TIGHT BOUNDS FOR CHEVALLEY--WARNING--AX--KATZTYPE ESTIMATES, WITH IMPROVED APPLICATIONS

OSCAR MORENO, KENNETH W. SHUM, FRANCIS N. CASTROAND P. VIJAY KUMAR

1. Introduction

In 1935, C. Chevalley proved a conjecture by E. Artin: if F ðX1; . . . ; XmÞ is ahomogeneous polynomial of total degree d < m over a 7nite 7eld Fq having q ¼ pfelements, then F has a non-trivial zero. Chevalley showed the result to hold evenwhen the homogeneity hypothesis is replaced by the weaker requirement thatF ðX1; . . . ; XmÞ has no constant term. Almost immediately, E. Warning showed thatthe characteristic of the 7eld divides the number of zeros. A proof of the Chevalley--Warning theorem can be found in [2] and [9]. In [2], Ax followed Dwork’s ideas [4]and developed a p-adic technique which revealed previously unseen solutions. Heproved that if b is equal to dm=de � 1, where dae is the smallest integer larger thanor equal to a, then the number of zeros of F is divisible by qb. Ax’s results werebased on an estimate of the p-divisibility of the exponential sum associated to F .

Using the principle of inclusion and exclusion, Ax extended his result to a systemof polynomials. Let

FiðX1; . . . ; XmÞ; for i ¼ 1; . . . ; r;

be a collection of polynomials over Fq of degree di respectively. Set

� ¼�m�

Pri¼1 diPr

i¼1 di

�:

Then the number of zeros of the system of polynomials is divisible by q�. In 1971,N. Katz [7] improved this theorem by showing that the number of zeros is divisibleby q�, where

� ¼�

m�Pri¼1 di

max16 i6 rfdegðFiÞg

�:

The results of both Ax and of Katz were obtained by 7rst estimating the divisibilityof exponential sums associated to the polynomials.

The result described below, due to Moreno and Moreno [17, 19], often providesan improved divisibility estimate when compared with the result of Ax and Katz.Let p be prime. Every non-zero rational number s can be written in the form pia=b,where neither a nor b is divisible by p. The integer i is called the valuation,written as ordpðsÞ, of s. By de7nition, ordpð0Þ ¼ 1. Given an integer,

n ¼ a0 þ a1pþ a2p2 þ . . .þ alpl; where ai 2 f0; 1; . . . ; p� 1g;

we denote its p-weight by �pðnÞ ¼Pli¼0 ai.

Received 5 December 2001; revised 5 March 2003.

2000 Mathematics Subject Classi�cation 11L07 (primary), 11G25 (secondary).

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Proc. London Math. Soc. (3) 88 (2004) 545--564 q 2004 London Mathematical SocietyDOI: 10.1112/S002461150301462X

A monomial Xe11 . . .Xemm in this paper is abbreviated by Xe. The p-weight degreewpðXeÞ of a monomial is de7ned by �pðe1Þ þ . . .þ �pðemÞ. The p-weight degreewpðF Þ of a polynomial F ðX1; . . . ; XmÞ ¼

Pi aiX

ei (with ai 6¼ 0) is then de7ned bywpðF Þ ¼ maxi wpðXeiÞ. The bound of Moreno and Moreno improving that of Axand Katz states that the number of zeros of F1 ¼ F2 ¼ . . . ¼ Fr ¼ 0 is divisible byp�, where

� ¼�f

m�Pri¼1 wpðFiÞ

max16 i6 rfwpðFiÞg

� ��:

This bound improves upon the Ax--Katz result when the characteristic p is smallin comparison with the degrees di.

Moreno and Moreno’s improvement to the Chevalley--Warning theorem [17, 19],that we improve in x 5.3, says that ifm�

Pri¼1 wpðFiÞ > 0 and none of the equations

in the system have constant term, then there is a non-trivial solution.Some new interesting techniques were introduced by Adolphson and Sperber

[1]. Suppose F ðX1; . . . ; XmÞ ¼Pi aiX

ei (with ai 6¼ 0). Let JðF Þ be the Newtonpolyhedron, which is the convex hull in Rm of the set feig [ fð0; . . . ; 0Þg. Let!ðF Þ be the smallest positive rational number such that !ðF ÞJðF Þ contains apoint of Nmnð0; . . . ; 0Þ, where N is the set of non-negative integers. Adolphson andSperber [1] proved that if F is not a polynomial in some proper subset of thevariables X1; . . . ; Xm, then the exponential sum

SðF Þ ¼X

x1;...;xm2Fq

�ðF ðx1; . . . ; xmÞÞ;

where � is a non-trivial additive character de7ned over Fq, satis7es

ordpðSðF ÞÞ> f!ðF Þ:When this bound is converted into a bound on the p-divisibility of the numberof zeros of a set of multivariable polynomials, the result improves upon theAx--Katz result.

The results of Moreno and Moreno and of Adolphson and Sperber are not directlycomparable. On the other hand, by combining the Moreno--Moreno reduction tothe ground 7eld method [19] with the Newton polyhedra techniques as introducedby Adolphson and Sperber, we proved that the divisibility obtained from the newNewton polyhedron gives an improvement to the result of Moreno and Moreno [15].

In this paper, we provide a tight bound to the p-divisibility of the exponentialsum associated to a multivariable polynomial F ðX1; . . . ; XmÞ. If F has expansion

F ðX1; . . . ; XmÞ ¼XNj¼1

ajXej ; where 0 6¼ aj 2 Fq;

then the bound depends only upon the exponents ej of the monomials Xej

appearing above with non-zero coeKcient, that is, with aj 6¼ 0. It is tight (see Theorem7 and Theorem 9) in the following strong sense: if the monomials Xej are 7xed andthe coeKcients aj varied, then there is some choice of coeKcients which will yield apolynomial for which the bound is tight. This result was motivated by and is alongthe lines of the result on the divisibility of the Hamming weights of code-words dueto McEliece [10].

A polynomial representation for the additive character used to de7ne theexponential sum is provided in x 2. The divisibility properties of the coeKcients of

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O. MORENO, K. W. SHUM, F. N. CASTRO AND P. V. KUMAR546

this polynomial are also stated there. The bound on the p-divisibility of theexponential sum is provided in x 3. A proof of tightness is also given.

Since the tight bound involves the solution of a set of modular equations which arenot always easy to solve, several techniques are introduced in order to give concreteapproximate solutions and they are given in x 4. The 7rst approximation technique(x 4.2) replaces our problem of solving a linear modular system of equations withthat of solving an integer linear programming problem. Actually using this wealso manage to obtain the Moreno--Moreno result as a corollary. In x 4 we alsopresent several approximations that, for many classes of examples, improve uponthe results of Ax and Katz, Moreno and Moreno, and of Adolphson and Sperber(see Theorem 18, and also Theorem 27 in x 5).

