

Theorem 5. For _every

_field

K such that

char K &#x3E; d and card K d + 2

we have

Theorem 6. For _every

_{algebraically}

closed

_{field K, if}

char K = 0 _or

char K &#x3E;

_d,

then

The

_proof

of Theorem 1 is based on two lemmas.

Lemma 1.

_{Letn 2::: 2~}

₍₁

i

_~

be a subset

_{of K of cardinality}

_d+l.

Then F = 0 is the

only polynomial

in

~[~i,... ?

of degree

_{at most} d _in

each variable such that

_{F(ai, ~2,...,}

= 0

for

all

(al, a2, ... ,

Proof.

See

_[1],

Lemma 2.2.

Lemma 2. Let

_for

each k =

0,1, ... ,

_{n -} 2 elements

of

K

(0 ~

_l

d)

be distinct and let

_for

a

_{positive integer q}

(d

+

_1)wl

be the

_expansion

_of

_{q -} 1 in base d + 1.

Define

Then det 0.

Proof.

Let us

_put

in Lemma 1:

Tï

=

{,Qi-i,c : ~ l

d} (1

i n -

_1).

By

the lemma the

_{only polynomial}

F E

_{K(xl, ... ,}

_xn-11

of

_degree

at most

d in each variable such that

Now,

all the vectors

_{~lo, ... , ln_2~}

E

_{O,

_{ll... ,}

dln-1

can be ordered

lex-icographically,

so that the vector

_{~lo, ... , l~_2~}

_occupies

the

_position

1 +

n-2

E

+

1 ) i

and then the

_system

of

_equations

₍₃₎

reads

i=O

Also the

_polynomial

F can be written as

and

₍₃₎

can be rewritten as

The fact that the

_only

solution of this

_system

is

corresponding

to F =

_0,

implies

in view of

(2)

that

But then also det = det

,-~

0.

Proof of

Theorem. 1. Let us choose in

5~

distinct elements

_{/3"_2,0, ... ,}

(2 v n).

By

Lemma 2

7

hence the matrix B

_consisting

of the rows r for which

1 .

n

+ d -

1

d is of rank

_equal

to the number of such rows M

= n + n - d 1 1 .

T herefo r e B has M

_{linearly independent}

columns _{81, 82, ... , sM.} We

_put

Let

(note

that the multinomial coefficient is

_non-zero).

For each 1 d we determine

( 1 ~

M)

from the

system

of

equations

which can be rewritten as

By

the choice of _{sl, ... ,} sM the matrix of this

system

has rank

equal

to the number of

_equations,

hence the

_system

is solvable for E K. We set

and

₍₁₎

follows from

₍₆₎

and

_(7).

Proof of Corollary.

In view of Theorem 1 it is sufficient to show that

M(n, d, K)

= oo if n &#x3E; 1 and

Let us consider an

_arbitrary

polynomial F

of the form

(6)

in which _ad,O,...,o

~

0 and _{ap-1,1,o,...,o}

_~

0. If K is infinite such

_polynomials

exist in _{every open}

subset of

_{P (n, d, K) .}

If

₍₁₎

_holds,

then the

_part

_Fd-p

of

_degree

_p of F satisfies

-which is

_impossible,

since

xpl-l X2

occurs with a non-zero coefficient on the

left-hand

_side,

but not on the

_right.

Proof of

Theorem 2. The dimension of the set of all _n-ary

_polynomials

of

_degree

not

_exceeding

d and

_greater

than e is

(n!d) -

( e

On the

d

where

_f

_{= L}

is at most d - e + n - 1 since the vectors a can be

I=e+l

normalized

_by

_taking

the first

_{non-vanishing}

coordinate

_equal

to 1. This

gives

the _upper bound

_m(n

+ 6! 2013 1 -

_e)

for the dimension of the set of all

m

polynomials

of the form

_E

_{f J1. (OoJ1.x)}

_and,

_by

the definition of

_{m (n,}

_d,

_K)

IA= 1

which

_implies

the first

_part

of the theorem.

In order to _prove the second

_part

let us observe that the number of

nor-malized vectors a is

q-1 ,

while the number of non-zero

polynomials

Hence we obtain at most

polynomials

of the form

_{f (ax)}

and at most

polyno-m

mials of the

_{form 1:}

On the other

_hand,

the number of _n-ary

p.=1

polynomials

over

_1FQ

of

degree

not

_exceeding

d and

_greater

than e is

By

the definition of

_M(n,

_d,

this

_gives

which

_implies

the second

_part

of the theorem.

