Wednesday, March 22, 2017

Warning! — Theorems ahead

Don't worry, no danger ahead! — This is just a short post about the German mathematician Ewald Warning and the theorems that bear his name.

It seems that Ewald Warning's name will be forever linked with that of Chevalley, for the Chevalley-Warning theorem is one of the rare modern results that can be taught to undergraduate students; in France, it is especially famous at the Agrégation level. (Warning published a second paper, in 1959, about the axioms of plane geometry.)

Warning's paper, Bemerkung zur vorstehenden Arbeit von Herrn Chevalley (About a previous work of Mr Chevalley), has been published in 1935 in the Publications of the mathematical seminar of Hamburg University (Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg), just after the mentioned paper of Chevalley. Emil Artin had a position in Hamburg at that time, which probably made the seminar very attractive; as a matter of fact, the same 1935 volume features a paper of Weil about Riemann-Roch, one of Burau about braids, one of Élie Cartan about homogeneous spaces, one of Santalo on geometric measure theory, etc.

1. The classic statement of the Chevalley—Warning theorem is the following.

Theorem 1. – Let pp be a prime number, let qq be a power of pp and let FF be a field with qq elements. Let f1,,fmf_1,\dots,f_m be polynomials in nn variables and coefficients in FF, of degrees d1,,dmd_1,\dots,d_m; let d=d1++dmd=d_1+\dots+d_m. Let Z=Z(f1,,fm)Z=Z(f_1,\dots,f_m) be their zero-set in FnF^n. If d<nd<n, then pp divides Card(Z)\mathop{\rm Card}(Z).

This is really a theorem of Warning, and Chevalley's theorem was the weaker consequence that if ZZ\neq\emptyset, then ZZ contains at least two points. (In fact, Chevalley only considers the case q=pq=p, but his proof extends readily.) The motivation of Chevalley lied in the possibility to apply this remark to the reduced norm of a possibly noncommutative finite field (a polynomial of degree dd in d2d^2 variables which vanishes exactly at the origin), thus providing a proof of Wedderburn's theorem.

a) Chevalley's proof begins with a remark. For any polynomial fF[T1,,Tn]f\in F[T_1,\ldots,T_n], let ff^* be the polynomial obtained by replacing iteratively XiqX_i^q by XiX_i in ff, until the degree of ff in each variable is <q<q. For all aFna\in F^n, one has f(a)=f(a)f(a)=f^*(a); moreover, using the fact that a polynomial in one variable of degree <q<q has at most qq roots, one proves that if f(a)=0f(a)=0 for all aFna\in F^n, then f=0f^*=0.

Assume now that ZZ contains exactly one point, say aFna\in F^n, let f=(1fjq1)f=\prod (1-f_j^{q-1}), let ga=(1(xiai)q1)g_a=\prod (1-(x_i-a_i)^{q-1}). Both polynomials take the value 11 at x=ax=a, and 00 elsewhere; moreover, gag_a is reduced. Consequently, f=gf^*=g. Then
(q1)n=deg(ga)=deg(f)deg(f)=(q1)deg(fj)=(q1)d, (q-1)n=\deg(g_a)=\deg(f^*)\leq \deg(f)=(q-1)\sum \deg(f_j)=(q-1)d,
contradicting the hypothesis that d<nd<n.

b) Warning's proof is genuinely different. He first defines, for any subset AA of FnF^n a reduced polynomial gA=aAga=aA(1(xiai)q1)g_A=\sum_{a\in A} g_a=\sum_{a\in A}\prod (1-(x_i-a_i)^{q-1}), and observes that gA(a)=1g_A(a)=1 if aAa\in A, and gA(a)=0g_A(a)=0 otherwise.
Take A=ZA=Z, so that f=gZf^*=g_Z. Using that deg(f)deg(f)=(q1)d\deg(f^*)\leq \deg(f)=(q-1)d and the expansion
(xa)q1=i=0q1xiaq1i (x-a)^{q-1} = \sum_{i=0}^{q-1} x^i a^{q-1-i}, Warning derives from the equality f=gZf^*=g_Z
the relations
 aZa1ν1anνn=0, \sum_{a\in Z} a_1^{\nu_1}\dots a_n^{\nu_n}=0,
for all (ν1,,νn)(\nu_1,\dots,\nu_n) such that 0νiq10\leq \nu_i\leq q-1 and νi<(q1)(nd)\sum\nu_i <(q-1)(n-d). The particular case ν=0\nu=0 implies that pp divides Card(Z)\mathop{\rm Card}(Z). More generally:

Proposition 2. — For every polynomial ϕF[T1,,Tn]\phi\in F[T_1,\dots,T_n] of reduced degree <(q1)(nd)<(q-1)(n-d), one has aAϕ(a)=0\sum_{a\in A} \phi(a) = 0.

c) The classic proof of that result is even easier. Let us recall it swiftly. First of all, for every integer ν\nu such that 0ν<q0\leq \nu <q, one has aFaν=0\sum_{a\in F} a^\nu=0. This can be proved in many ways, for example by using the fact that the multiplicative group of FF is cyclic; on the other hand, for every nonzero element tt of FF, the change of variables a=tba=tb leaves this sum both unchanged and multiplied by tν,t^\nu, so that taking tt such that tν1t^\nu\neq 1, one sees that this sum vanishes. It follows from that that for every polynomial fF[T1,,Tn]f\in F[T_1,\dots,T_n] whose degree in some variable is <q1\lt q-1, one has aFnf(a)=0\sum_{a\in F^n} f(a)=0. This holds in particular if the total degree of ff is <(q1)n\lt (q-1)n.
Taking ff as above proves theorem 1.