In x 5, we apply our method and 7nd bounds on the divisibility for the number ofsimultaneous zeros of a collection of polynomials. An important consequence of oneof our main results (Theorem 25) is a method that we present in x 5.2 to 7nd classesof examples for which bounds are tight. In particular we give classes of examplesfor which the Moreno--Moreno bound is tight (Corollary 26). We want to point outthat our example of x 5.3 illustrating how our method gives improvement to theChevalley--Warning theorem is of importance for the applications.

In this paper a system of techniques is introduced to study the divisibilityproperties of exponential sums as well as that of systems of equations. We giveseveral important applications and, in particular, Corollary 26 and Theorems 27 and28. On the other hand, other important applications are possible and in particularwe can improve p-adic Serre bounds of the type introduced by Moreno and Morenoin [20]. Also we can apply our methods to the study of singular curves and varieties.This will be done elsewhere.

Finally we wish to point out that the Moreno--Moreno result has found applica-tions in coding theory [13] and also we have applied the results of this paper tocompute the covering radius of certain codes [12], and to Waring’s problem in [14].Consequently, it is most likely that the results of this paper, as well as the generaltechniques that we introduce here, will also 7nd more applications in coding theoryand other similar areas.

2. A polynomial representation for an additive character

Notation. Let q ¼ pf be a power of a prime and Fq be the 7nite 7eld with qelements. Let Qp be the p-adic 7eld with ring of integers Zp. Let K be the uniqueunrami7ed extension over Qp of degree f . Then K can be obtained by adjoining aprimitive ðpf � 1Þth root of unity in Qp, the algebraic closure of Qp. The residueclass 7eld is isomorphic to Fq and let T denote the Teichm€uuller representatives of Fqin K. Let ! be a primitive pth root of unity in Qp, so Kð!Þ is a rami7ed extensionover K of degree p� 1. For a 2 Zp, de7ne !

a ¼ !amod p. De7ne

" ¼ ! � 1: ð1Þ

Let OK be the ring of integers in K, and P be the unique maximal ideal. Let OKð!Þdenote the ring of integers of Kð!Þ and Q be the corresponding maximal idealgenerated by ". Both OK=P and OKð!Þ=Q are isomorphic to Fq. (In characteristic2, the primitive square root of unity, namely �1, is in Qp, and Kð!Þ is the same asK. In this case, " should be set equal to 2, and with this all statements in the restof this paper hold with " replaced by 2.)

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CHEVALLEY--WARNING--AX--KATZ TYPE ESTIMATES 547

Given 06 j, let ji be integers, 06 ji < p, such that j ¼Pn�1i¼0 jip

i. Set

�pðjÞ ¼Xn�1

i¼0

ji and #pðjÞ ¼Yn�1

i¼0

ji!:

The subscript p will be dropped if it is clear from the context. Note that ð�pðjÞ � jÞis divisible by ðp� 1Þ.

LEMMA 1 [2]. There is a unique polynomial

CðXÞ ¼Xq�1

‘¼0

cð‘ÞX‘ 2 Kð!Þ½X�

of degree q � 1 such that

CðtÞ ¼ !TrK=Qp ðtÞ; for all t 2 T : ð2Þ

Moreover, the coe"cients of CðXÞ satisfy

cð0Þ ¼ 1; ð3aÞðq � 1Þcðq � 1Þ ¼ �q; ð3bÞ

ðq � 1ÞcðjÞ ¼ gðjÞ for 0 < j < q � 1; ð3cÞ

where gðjÞ is the Gauss sum,

gðjÞ ¼Xt2T �

t�j!TrK=Qp ðtÞ:

Remark 1. Note that cð‘Þ ¼ cð‘pÞ. If ‘1 and ‘2 lie in the same cyclotomic cosetmod q � 1, then cð‘1Þ ¼ cð‘2Þ.

PROPOSITION 2 [5]. The numbers p and "p�1 are associates in the ring OKð!Þ.

COROLLARY 3. The number " is not invertible in OKð!Þ. The ideal P in OKlies over pZ.

THEOREM 4 (Stickelberger [11, 8]). For 06 j < q � 1,

gðjÞ#pðjÞ"�pðjÞ

� �1 mod ": ð4Þ

Using Stickelberger’s theorem, we can calculate the valuation of the coeKcientsof CðXÞ. The following lemma is an immediate consequence of Lemma 1 andStickelberger’s theorem.

LEMMA 5. For ) 2 Kð!Þ, let ord"ð)Þ be the valuation with respect to ". Then

ord"ðcðjÞÞ ¼ �pðjÞ: ð5Þ

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3. Bound on p-divisibility of an exponential sum

3.1. Preliminaries

The non-zero elements in the Teichm€uuller set T satisfy the following orthogonalrelation: X

t2T �tj ¼ q � 1 if j � 0 mod q � 1;

0 otherwise.

�ð6Þ

If we sum over all elements in T and use the convention 00 ¼ 1, we get

Xt2Ttj ¼

q if j ¼ 0,

q � 1 if j 6¼ 0 and j � 0 mod q � 1,

0 otherwise.

8<:

This translates in the multivariable case to the following result.

LEMMA 6. Suppose e1; e2; . . . ; em are non-negative integers and s of them arenon-zero, then

Xt2T m

te11 te22 . . . temm ¼ ðq � 1Þsqm�s if ei are all divisible by q � 1,

0 otherwise.

�ð7Þ

A vector e ¼ ðe1; . . . ; emÞ is said to be divisible by an integer n if every entry ofe is divisible by n. The lemma above can be rewritten as

Xt2T m

te ¼ ðq � 1Þsqm�s if ðq � 1Þ j e,0 otherwise.

�

DEFINITION. The map

* : ) 7! !TrK=Qp ð)Þ

is a non-trivial additive character that is trivial on P. Since Fq ffi OK=P, * can beviewed as a character on the additive group of Fq. If x 2 Fq and t is thecorresponding Teichm€uuller representative of x,

*ðxÞ ¼ !TrK=Qp ðtÞ ¼ CðtÞ; ð8Þwhere CðtÞ is de7ned in Lemma 1.

3.2. The bound

THEOREM 7. Let Fq be the �nite �eld of size pf and let F be an N-termpolynomial in m variables over Fq given by

F ðX1; . . . ; XmÞ ¼XNj¼1

ajXej ; where 0 6¼ aj 2 Fq: ð9Þ

The exponents ej are integer vectors with non-negative entries and

Xe ¼ Xe11 Xe22 . . .Xemm :

For a vector v, let suppðvÞ be the number of non-zero entries in v.