Proof of

Theorem 3. Let

_Fo,

the

_leading

_quadratic

form of

_F,

be of rank r.

By

_Lagrange’s

theorem there exist

_{linearly independent}

vectors

~a~ 1, ... , apn]

in Kn

_{( 1}

_p

_r)

such that

We set _au = 0 for _r

ti

~

n and choose n - r vectors

_{[0:1£1’...’}

in Kn

₍₁

_{:s; J..t}

_r)

such that Then there exist E K

and

₍₁₎

follows with M = _n .

n

On the other

_hand,

the

_polynomial

F

₌

where 0 is

_clearly

v=1

not

_{representable}

in the form

_{( 1 )}

with M n.

For the

_proof

of Theorem 4 we need

Lemma 3. We have the

_identity

where TZ is the i-th

fundamental

_symmetric

functiort of

~1, ... , zd, To = 1.

Proof .

See

_[7],

_p. 333.

Lemma 4. Let _ap’

_{(1 ~ p,}

_d)

be

_{arbitrary pairwise linearly independent}

vectors in

K2.

_If

char K = 0 or char K &#x3E;

_d,

_for

_every

_polynomial

F E

K[XI,X2]

of degree

at most d - 1 there exist

_polynomials

_f~,

E

_K[z]

such that

Proof .

Since _{ce IA} are

_pairwise

_{linearly independent}

we _may assume that

either

(i)

O:¡.¡.l =

1,

c~2 are all distinct

(1 ~

d),

or

(ii)

_all =

1,

a,2 are all distinct

(1 G ~c

d),

_adl =

_0,

ad2 = 1.

Let now

(note

that the binomial coefficient is

_non-zero).

In the case

(i)

for each

since the rank of the matrix of the coefficients

_equals

the number of equa-tions. Then we set

n-1

In the case

(ii)

for each I d we can solve for in K the

_system

of

equations

and then we set

Proof of

Theorem

_4.

In view of Theorem 3 we _may assume d &#x3E; 3. We shall prove first that

M(2, d, K) d.

Let F E

_P(2,

_d,

_K)

and let

_Fo

be the

_{highest homogeneous}

_part

of F.

Supposing

that we have

_represented

_Fo

in the form

(1)

with M = d _we

may assume that _ap

(1 ~ JJ

d)

are

_pairwise

linearly independent

and

then

_apply

Lemma 4 to

_represent

_F-Fo

in the form

₍₁₎

with the same _{a IA} *

Therefore,

it is

_enough

to find a

_{representation}

(1)

for F

_homogeneous

of

degree

d.

_By

Lemma 1 there exist _{Cll, C21} in K such that 0.

Replacing

F

_by

_F(cllxl

+

_C22~2)~

where _{c12, c22} are chosen

in K so that _{CllC22 -}

c12c21 ~

0 we _may assume that the coefficient of

xd

in

_{F(xi, X2)}

is non-zero. Let then

and let us consider the

_polynomial

Since 0 the

_polynomial

G is not

_identically

0 and we have for each

-Since card K &#x3E; 2d -

_2,

_by

Lemma 1 there exist elements

_{{31, ... , #d_2}

of K such that

We now

_put

which makes _sense, since

_by

₍₉₎

the denominator is non-zero.

Again

by

(9)

we have

_{,Qi ,-~}

_{3j

for 1 i

_j

d. Hence

However,

by

(10)

and Lemma

_3,

Hence the

_system

of

_equations

We set

and obtain

₍₁₎

from

₍₈₎

and

_(11).

It remains to show that

_{M(2, d, K) &#x3E;}

d. Let us consider the

equation

In order to _prove that it is

_impossible

for every a E K it is

clearly

sufficient to consider

_fp

=

bp.zd,

all =

_1,

ap2 distinct.

Comparing

the coefficients of on both sides of

(12)

we obtain

The determinant of this

_system

is

(1 ~

d)

and

_by

₍₁₂₎

, hence

b~ -

0

a contradiction. This

_argument

is valid without the

assumption

on

For the

_proof

of Theorem 5 we need

Lemma 5. Let _{al, ... , ak} be distinct elements

_of

_Pp,

k &#x3E; _{p -} 3. Then

Proof .

