A NEW PROOF OF WARNING’S SECOND THEOREM
SHAMIL ASGARLI
Abstract. We give an elementary proof of Warning’s second theorem on the number of solutions
to the system of polynomial equations over finite fields.
1. Introduction.
A basic result in linear algebra states that if a system of r linear equations in n variables has a
solution, and n > r, then the system has another solution. It is natural to ask if this statement
extends to a higher degree system of polynomials. The following result was conjectured by Emil
Artin and was first proved by Chevalley [3].
Theorem 0. Let q = pk be a prime power. Suppose that f1 , . . . , fr ∈ Fq [x1 , . . . , xn ] are polynomials
such that
r
X
n>
deg(fi ).
i=1
Let Z = {a ∈ Fnq : fi (a) = 0 for each i} to be the common zero locus. If Z 6= ∅, then #Z ≥ 2.
Around the same time, Warning [9] strengthened Chevalley’s theorem in two directions. Using the
same hypothesis as above, we can state them as follows:
Theorem 1 (Warning’s first theorem). p | #Z.
Theorem 2 (Warning’s second theorem). If Z 6= ∅, then #Z ≥ q n−d , where d =
Pr
i=1 deg(fi ).
The analogue of this in linear algebra (when deg fi = 1 for all i) is well known: if a system
of r linear equations in n variables has a solution with n > r, then the solution set is at least
(n − r)-dimensional.
There are stronger results regarding divisibility of the number of solutions, originating
Pin the work
of Ax [2]. Under the same hypothesis above, Ax proved that q b | #Z whenever n > b ri=1 deg(fi ).
In particular, we get that q | #Z in all cases, which is a deeper result than Theorem 1. This
theorem was later strengthened by Katz [7].
It is also worth mentioning the rich combinatorial aspects of these results. First, Chevalley’s
theorem has a proof using Alon’s combinatorial nullstellensatz [1]. By refining this method, Clark
[4] proves a restricted-variable generalization of Theorem 1. In the same spirit, Theorem 2 has been
vastly generalized in [5].
The main goal of this note is to give an elementary proof of Theorem 2 using a counting argument.
The key ingredient is a certain incidence correspondence between the points and hyperplanes in
Fnq . We should mention that Warning’s original proof used a different counting argument, which
was recently refined by Heath-Brown [6]. In fact, Warning’s proof of Theorem 2 worked over Fq
directly, while ours works only over Fp (see Section 3). In Section 4, we explain how to get the
statement over Fq using restriction of scalars.
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2. Warning’s first theorem.
In our proof of Theorem 2 (see Section 3), we will need Theorem 1 as the base case of the induction.
To make the exposition self-contained, we include the proof of Theorem 1 below. This proof is
adapted from the one in [2].
Proof of Theorem 1. Let f1 , f2 , . . . , fr ∈ Fq [x1 , . . . , xn ] be polynomials satisfying
For any a ∈ Fnq , we have
(
1 if fi (a) 6= 0,
fi (a)q−1 =
0 if fi (a) = 0.
Pr
i=1 deg(fi )
< n.
As a result, the quantity
N (f1 , ..., fr ) =
r
XY
(1 − fi (a)q−1 )
a∈Fn
q i=1
Q
counts the cardinality of Z = {a ∈ Fnq : fi (a) = 0 for each i} modulo p. The polynomial ri=1 (1 −
u1 u2
q−1
u
un
fi (x)
Pr ) is a linear combination of monomials x =u x1 x2 · · · xn with degree at most (q −
1) i=1 deg(fi ) < n(q − 1). For each such monomial x ,
X
a∈Fn
q
au =
n X
Y
i=1 ai ∈Fq
aui i =
n
Y
Y (ui ),
i=1
P
where Y (ui ) := ai ∈Fq aui i . If ui is a positive multiple of q − 1, then Y (ui ) = q − 1. Otherwise, we
can find some b ∈ F∗q with bui 6= 1. Then
X u
X
X u
Y (ui ) =
ai i =
(bai )ui = bui
ai i = bui Y (ui )
ai ∈Fq
ai ∈Fq
ai ∈Fq
so that Y (ui ) = 0. Consequently,
(
q − 1 if ui is a positive multiple of q − 1,
Y (ui ) =
0
otherwise.
P
Pr
Since (q − 1) i=1 deg(fi ) < n(q − 1), at least one ui is smaller than q − 1. Therefore, a∈Fnq au = 0
and so N (f1 , . . . , fr ) = 0, implying that p | #Z.
3. Warning’s second theorem (special case).
The goal of this section is to prove Warning’s second theorem in the special case when q = p is a
prime number.
P
Proof of Theorem 2 (special case q = p). Let m = n − ri=1 deg(fr ). By hypothesis, m > 0. We
want to show that if a system of equations f1 = f2 = · · · = fr = 0 has a solution, then it has at
least pm solutions. We will proceed by induction on m. After translating the variables, we may
assume that f1 (0) = · · · = fn (0) = 0, because the degrees of fi and the size of the zero locus of
{fi }ri=1 do not change after a linear change of variables.
Base case m = 1. By Theorem 1, p | #Z. Since 0 ∈ Z, we get #Z ≥ p.
Inductive step. Assume m ≥ 2 and suppose the result is true for m − 1.
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Before we delve into the proof, we need a few preliminaries. A hyperplane H is a codimension
one Fp -subspace of Fnp . Each hyperplane H is the zero locus of some nonzero homogeneous linear
n −1
hyperplanes in
polynomial LH . Note that LH is unique up to scaling. As a result, there are pp−1
n
Fp . For a given hyperplane H, let
ZH = {a ∈ Fnp | f1 (a) = · · · = fr (a) = LH (a) = 0}.