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CHEVALLEY--WARNING--AX--KATZ TYPE ESTIMATES 549

If SðF Þ is the exponential sum

SðF Þ ¼X

ðx1;...;xmÞ2Fqm*ðF ðx1; . . . ; xmÞÞ ð10Þ

then

ord"ðSðF ÞÞ>L;where

L ¼ min‘1;...;‘N

�XNj¼1

�pð‘jÞ þ fðp� 1Þðm� suppð‘1e1 þ . . .þ ‘NeNÞÞ�; ð11Þ

where the minimum is taken over all ðl1; . . . ; lNÞ 2 f0; 1; . . . ; q � 1gN such that q � 1divides ‘1e1 þ . . .þ ‘NeN .

Proof. Let aj 2 T be the Teichm€uuller representative of coeKcient aj 2 Fq. Theexponential sum SðF Þ is lifted to

SðF Þ ¼Xx

pYNj¼1

*ðajxejÞ ¼Xt

YNj¼1

CðajtejÞ:

The summation is over t 2 T m. Expanding the expression and rearranging SðF Þas a polynomial in the aj, we have

SðF Þ ¼Xt

YNj¼1

Xq�1

‘j¼0

cð‘jÞa‘jj t‘j ej

¼Xq�1

‘1¼0

. . .Xq�1

‘N¼0

Xt

YNj¼1

cð‘jÞ� YN

j¼1

a‘jj

�t‘1e1þ...þ‘NeN

¼Xq�1

‘1¼0

. . .Xq�1

‘N¼0

YNj¼1

cð‘jÞ� X

t

t‘1e1þ...þ‘NeN� YN

j¼1

a‘jj

�: ð12Þ

We can restrict the summation to those ð‘1; . . . ; ‘NÞ such that q � 1 divides vectorPj ‘j ej, since

Pt t‘1e1þ...þ‘NeN is zero otherwise, by Lemma 6.

Now 7x values of ‘1; ‘2; . . . ; ‘N such that ðq � 1Þ dividesPj ‘j ej. Let

s ¼ supp

�XNj¼1

‘j ej

�;

the number of entries that are positive multiples of q � 1. Then 06 s6m andXt

t‘1e1þ...þ‘NeN ¼ ðq � 1Þsqm�s ð13Þ

by Lemma 6. Its valuation with respect to " is

ord"

�Xt

t‘1e1þ...þ‘NeN�

¼ fðp� 1Þðm� sÞ: ð14Þ

On the other hand, we know the valuation of cð‘jÞ from Lemma 5,

ord"

�YNj¼1

cð‘jÞ�

¼XNj¼1

ord"ðcð‘jÞÞ ¼XNj¼1

�ð‘jÞ:

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Combining the two equations above shows that the coeKcient ofQNj¼1 a

‘jj in

equation (12) has valuation,

ord"

�YNj¼1

cð‘jÞYt

t‘1e1þ...þ‘NeN�

¼XNj¼1

�ð‘jÞ þ fðp� 1Þðm� sÞ: ð15Þ

The value of L de7ned in (11) is the minimum "-divisibility of the coeKcients ofa‘11 . . . a‘NN such that q � 1 divides

PNj¼1 ‘jej.

Since aj 6¼ 0, we have ord"ðajÞ ¼ 0. Applying the triangular inequality to SðF Þin equation (12) yields

ord"ðSðF ÞÞ> min‘1;...;‘N

ord"

�cð‘1Þ . . . cð‘NÞ

Xt

t‘1e1þ...þ‘NeN�

¼ L: �

�

Remark 2. Let ei ¼ ðei1; . . . ; eimÞ. The computation of L amounts to 7ndingthe solutions to the following system of modular equations:

‘1e11 þ ‘2e21 þ . . .þ ‘NeN1 � 0 mod q � 1;

..

.

‘1e1m þ ‘2e2m þ . . .þ ‘NeNm � 0 mod q � 1:

9>=>; ð16Þ

Usually, all variables X1; . . . ; Xm appear in the polynomial F , that is, thedegree in each Xi is positive. The statement in the main theorem can be simpli7edunder this assumption.

THEOREM 8. Under the conditions in Theorem 7 and the assumption that allvariables X1; . . . ; Xm appear in F , the "-divisibility of the exponential sum SðF Þis at least

L ¼ min‘1;...;‘N

f�pð‘1Þ þ . . .þ �pð‘NÞg; ð17Þ

where the minimum is taken over all the ‘j such that

q � 1 j ð‘1e1 þ . . .þ ‘NeNÞ and suppð‘1e1 þ . . .þ ‘NeNÞ ¼ m:

Proof. The theorem says that to compute the minimum L, it suKces to considerthe ‘j such that

PNj¼1 ‘j ej has full support. This is because if the ith entry of

‘1e1 þ . . .þ ‘NeN is zero as an integer, then by assumption we can 7nd a monomial,say the j0th monomial, such that ej0i is non-zero and hence ‘j0 must be zero. Bychanging lj0 by a suitable multiple of ðq � 1Þ and leaving the other lj 7xed we get anew combination l0, which retains the property that ðq � 1Þ divides

Pl0jej but for

which the support ofPlj ej is enlarged to contain j0. �

3.3. Showing the bound to be tight

We follow the approach in [3], to show that the bound in Theorems 7 and 8above are tight.

THEOREM 9. Let G be the class of polynomials generated by the monomialsXe1 ; . . . ;XeN :

G ¼ fa1Xe1 þ . . .þ aNXeN : ai 2 Fqg:

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CHEVALLEY--WARNING--AX--KATZ TYPE ESTIMATES 551

With L as in (11), there is at least one polynomial F in G such that SðF Þ is divisibleby "L but not divisible by "Lþ1.

Proof. Divide SðF Þ in (12) by "L. Then SðF Þ="L is in the valuation ring OKð!Þ.Reduction modulo the maximal ideal Q in OKð!Þ yields a polynomial in variablesa1 . . . aN :

SðF Þ"L

¼Xq�1

‘1¼0

. . .Xq�1

‘N¼0

1

"L

YNj¼1

cð‘jÞ� X

t

t‘1e1þ...þ‘NeN� YN

j¼1

a‘jj

�mod Q ð18Þ

with coeKcients in Fq. Since the minimum L is attained by some combination of

the ‘j, there is a term a‘11 . . . a‘NN whose coeKcient after division by "L is a unit inOKð!Þ. Therefore, there is at least one non-zero coeKcient.