2. On the other hand there is a second Warning theorem, which seems to be absolutely neglected in France. It says the following:

Theorem 3. — Keep the same notation as in theorem 1. If ZZ is nonempty, then Card(Z)qnd\mathop{\rm Card}(Z)\geq q^{n-d}.

To prove this result, Warning starts from the following proposition:

Proposition 4. – Let L,LL,L' be two parallel subspaces of dimension dd in FnF^n. Then Card(ZL)\mathop{\rm Card}(Z\cap L) and Card(ZL)\mathop{\rm Card}(Z\cap L') are congruent modulo pp.

Let r=ndr=n-d. Up to a change of coordinates, one may assume that L={x1==xr=0}L=\{x_1=\dots=x_{r}=0\} and L={x11=x2==xr=0}L'=\{x_1-1=x_2=\dots=x_{r}=0\}. Let
ϕ=1x1q11x1(1x2q1)(1xrq1). \phi = \frac{1-x_1^{q-1}}{1-x_1} (1-x_2^{q-1})\cdots (1-x_{r}^{q-1}).
This is a polynomial of total degree is (q1)r1<(q1)(nd)(q-1)r-1<(q-1)(n-d). For aFna\in F^n, one has ϕ(a)=1\phi(a)=1 if aLa\in L, ϕ(a)=1\phi(a)=-1 if aLa\in L', and ϕ(a)=0\phi(a)=0 otherwise. Proposition 4 thus follows from proposition 2. It is now very easy to prove theorem 3 in the particular case where there exists one subspace LL of dimension dd such that Card(ZL)≢0(modp)\mathop{\rm Card}(Z\cap L)\not\equiv 0\pmod p. Indeed, by proposition 4, the same congruence will hold for every translate LL' of LL. In particular, Card(ZL)0\mathop{\rm Card}(Z\cap L')\neq0 for every translate LL' of LL, and there are qndq^{n-d} distinct translates.

To prove the general case, let us choose a subspace MM of FnF^n of dimension sds\leq d such that Card(ZM)≢0(modp)\mathop{\rm Card}(Z\cap M)\not\equiv 0\pmod p, and let us assume that ss is maximal.
Assume that s<ds< d. Let t{1,,p1}t\in\{1,\dots,p-1\} be the integer such that Card(ZM)t(modp)\mathop{\rm Card}(Z\cap M)\equiv t\pmod p. For every (s+1)(s+1)-dimensional subspace LL of FnF^n that contains MM, one has Card(ZL)0(modp)\mathop{\rm Card}(Z\cap L)\equiv 0\pmod p, by maximaility of ss, so that Z(LM)Z\cap (L\setminus M) contains at least ptp-t points. Since these subspaces LL are in 1-1 correspondence with the lines of the quotient space Fn/MF^n/M, their number is equal to (qns1)/(q1)(q^{n-s}-1)/(q-1). Consequently,  Card(Z)=Card(ZM)+LCard(Z(LM))t+(pt)qns1q1qns1qnd, \mathop{\rm Card}(Z) = \mathop{\rm Card}(Z\cap M) + \sum_L \mathop{\rm Card}(Z\cap (L\setminus M)) \geq t + (p-t) \frac{q^{n-s}-1}{q-1} \geq q^{n-s-1}\geq q^{n-d}, as was to be shown.

3. Classic theorems seem to an everlasting source of food for thought.

a) In 1999, Alon observed that Chevalley's theorem follows from the Combinatorial Nullstellensatz he had just proved. On the other hand, this approach allowed Brink (2011) to prove a similar result in general fields FF, but restricting the roots to belong to a product set A1××AnA_1\times\dots\times A_n, where A1,,AnA_1,\dots,A_n are finite subsets of FF of cardinality qq. See that paper of Clarke, Forrow and Schmitt for further developments, in particular a version of Warning's second theorem.

b) In the case of hypersurfaces (with the notation of theorem 1, m=1m=1), Ax proved in 1964 that the cardinality of ZZ is divisible not only by pp, but by qq. This led to renewed interest in the following years, especially in the works of Katz, Esnault, Berthelot, and the well has not dried up yet.

c) In 2011, Heath-Brown published a paper where he uses Ax's result to strengthen the congruence modulo pp of proposition 4 to a congruence modulo qq.

d) By a Weil restriction argument, a 1995 paper of Moreno-Moreno partially deduces the Chevalley-Warning theorem over a field of cardinality qq from its particular case over the prime field. I write partially because they obtain a divisibility by an expression of the form pfαp^{\lceil f \alpha\rceil}, while one expects qα=pfαq^{\lceil \alpha}=p^{f\lceil\alpha\rceil}. However, the same argument allows them to obtain a stronger bound which does not involve not the degrees of the polynomials, but the pp-weights of these degrees, that is the sum of their digits in their base pp expansions. Again, they obtain a divisibility by an expression of the form p fβp^{\lceil f\beta\rceil}, and it is a natural question to wonder whether the divisibility by pfβp^{f\lceil\beta\rceil} can be proved.