For the new system f1 , ..., fr , LH , we have
n−
r
X
deg(fr ) − 1 = m − 1.
i=1
The induction hypothesis implies that #ZH ≥ pm−1 .
Let S = Z \ {(0, ..., 0)}. If #S ≥ pm − p, then #Z > pm − p, which combined with p | #Z, implies
the desired result #Z ≥ pm . Assume, to the contrary, that #S < pm − p. In fact, we will assume
that #S ≤ pm − p. Consider the incidence set
E = {(x, H) | x ∈ S and H hyperplane with x ∈ H}.
Let’s count E in two different ways. For a fixed x ∈ S, the number of (x, H) ∈ E can be at most
the number of hyperplanes H containing x. The latter is the same as the number of hyperplanes in
n−1
the quotient space Fpn /hxi which has dimension n − 1. So each x ∈ S contributes at most p p−1−1
elements to E. Since #S ≤ pm − p, we obtain
(1)
#E ≤ (pm − p)
pn−1 − 1
.
p−1
On the other hand, as we saw above, each hyperplane H satisfies #ZH ≥ pm−1 , and so it contributes
at least pm−1 − 1 elements to E. This gives us
(2)
#E ≥ (pm−1 − 1)
pn − 1
.
p−1
Combining the inequalities (1) and (2), we get
(pm − p)
pn−1 − 1
pn − 1
≥ (pm−1 − 1)
.
p−1
p−1
After dividing both sides by pm−1 − 1 (which is allowed since m ≥ 2), this becomes
p(pn−1 − 1)
pn − 1
≥
,
p−1
p−1
that is, pn − p ≥ pn − 1, which is a contradiction.
4. Warning’s second theorem (general case).
It turns out that Warning’s second theorem over Fp implies Warning’s second theorem over Fq .
This type of reduction was used in [8] in a more general setting where the degrees are replaced by
their p-weights. In our situation, the proof is an example of a “restriction of scalars” argument.
We are grateful to the referee for explaining this step.
P
Proof of Theorem 2 (general case). Suppose f1 , . . . , fr ∈ Fq [x1 , . . . , xn ] and n > ri=1 deg(fi ). We
can view Fq = Fpk as a k-dimensional vector space over Fp . Let α1 , α2 , . . . , αk be a basis for this
3
P
vector space. Next, formally replace each xi with kj=1 αj xij . If f ∈ Fq [x1 , . . . , xn ], then f = 0
becomes


k
k
k
X
X
X
f
αj x1j ,
αj x2j , . . . ,
αj xnj  = 0.
j=1
j=1
j=1
After expanding the polynomial and collecting the coefficients, this equation can be written as
α1 g1 ({xij }) + · · · + αk gk ({xij }) = 0, where each g` ({xij }) is a polynomial in variables xij with
coefficients in Fp . As a result, {a ∈ Fnq : f (a) = 0} is in bijection with the set {b ∈ Fkn
p : gj (b) =
0 for each j = 1, ..., k}. By the same reasoning,
Z = {a ∈ Fnq : fi (a) = 0 for each i = 1, . . . , r}
is in bijection with
(i)
Z 0 = {b ∈ Fkn
p : gj (b) = 0 for each j = 1, . . . , k and for each i = 1, . . . , r},
(i)
j=1,...,k
where {gj }i=1,...,r
is a collection of kr polynomials in kn variables over Fp . Note that
r X
k
X
i=1 j=1
where d =
Pr
i=1 deg fi .
(i)
deg gj =
r X
k
X
deg fi = kd,
i=1 j=1
By Theorem 2 (over Fp ), #Z 0 is at least pkn−kd = q n−d .
Acknowledgment. I would like to thank my advisor Brendan Hassett for reading and checking
the proof. I also owe thanks to Peter McGrath, who first encouraged me to publish this note.
Finally, I am grateful to the two anonymous referees for providing me with such a detailed and
insightful report.
References
[1] N. Alon, Combinatorial Nullstellensatz, Combin. Probab. Comput., 8 (1999) 7–29.
[2] J. Ax, Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964) 255-261.
[3] C. Chevalley, Démonstration d’une hypothèse de M. Artin, Abh. Math. Sem. Univ. Hamburg 11 no. 1 (1935)
73–75.
[4] P. L. Clark, The Combinatorial Nullstellensätze revisited, Electron. J. Combin. 21 no. 4 (2014) #P4.15. http:
//www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p15.
[5] P. L. Clark, A. Forrow, J. R. Schmitt, Warning’s second theorem with restricted variables, Combinatorica (2016),
1–21, http://dx.doi.org/10.1007/s00493-015-3267-8.
[6] D. R. Heath-Brown, On Chevalley-Warning theorems Uspekhi Mat. Nauk 66 (2011) 223–232; translation in:
Russian Math. Surveys 66 (2011) 427–436.
[7] N. M. Katz, On a theorem of Ax, Amer. J. Math. 93 (1971) 485–499.
[8] O. Moreno, C. J. Moreno, Improvements of the Chevalley-Warning and the Ax-Katz theorems, Amer. J. Math.
117 (1995) 241–244.
[9] E. Warning, Bemerkung zur vorstehenden Arbeit von Herrn Chevalley, Abh. Math. Sem. Univ. Hamburg 11 no.
1 (1935) 76–83.
Shamil Asgarli (shamil [email protected]) is a math PhD student at Brown University.
Box 1917, Department of Mathematics, Brown University, Providence RI 02912
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