The degree of each aj is less than or equal to q � 1 and is strictly less than thesize of Fq. By the next lemma there is some combination of the aj, not all zero,

such that SðF Þ="L is non-zero as an element in Fq. Thus SðF Þ="L 6� 0modQ. Itfollows that for some choice of aj, SðF Þ="L is not divisible by ", and hence SðF Þis not divisible by "Lþ1. �

We have used the following lemma, easily proved by induction on m.

LEMMA 10 [6]. Let F ðX1; . . . ; XmÞ be a polynomial over Fq of degree lessthan q in each variable. If F vanishes on all ðx1; . . . ; xmÞ 2 Fq

m, then it is thezero polynomial.

Remark 3. The theorem also applies to complex exponential sums. Let p beprime and " 2 Qð,Þ be a prime element lying above p, where , is a complex pthroot of unity. Let ordp and ord" be the valuations corresponding to Q and Qð,Þ(respectively). Then

ord"

�X,TrFq=Fp ðF ðx1;...;xmÞÞ

�>L;

where the sum is taken over ðx1; . . . ; xmÞ 2 Fqm and L is determined by (11).

Remark 4. The single-variable case of Theorem 8 reduces to McEliece’stheorem [10].

Example 1. Let F ðX; Y Þ ¼ aX4Y 3 þ bY 2 be a polynomial over F9,, ¼ 1

2 ð�1þffiffiffiffiffiffiffi�3

pÞ and " ¼ , � 1. Note that ð"Þ is a prime ideal in Z½,� lying

above the ideal (3).The exponential sum

SðF Þ ¼XX;Y 2F9

,TrðF ðX;Y ÞÞ

has valuation ord"ðSðF ÞÞ> 3. In fact, ‘1 ¼ 2 and ‘2 ¼ 1 achieve the minimumvalue of L in equation (17) (since 2ð4; 3Þ þ 1ð0; 2Þ ¼ ð8; 8Þ). The bound is tightbecause when a ¼ b are primitive elements in F9,

SðF Þ ¼ 9þ 24ffiffiffiffiffiffiffi�3

p¼ "3ð4�

ffiffiffiffiffiffiffi�3

pÞ

and ord"ðSðF ÞÞ ¼ 3.

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4. Approximations to the bound and applications

4.1. Properties of the p-weight function

Though the calculation of L in (11) is computational, it is possible to derivesome lower bounds which are still useful. For this, we need some properties of thep-weight function �p. We will replace �p by � when p is clear from the context.

PROPOSITION 11. For non-negative integers a, b, and �,(i) �pðapkÞ ¼ �pðaÞ, for k > 0,(ii) �pðaÞ þ �pðbÞ> �pðaþ bÞ,(iii) �pðaÞ�pðbÞ> �pðabÞ,(iv) if q is a power of p, and � is a positive multiple of q � 1, then �pð�Þ> �pðq � 1Þ.

Note that �pðq � 1Þ ¼ fðp� 1Þ ¼ ord"ðqÞ:

Proof. Part (i) is obvious. For (ii), we simply note that �ðaÞ is the sum of thedigits of a written in base p; and when we add a to b in base p arithmetic, the sumof the digits is reduced by p� 1 whenever there is a carry-over. This also tells usthat �ðaÞ�ðbÞ> �ðabÞ.

For part (iv), it is enough to show that if a > 1 then there is an a0 witha > a0 > 1 and �ðaðq � 1ÞÞ> �ða0ðq � 1ÞÞ. But we may write aðq � 1Þ ¼ bq þ c with06 c6 q � 1 and now �ðbq þ cÞ ¼ �ðbÞ þ �ðcÞ, so �ðaðq � 1ÞÞ ¼ �ðbÞ þ �ðcÞ >�ðbþ cÞ ¼ �ðða� bÞðq � 1ÞÞ with a > a� b> 1. �

4.2. Modular equation replaced by a non-modular condition

In this section, we assume that the polynomial F contains all m variables, that is,that the degree in each variable is strictly greater than zero. In order to approximatethe minimum "-divisibility L determined by Theorem 8, we apply the p-weightfunction �p to each equation in (16) and obtain the inequalities,

�pð‘1Þ�pðe11Þ þ �pð‘2Þ�pðe21Þ þ . . .þ �pð‘NÞ�pðeN1Þ> ðp� 1Þf;...

�pð‘1Þ�pðe1mÞ þ �pð‘2Þ�pðe2mÞ þ . . .þ �pð‘NÞ�pðeNmÞ> ðp� 1Þf:

9>=>; ð19Þ

If L0 is the minimum ofPj �pðljÞ subject to the above constraints, then L0 is a

lower bound to L. Note that the calculation of L0 does not involve modulararithmetic and is an integer programming optimization.

THEOREM 12. Let notation be as in Theorem 7, and assume that the degree ofthe polynomial F in each variable is greater than 0. Then, if L0 is the minimum ofPNj¼1 zj, taken over all the zj such that

�pðe11Þz1 þ �pðe21Þz2 þ . . .þ �pðeN1ÞzN > ðp� 1Þf;...

�pðe1mÞz1 þ �pðe2mÞz2 þ . . .þ �pðeNmÞzN > ðp� 1Þf;06 zj6 fðp� 1Þ; the zj are integers;

9>>>>=>>>>;

ð20Þ

we have ord"ðSðF ÞÞ>L0.

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CHEVALLEY--WARNING--AX--KATZ TYPE ESTIMATES 553

If we sum the m inequalities in (20), we have

wpðXe1Þz1 þ . . .þ wpðXeN ÞzN >mðp� 1Þf:

Suppose the p-weight of F is denoted by wpðF Þ ¼ l. We can replace all coeKcientsabove by l and recover the Moreno--Moreno bound given in [17, 19].

THEOREM 13. If F ¼ a1Xe1 þ . . .þ aNXeN is a polynomial in m variablesover Fp f , the exponential sum SðF Þ is divisible by "dmfðp�1Þ=le. In particular, if thecharacteristic is 2, SðF Þ is divisible by 2dmf=le.

4.3. Some conditions under which the exponential sum is divisible by q

COROLLARY 14. Let F ðXÞ be an m-variable polynomial over Fq. If the degreeof each monomial of F is a power of p, then the exponential sum SðF Þ de�nedin (10) is divisible by q.

Example 2. Let F ðX1; X2Þ ¼ )X71X2 þ *X1X

32 over F2f . By the Moreno--

Moreno theorem (Theorem 13), the 2-weight degree of F is 4 and ord2ðSðF ÞÞ> 12 f:

Using the Adolphson--Sperber result, we have !ðF Þ ¼ 25, because

14 ð7; 1Þ þ 3

4 ð1; 3Þ ¼ð 52 ; 52 Þ. Hence, ord2ðSðF ÞÞ> 2

5 f .Applying Corollary 14, we have ord2ðSðF ÞÞ> f .