If k = _{p - 1} we use the

identity

If k = _{p - 2} we _argue

_by

induction.

_{For j}

= 0 the _{statement is}

true,

for

hence, by

induction

If k = _{p -} 3 we _argue

_{again by}

induction.

_{If j}

= 0 the statement is _true. If

k

_{&#x3E; j}

&#x3E; 1 we have the

identity

hence,

by

induction

Proof of

Theorem 5.

_By

the last statement in the

_proof

of Theorem 4 we

have

_M(2,

_d,

_{K) &#x3E;}

_d,

thus it is remains to _prove the reverse

_inequality.

Let F E

_P(2,

_d,

_K).

_By

Lemma 4 we _may assume that F is

homogeneous.

Let

and consider first card K = _p = d ₊ 1.

Let us assume first that the

_mapping

_Pp

-

_Fp

_{given by t}

H

_{f (t)}

=

p-i ,

E

ap-l-it’

is not

_injective.

Then there exist _{r, s} such that

_{r #}

s and

i=O

f (r)

=

f (s),

hence

71= I

Setting

a12 =

0,

~a22, ... ,

ap_2,2~ _ ~ ~

jr, s}

we have

_by

Lemma 5

hence,

by

(14),

Since det

_(~2)

_0,

this suffices for

_solvability

over

Fp

of the

_system

of

_equations

-Then we obtain from

(13) - (15)

that

Assume now that the

_{mapping Pp -+}

_IFP

_{given by}

t ~

_{f (t)}

is

_injective.

We shall consider three cases

In the case

(i),

let

so that

Setting

a12 =

0,

{a22, ... ,

ap_1,2} _ ~

B Irl

we have

by

Lemma 5

hence, by

(16),

Since det

₍₂₎

_0,

this suffices for

_solvability

over

_IFp

of the

_system

of

_equations

Then we obtain from

(13)

and

(17)

that

In the case

(ii),

let

so that

Setting

all =

0,

{a21, ... ,

B {r}

_we have

_by

Lemma 5

hence, by

(18),

and, by

Lemma

_3,

Since det

_(~1)

0_ jGp-1

=1=

0,

this suffices for

solvability

over

_Fp

of the

_system

of

_equations

In the case

(iii),

since

we have _ao =

Op-i. Hence the first and the last row of the determinant

are

equal

and the determinant vanishes.

Since

0,

this suffices for

_solvability

over

_Fp

of the

i

system

of

_equations

Then we obtain from

(13)

and

(20)

Consider now the _case, where

Again,

let us assume first that the

mapping F; --+

IF~

given by

t ~

f (t)

=

is not

_injective.

Then there exist r, s

_{E PP}

such that

hence

Setting

we have

by

Lemma 5

and,

by

Lemma

_3,

Since det

(~2)

o~jp-3 7~

0,

this suffices for

_solvability

over

_Fp

of the

_system

&#x3E;

of

_equations

-Then we obtain from

(13)

and

(22)

that

Assume now that the

_{mapping Pp}

-

_IFp

_{given by}

t 1-4

_{f (t)}

is

_injective.

We shall consider two cases

In the case

(iv)

let 0 =

f (r),

_{r E so} that

Setting

we have

by

Lemma 5

hence,

by

(23)

Since det

_(C~,,2)

_{Ojp-2 0}

_0,

this suffces for

_solvability

over

_1Fp

of the

_system

of

_equations

Then we obtain from

(13)

and

(25)

that

In the case

(v) t

-&#x3E;

f (t)

is a

bijective mapping

_{ofF; onto F;.}

If the

_mapping

t H

_{t f (t)}

had the same

_property,

we should obtain

which is

_impossible.

Hence there exist _{r, s}

_{E IF;}

such that

_{r 0 s and}

Setting

we have

_by

Lemma 5

hence, by

(26), (24)

holds and we conclude the

_argument

as in the case

(iv).

The

_proof

of Theorem 5 is

_complete.

Proof

of

Theorem 6. We shall prove first that

hence on

_taking

e = d + 1 - u we obtain from Theorem 2

which

_gives

_(27).

In order to show that

we notice that

Let us consider

independent

variables where

and the matrix B = where

1~~

being

determined

_by

the

_inequality

Let

Bv

be the minor of the matrix B obtained

_{by omitting}

the v-th column and D be the discriminant of the

_polynomial

Polynomials

Bo

and D in the variables _aid are not

identically

zero.