COROLLARY 15. Let F ðXÞ be an m-variable homogeneous polynomial over Fq.If the degree of F is relatively prime to q � 1, then the exponential sum SðF Þ de�nedin (10) is divisible by q.

Example 3. Let l1; l2 2 N and ðl1 þ l2; q � 1Þ ¼ 1. Suppose that

F ðX1; . . . ; XmÞ ¼ )1Xl11 Xl22 þ )2Xl12 X

l23 þ . . .þ )m�1X

l1m�1X

l2m þ )mXl1mXl21

has coeKcients in Fq. Then q divides SðF Þ.

COROLLARY 16. Let F ðXÞ be an m-variable polynomial over Fq. Let d1; . . . ; dNbe the degree of each monomial of F , and let d be a common divisor of d1; . . . ; dN ,such that dj=d is a power of p for j ¼ 1; . . . ; N and such that ðd; q � 1Þ ¼ 1. Thenthe exponential sum SðF Þ in (10) is divisible by q.

In fact, Corollaries 14 and 15 are consequences of Corollary 16. We will onlyprove Corollary 16.

Proof. We want to 7nd a lower bound ofPjð�pð‘jÞÞ in (11). The ‘j satisfy a

system of modular equations (16). Summing the equations, we get

‘1d1 þ . . .þ ‘NdN � 0 mod q � 1:

Since the common divisor d is relatively prime to q � 1, we have

ðq � 1Þ�����‘1d1dþ . . .þ ‘N

dNd

�:

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Let dj=d ¼ pkj . ThenXNj¼1

�ð‘jÞ ¼XNj¼1

�ð‘jpkjÞ> ��X

j

‘jpkj

�> �ðq � 1Þ ¼ ðp� 1Þf: �

�In the system of equations (16), one can select a subset of k equations and 7nd theminimum subject to these k equations. This idea extends Corollary 16 as follows.

COROLLARY 17. Let F ðXÞ ¼Pi aiX

ei be an m-variable polynomial over Fq.Let G be a subset of variables, D ¼ fXj1 ; . . . ; Xjkg, and let di, for 16 i6N , bethe sum of the degrees of variables in G in the ith monomial in D. If d is a commondivisor of d1; . . . ; dN , di=d is a power of p, and ðd; q � 1Þ ¼ 1, then the exponentialsum SðF Þ is divisible by q.

Example 4. Let F ðX1; X2; X3Þ ¼ )X151 X

33 þ *X3

1X32 þ 2X3

2X33 be a poly-

nomial over F2f . The 2-weight degree of F is 6, and the Moreno--Moreno resultgives the estimate ord2ðSðF ÞÞ> 1

2 f . Using Adolphson--Sperber theory, we haveord2ðSðF ÞÞ> 7

18 f . Assume 3 does not divide 2f � 1, and use Corollary 17 withG ¼ fX2; X3g. Then d1, d2 and d3 are 3, 3 and 6, respectively, and d ¼ 3.Corollary 17 now shows that ord2ðSðF ÞÞ> f .

4.4. Other approximations to the divisibility

The next few theorems require the following de7nitions.

DEFINITION. The relative p-weight degree with respect to the variable Xi of themonomial Xe ¼ Xe11 . . .Xeii . . .Xemm is wp;XiðX

eÞ ¼ �pðeiÞ. The relative p-weightdegree with respect to the variable Xi of the polynomial F ðXÞ ¼

Pj ajX

ej (withaj 6¼ 0) is wp;XiðF Þ ¼ maxj wp;XiðX

ejÞ.

THEOREM 18. Let F ðXÞ ¼PNj¼1 ajX

ej be a polynomial in m variables,

0 6¼ aj 2 Fq, q ¼ pf and Xei ¼ Xei11 . . .Xeimm . Assume that all X1; . . . ; Xm appearin F ðXÞ. We have the following lower bound to the "-divisibility of the exponentialsum SðF Þ.

(i) Let t ¼ min16 i6m wp;XiðF Þ. Thenord"ðSðF ÞÞ> ðp� 1Þf=t:

(ii) Set

�ðajXejÞ ¼Xmi¼1

wp;XiðajXejÞ

wp;XiðF ðXÞÞ ;

and h ¼ max16 j6N �ðajXejÞ. Then,

ordpðSðF ÞÞ>ðp� 1Þfh

Xmj¼1

1

wp;XjðF ðXÞÞ :

Proof. We will apply Theorem 12 and 7nd lower bounds to the zj that satisfyinequalities in (20).

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Replace the coeKcients in the ith inequality by wp;XiðF Þ. Thenwp;XiðF Þz1 þ . . .þ wp;XiðF ÞzN > ðp� 1Þf;

z1 þ . . .þ zN >ðp� 1Þfwp;XiðF Þ

:ð21Þ

holds for all i ¼ 1; 2; . . . ;m. We get part (i) by taking the maximum lower bound.To prove part (ii), divide the ith inequality by wp;XiðF Þ. The system of

inequalities now reads

1

wp;X1ðF Þ ð�pðe11Þz1 þ �pðe21Þz2 þ . . .þ �pðeN1ÞzNÞ>

ðp� 1Þfwp;X1

ðF Þ ;

..

.

1

wp;XmðF Þð�pðe1mÞz1 þ �pðe2mÞz2 þ . . .þ �pðeNmÞzNÞ>

ðp� 1Þfwp;XmðF Þ

;

06 zj6 fðp� 1Þ; the zj are integers.

9>>>>>>>>>=>>>>>>>>>;

After summing the inequalities, the maximum of the coeKcients of the zj isequal to h. We get

Xmi¼1

wp;Xiða1Xe1Þ

wp;XiðF Þz1 þ . . .þ

Xmi¼1

wp;XiðaNXeN Þ

wp;XiðF ÞzN > ðp� 1Þf

Xmi¼1

1

wp;XiðF Þ;

z1 þ . . .þ zN >ðp� 1Þfh

Xmi¼1

1

wp;XiðF Þ: �

�

Example 5. Let

F ðX1;X2; X3Þ ¼ )1X151 X

152 þ )2X3

3X1 þ )3X53X1 þ gðX1;X2; X3Þ

be a polynomial over F2f , where gðX1; X2; X3Þ satis7es w2;XiðgÞ6 3 for i ¼ 1; 2and w2;X3

ðgÞ6 2. The Moreno--Moreno result implies that ord2ðSðF ÞÞ> 38 f ,

and the Adolphson--Sperber result implies that ord2ðSðF ÞÞ> 38 f. Using

Theorem 18(i), we have ord2ðSðF ÞÞ> 12 f since t ¼ 2.

Example 6. Let F ðX1; X2; X3Þ ¼ X1X2X3 þX151 þX7

2 þX73 over F2f . Then

ord2ðSðF ÞÞ> fXmi¼1

1

w2;XiðF Þ¼ 11

12 f

from Theorem 18(ii). The Moreno--Moreno result (Theorem 13) implies thatord2ðSðF ÞÞ> 3

4 f .

Now we have another application of Theorem 8. The case for one variable istreated in [18] (it is a generalization of Theorem 3 in [18]).

COROLLARY 19. Let F ðX1; . . . ; XmÞ be a polynomial over Fq, with q ¼ pf .Suppose that s < f , s j f , and F is of the form

F ðX1; . . . ; XmÞ ¼ aXps�1

1 . . .Xps�1m þ gðX1; . . . ; XmÞ;

and wp;XiðgÞ6 �pðps � 1Þ for all i. Then,

ord"ðSðF ÞÞ> f=s:

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Moreover, if wpðgðX1; . . . ; XmÞÞ < wpððX1 . . .XmÞps�1Þ, then

ord"ðSðF ÞÞ ¼ f=s if TrFps =Fpðaðp f�1Þ=ðps�1ÞÞ 6¼ 0

and

ord"ðSðF ÞÞ > f=s if TrFps =Fpðaðp f�1Þ=ðps�1ÞÞ ¼ 0:

Proof. We want to show that L0 in Theorem 12 is equal to f=s. Replacing allthe coeKcients in (20) by the upper bound sðp� 1Þ, we have

sðp� 1Þz1 þ sðp� 1Þz2 þ . . .þ sðp� 1ÞzN > ðp� 1Þf;z1 þ . . .þ zN > f=s:

Therefore, L0> f=s. But ðz1; z2; . . . ; zNÞ ¼ ðf=s; 0; 0; . . . ; 0Þ achieves the lower

bound and thus equality holds.For the second part of the statement, we will 7nd all solutions to (16) that meet

the minimum L ¼ f=s. Let gðXÞ ¼ a2Xe2 þ . . .þ aNXeN . Suppose that ‘1; . . . ; ‘Nsatisfy (16), and �pð‘1Þ þ . . .þ �pð‘NÞ ¼ f=s. Summing the inequalities in (19)we get

wpðaXps�1

1 . . .Xps�1m Þ�pð‘1Þ þ wpða2Xe2Þ�pð‘2Þ þ . . .þ wpðaNXeN Þ�pð‘NÞ

>mfðp� 1Þ:

Suppose that

0 < wpðajXejÞ < wpðaXps�1

1 . . .Xps�1m Þ:

If ‘j > 0 for any j ¼ 2; 3; . . . ; N , we get the strict inequality

msðp� 1Þð�pð‘1Þ þ . . .þ �pð‘NÞÞ > mfðp� 1Þ:

This contradicts the minimality ofPj �pðjÞ. Therefore ‘2 ¼ . . . ¼ ‘N ¼ 0. The

modular equations (16) now reduce to

ðps � 1Þ‘1 � 0 mod q � 1;

that is, ‘1 is equal to kðpf � 1Þ=ðps � 1Þ for some integer k, with 0 < k < ps.When we add kþ kps þ kp2s þ . . ., there is no carry-over. The additive property of �p(see Proposition 11) now shows that

�pðkðpf � 1Þ=ðps � 1ÞÞ ¼ �pðkþ kps þ . . .þ kpðf=s�1ÞsÞ ¼ fs�pðkÞ:

But we assume that it achieves the minimum value and is equal to f=s. This impliesthat �pðkÞ ¼ 1 and k must be a power of p. The possible values of k are 1; p; . . . ; ps�1.

There are exactly s non-zero terms in equation (18). They have the samecoeKcient, by Remark 1. Let Q denote the unique maximal ideal in OKð!Þ. NowSðF Þ="f=s is zero modulo Q if and only if

aðpf�1Þ=ðps�1Þ þ apðp

f�1Þ=ðps�1Þ þ . . .þ aps�1ðp f�1Þ=ð ps�1Þ 6¼ 0:

The left side of the above equation is precisely equal to TrFps =Fpðaðp f�1Þ=ðps�1ÞÞ. �

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Example 7. Let f ¼ 4, s ¼ 2. Then

SðX8Y 8 þXY þX þ Y Þ ¼Xx;y2F

34

eð2"i=3ÞTrðx8y8þxyþxþyÞ

¼ 267

2þ 117

ffiffiffiffiffiffiffi�3

p

2¼ 3

�89

2þ 39

ffiffiffiffiffiffiffi�3

p

2

�:

4.5. Approximations based on a ‘packing’ technique

There is another technique called the packing method, introduced in [16]. Thepacking method is a combinatorial technique to solve modular equations of the typein (16). In [16], the following theorem was proved using the packing technique.

THEOREM 20. Let F2f be a �nite �eld with 2f elements. Let

F ðXÞ ¼ )1X23 þ )2Xe2 þ . . .þ )rXer

over F2f . We assume that, for 26 j6 r, the lj are strictly less than 23, but notequal to 13 or 15, and the binary weights of the ej are at most 3. Then

ord2ðSðF ÞÞ ¼Xx2F

2f

ð�1ÞTrð)1X23þ)2Xe2þ...þ)rXer Þ

>13 f:

Proof. In [16], Moreno et al. proved that

�2ð‘1Þ þ . . .þ �2ð‘rÞ> 13 f

if 23‘1 þ e2‘2 þ . . .þ er‘r � 0 mod 2f � 1. Now, we combine this with Theorem 8to obtain

ord2ðSðF ÞÞ ¼XX2F

2f

ð�1ÞTrð)1X23þ)2Xl2þ...þ)mXlm Þ

>13 f: �

�

A generalization is the following corollary.

COROLLARY 21. Let

F ðX1; . . . ; XmÞ ¼ )X231 . . .X23

m þ *X1 . . .Xm þ gðX1; . . . ; XmÞover F2f , where the degree of the variable Xi of g is less than or equal to 23for i ¼ 1; . . . ;m, and 15 and 13 do not appear in any exponent of the variables ofg. Then,

ord2

� XX1;...;Xm2F2f

ð�1ÞTrðF ðX1;...;XmÞÞ�>

13 f:

The above theorem improves the Moreno--Moreno and Adolphson--Sperbertheorems for this family of polynomials.