In order to see that 0 let us order all variables _aid

_{linearly assuming}

aki if either ~ + l or

_{2 -+- j}

=

k + 1 and j

l. Then all

products

of aid are ordered

lexicographically.

The

product

occuring

in the

_expansion

of

Bo

precedes

in the

_{lexicographic}

order _any other term in this

_expansion,

hence it does not cancel and 0. On the other hand

-

Now,

the discriminant of the

_polynomial

is not

_identically

0 as a function

of t"

(it

is different from 0 for

_tv

= 0 for

v m - _p,

_tm-p

=

1),

hence

Do

= 0

implies

_an

algebraic dependence

_over

the

_prime

field II of K between

_{(1 v m -}

_p).

Let

We assert that for

This is

_obviously

true for i m - 1. Assume that it is true for all i

_j,

where

_{m j m -f- 1 - 1.}

_Since,

_by

the Cramer

formulae,

for _p =

.i_; - _ , ...

and all a’s

_occuring

on the left-hand side have the second index at most

it follows that E Q

(28)

is

_complete.

But then

and the inductive

_proof

of

implies

while the number of

_independent

variables

1, m j m ~- k - 1)

equals

We now assert that if for a

polynomial

we have

BoD =1=

0,

then there exist

such that

Indeed, using

the notation introduced earlier we take for

the m distinct zeros of the

_polynomial

Now for each 1 d we solve the

_system

of

equations

for in K and assert that the solution satisfies the

_larger

_system

The

_proof

is

_by

induction on

_j.

We assume that

(32)

is true for

_{all j}

_i,

i &#x3E; m and obtain

However the sum on the

right-hand

side of

(33)

coincides with the sum on

the left-hand side of

₍₂₉₎

on

_putting

there k

=1-m+l,

_a =

i-rrz+~2~+1

(kt1).

Hence

_by

₍₂₉₎

which proves

(32).

we obtain

(31)

from

(30)

and

(32).

Example.

Each three of the vectors

_{~1, 0, 0~,}

_{(0,1, 0~, [0,0,1],}

_(3,1,1~,

[1, 3, 1], (3, 3, 2~

are

linearly independent

over

Q,

nevertheless for all

poly-nomials

_fi

E

_Q[z]

₍₁

-i

₆₎

we have

Indeed,

it is

_enough

to consider the case

_fi

=

biz2 (1 i G 6).

Assuming

the

_equality

in

₍₃₃₎

we obtain

_comparing

the coefficients of _{XIX2, XlX3} and

which is

_impossible,

since

I conclude

_{by expressing}

_my thanks to U. Zannier for a remark

_helpful

in the

_proof

of Theorem 5.

Note added in

_proof.

M. Kula has checked that

_{M(2,d,K) =}

d in the

simplest

cases not covered

_by

Theorems 4 and 5: d = 7 or

_8,

K =

References

[1] N. ALON, M. B. NATHANSON, I. RUZSA, The _polynomial method and restricted sums _of

congruence classes. J. Number Theory 56 (1996), 404-417.

[2] P. DIACONIS, M. SHAHSHAHANI, On nonlinear functions of linear combinations. SIAM J. Sci. Stat. Comput. 5 _(1984), 175-191.

[3] W. J. _{ELLISON, A ’Waring’s problem’ for homogeneous forms.} Proc. _Cambridge Philos. Soc. 65 _(1969), 663-672.

[4] U. _{HELMKE, Waring’s problem for binary forms.} J. Pure _{Appl. Algebra} 80 _(1992), 29-45.

[5] A. _IARROBINO, Inverse _{system of} a _{symbolic power} II. The _{Waring problem for forms.} J.

Algebra 174 _(1995), 1091-1110.

[6] B. F. LOGAN, L. A. _{SHEPP, Optimal} reconstruction of a _{function from} its projections. Duke Math. J. 42 _(1975), 645-659.

[7] T. Mum, A treatise on the theory of determinants. Dover, 1960.

[8] B. _REZNICK, Sums of powers of complex linear forms, unpublished manuscript of 1992.

Andrzej SCHINZEL

Institute of Mathematics Polish Academy of Sciences 00-950 Warsaw

Poland