Example 8. Let

F ðX; Y Þ ¼ )X23Y 23 þ *X7Y 7 þ 5XYover F23f . Using the packing method, we have �2ð‘1Þ þ �2ð‘2Þ þ �2ð‘3Þ> f .Therefore, 2f divides SðF Þ ¼

PX;Y 2F

23fð�1ÞTrðF ðX;Y ÞÞ. Using the Moreno--Moreno

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result, we get ord2ðSðF ÞÞ> 34 f . Using the Adolphson--Sperber result, we get

ord2ðSðF ÞÞ> 331 f .

5. Estimating the number of zeros using the exponential sum bound

5.1. Estimation based on the bound

Let F be a polynomial in m variables as given in (9). The number of zeros of Fin Fq

m is denoted by NðF Þ. We begin with the following theorem.

THEOREM 22. Suppose F is a non-constant polynomial in m variables and Fdepends on all X1; . . . ; Xm:

F ðXÞ ¼XNi¼1

aiXei :

Set

L ¼ min‘1;...;‘N

�XNj¼1

�ð‘jÞp� 1

�: ð22Þ

The minimum is taken over all ð‘1; . . . ; ‘NÞ 2 f0; 1; . . . ; q � 1gN , subject to theconditions that both

XNj¼1

‘j and ‘1e1 þ . . .þ ‘NeN

are positive multiples of q � 1, and suppð‘1e1 þ . . .þ ‘NeNÞ ¼ m. Then, thenumber of zeros is divisible by pL�f .

Proof. The number of zeros of F ðXÞ ¼ 0, denoted by NðF Þ, can bedetermined by

qNðF Þ ¼X

ðx0;...;xmÞ2Fqmþ1

*ðx0F ðx1; . . . ; xmÞÞ;

where * is an additive character de7ned in x 2. We augment every vector ej to e0jby pre-appending a 1 in front of ej

ej ¼ ðej1; . . . ; ejmÞ �! ð1; ej1; . . . ; ejmÞ ¼ e0j:

It follows from Theorem 8 that ordpðqNðF ÞÞ is at least L. Hence NðF Þ isdivisible by pL�f . �

THEOREM 23. Let G be the class of polynomials generated by the monomialsXe1 ; . . . ;XeN :

G ¼ fa1Xe1 þ . . .þ aNXeN : ai 2 Fqg:

With L given in (22), there is at least one polynomial F in G such that NðF Þ isdivisible by pL�f but not divisible by pLþ1�f .

The proof is similar to the proof of Theorem 9.

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For zeros of several polynomials, the method can be extended as follows. LetF1; . . . ; Fr be polynomials in m variables over Fq:

Fk ¼XNki¼1

akiXeki ;

where aki may be equal to zero. Let NðF1; . . . ; FrÞ be the number of common zerosto the r polynomials. Introduce r auxiliary variables Y1; . . . ; Yr:

qrNðF1; . . . ; FrÞ ¼X

ðx1;...;xmÞ2Fqm

� Xy12Fq

*ðy1F1ðx1; . . . ; xmÞÞ�

. . .

� Xyr2Fq

*ðyrFrðx1; . . . ; xmÞÞ�

¼Xx

Xy

*ðy1F1ðxÞ þ . . .þ yrFrðxÞÞ:

The 7rst sum is over all x 2 Fqm and the second is over y 2 Fq

r.

LEMMA 24. Assume that the polynomials depend on all variables. ThenNðF1; . . . ; FrÞ is divisible by p�, with � ¼ L� rf , where

L ¼ min

�Xrk¼1

XNj¼1

�ð‘kjÞ=ðp� 1Þ�: ð23Þ

The minimum can be taken over all ‘11; . . . ; ‘rN , satisfying the condition that

‘11 þ ‘12 þ . . .þ ‘1N1;

‘21 þ ‘22 þ . . .þ ‘2N2;

..

.

‘r1 þ ‘r2 þ . . .þ ‘rNr ;‘11e11 þ ‘12e12 þ . . .þ ‘1N1

e1N1;

‘21e21 þ ‘22e22 þ . . .þ ‘2N2e2N2

;

..

.

‘r1er1 þ ‘r2er2 þ . . .þ ‘rNrerNrare all positive multiples of q � 1 and have full support.

Remark 5. The above lemma states that to compute the minimum L, it suKcesto consider the ‘kj such that

PNrj¼1 ‘kjekj (with k ¼ 1; . . . ; r) has full support. The

proof is similar to that of Theorem 8.

Remark 6. Proposition 11 and its proof show that L is an integer and is greaterthan or equal to rf .

Now we are ready to give our main theorem of this section.

THEOREM 25. Let G be the following class of polynomials:

G ¼ fa11Xe11 þ . . .þ a1N1Xe1N1 ; . . . ; ar1X

er1 þ . . . ; arNrXerNr : aij 2 Fqg:

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With L given in (23), there are polynomials F1; . . . ; Fr in G, such thatNðF1; . . . ; FrÞ is divisible by pL�fr but not divisible by pLþ1�fr.

5.2. Applications of our main theorem

In this section we will apply Theorem 25 and 7nd general classes of exampleswhere the Moreno--Moreno bound is tight. The method that we introduce in thissection is quite general and might be used for other bounds such as Ax--Katz andAdolphson--Sperber. Our method is as follows.

(1) First we have a family of polynomials (for all possible values of the coeKcientsas in Theorem 25) and must have a bound for the divisibility of the family. In thecase presented below we use the Moreno--Moreno bound (see the introduction).

(2) Next we 7nd a solution of our modular system, with minimum weight L (seeLemma 24).

(3) Finally we invoke our Theorem 25 and conclude that for some particularvalue of the coeKcients our bound of (1) is tight.

Now we apply the above method obtaining the following corollaries.

COROLLARY 26. Let

F1ðX1; . . . ; XmÞ ¼ a11X2i1�11 þ . . .þ a1mX2i1�1

m ¼ 0;

..

. ...

FrðX1; . . . ; XmÞ ¼ ar1X2ir�11 þ . . .þ armX2ir�1

m ¼ 0;

ð24Þ

over F2f , where ij j f , i1 6 i2 6 . . . 6 ir and m ¼Prj¼1 ij þ kir. Then there are

a11; . . . ; arm 2 F2f such that

ord2ðNðF1; . . . ; FrÞÞ ¼ðm�

Prj¼1 ijÞfir

:

In other words, there is at least one instance of the coe"cients where our system(24) is tight.

Proof. We will discuss the case when k ¼ 0 and at the end we will brieNyindicate how to handle the general case (for k 6¼ 0).

For point (1) we use the system (24). Note that the system

ð2i1�1 � 1Þj11 þ ð2i2�1 � 1Þj21 þ . . .þ ð2ir�1 � 1Þjr1 � 0 mod 2f � 1;

..

.

ð2i1�1 � 1Þj1m þ ð2i2�1 � 1Þj2m þ . . .þ ð2ir�1 � 1Þjrm � 0 mod 2f � 1;

ð25Þ

j11 þ j12 þ . . .þ j1m � 0 mod 2f � 1;

..

.

jr1 þ jr2 þ . . .þ jrm � 0 mod 2f � 1;

is the modular system corresponding to (24) (see Lemma 24). In point (2) we need asolution to the modular system (25) that has minimal weight. The vector with entries

vts ¼ 2t�1 2f � 1

2is � 1; for t ¼ 1; . . . ; is; s ¼ 1; . . . ; r; ð26Þ

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is a solution of the modular system (25) and has minimal weight. We can concludethat a solution of the above system that has minimal weight has weightL ¼

Pms¼1 isðf=isÞ ¼ rf .

Now we use point (3), and conclude that there are coeKcients ats in F2f suchthat ord2ðNðF1; . . . ; FrÞÞ ¼ rf � rf ¼ 0:

For the general case k 6¼ 0, we repeat k times the last ir entries of the solutionfound in the case k ¼ 0 (see (26)), that is,

ðv11; . . . ; vi11; 0; . . . ; 0; v12; . . . ; vi22; . . . ; v1m; . . . ; virmÞ: �

�It is possible to 7nd the coeKcients described in the corollary above (see [18]).

5.3. Estimation based on approximations to the exponential sum bound

The number of variables of a monomial aXe ¼ aXe11 . . .Xemm is 2ðaXeÞ ¼Pmi¼1 bi,

where bi ¼ 1 if ei > 0, and 0 otherwise. The maximum number of variables in amonomial for a polynomial F ðXÞ is 2ðF Þ ¼ maxj 2ðajXejÞ.

DEFINITION. De7ne the local p-weight degree of a set of polynomials to be

wp;XiðF1; . . . ; FrÞ ¼ maxj

fwp;XiðFjÞg:

Similar to Theorem (18)(ii), we have the following.

THEOREM 27. Let F1ðX1; . . . ; XmÞ; . . . ; FrðX1; . . . ; XmÞ, be r m-variable poly-

nomials over Fq, with q ¼ pf . Set2ðF1; . . . ; FrÞ ¼ max

16 i6 r2ðFiÞ:

Then,

ordpðNðF1; . . . ; FrÞÞ

>

�1

wp;X1ðF1; . . . ; FrÞ

þ . . .þ 1

wp;XmðF1; . . . ; FrÞ� r2ðF1; . . . ; FrÞ

�f

2ðF1; . . . ; FrÞ:

Example 9. Let FiðX1; . . . ; XmÞ ¼ Xai1 X

bi2 þ giðX1; . . . ; XmÞ for i ¼ 1; . . . ; r

be polynomials over F2f , where �2ðaiÞ ¼ �2ðbiÞ ¼ 3, 2ðgiÞ6 2, and w2;XiðgiÞ6 2,for i ¼ 1; . . . ; r. Using the Moreno--Moreno theorem, we get

ord2ðNðF1; . . . ; FrÞÞ> 16 ðm� 6rÞf:

The above theorem implies that

ord2ðNðF1; . . . ; FrÞÞ> ð 14 ðm� 4r� 2Þ þ 13 Þf:

Note that 14 ðm� 4r� 2Þ þ 1

3 >16 ðm� 6rÞ for every r and m. We get information

about NðF1; . . . ; FrÞ when m> 4rþ 1.

THEOREM 28. Let F1ðX1; . . . ; XmÞ; . . . ; FrðX1; . . . ; XmÞ be polynomials overFq with pf elements. For each monomial aXe11 . . .Xemm in F , let

�ðaXe11 . . .Xemm ; F Þ ¼Xmi¼1

wp;XiðaXe11 . . .Xemm Þ

wp;XiðF Þ:

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Then,

ordpðNðF1; . . . ; FrÞÞ>f

h

Xmi¼1

1

wp;XiðF1; . . . ; FrÞ� fr;

where h ¼ maxf�ðaX; FiÞ : aX is a monomial of Fi; i ¼ 1; . . . ; rg.

Proof. The proof is similar to the proof of Theorem 18. �

Example 10. Let

F1ðX1; X2; X3; X4Þ ¼ *1X71 þ )1X3

2 þ 21X33 þ 51X4

and

F2ðX1; X2; X3; X4Þ ¼ *2X1 þ )2X2 þ 22X3 þ 52X4

over Fq, with 2f elements. Then

ord2ðNðF1; F2ÞÞ> 13 f;

sinceP4i¼1 1=w2;XiðF1; F2Þ ¼ 1

3 þ 12 þ 1

2 þ 1 ¼ 73 and h ¼ 1. Note that the system of

modular equations associated to y1F1 þ y2F2 has six equations, but we only considerthe four equations associated to the variables X1; . . . ; X4.

Now we give an application of Theorem 12 to the number of zeros of F .

Example 11. Let a; b; c; d; e 2 F2f . Then,

F ðXÞ ¼ a1Xi1X2s1þ12 þ a2Xk1X2s2þ1

3 þ a3X2 þ a4X3 þ a5Xt1 ¼ 0

has non-trivial solutions whenever f is odd. Indeed, the modular system ofequations associated to NðF Þ is

iz1 þ kz2 þ tz5 � 0 mod q � 1;

ð2s1 þ 1Þz1 þ z3 � 0 mod q � 1;

ð2s2 þ 1Þz2 þ z4 � 0 mod q � 1;

z1 þ z2 þ z3 þ z4 þ z5 � 0 mod q � 1:

9>>>=>>>;

We only consider the second and third equations of the above modular system ofequations. Applying �2 to the two equations, we get

�2ðz1Þ þ �2ðz2Þ þ �2ðz3Þ þ �2ðz4Þ> f þ 1;

since 2 does not divide f.Therefore, ord2ðNðF ÞÞ > 0 and F ðXÞ ¼ 0 has non-trivial solutions. To apply

Moreno and Moreno’s improvement to Chevalley and Warning’s result would haverequired more variables than maxf�2ðjÞ þ 2; �2ðkÞ þ 2; �2ðtÞg, which in fact can bearbitrarily large.

Acknowledgement. We thank the referee for valuable suggestions.

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Oscar Moreno and Francis N. CastroDepartment of Mathematics and

Computer ScienceUniversity of Puerto RicoPO Box 23355San JuanPR 00931-3355Puerto Rico

o moreno@upr1.clu.edufcastro@goliath.cnnet.clu.edu

Kenneth W. Shumand P. Vijay Kumar

Communications Sciences InstituteEE-SystemsUniversity of Southern CaliforniaLos AngelesCA 90089-0272USA

kshum2003@yahoo.com.hkvijayk@usc.edu

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