A note on the Chevalley-Warning theorems. (English. Russian original) Zbl 1247.11089

Heath-Brown, D. R.

Russ. Math. Surv. 66, No. 2, 427-435 (2011); reprinted from Usp. Mat. Nauk 66, No. 2, 223-232 (2011); erratum Russ. Math. Surv. 79, No. 3, 563 (2024).

The main purpose of this paper is to strengthen the Chevalley-Warning theorems. Let \(\mathbf{f}=(f_1,\dots,f_r)\) be an \(r\)-tuple of polynomials in \(n\) variables over the field \(\mathbb{F}_q\). Let \(d_i\) be the total degree of \(f_i\) and write \(d=d_1+\ldots+d_r\). For any subset \(S\subset \mathbb{F}_q^n\) put \(z(\mathbf{f};S)=\{\mathbf{x}\in S:f_i(\mathbf{x})=0(1\le i\le r)\}\) and \({\mathcal N}(\mathbf{f};S)={\mathcal N}(S)=\#z(\mathbf{f};S)\).
His first theorem shows that \(\mathcal N(L_1)\equiv{\mathcal N}(L_2)\pmod q\) for any two parallel linear spaces \(L_1, L_2\subset \mathbb{A}^n(\mathbb{F}_q)\) of dimension \(d\) or more.
When \(n>d\) and \(z(\mathbb{A}^n(\mathbb{F}_q))\) is non-empty and is not a linear subspace of \(\mathbb{A}^n(\mathbb{F}_q)\), his second theorem says that
(i) \({\mathcal N}(\mathbb{A}^n(\mathbb{F}_q))>q^{n-d}\) for any \(q\);
(ii) \({\mathcal N}(\mathbb{A}^n(\mathbb{F}_q))\ge 2q^{n-d}\) if \(q\geq 4\);
(iii) \({\mathcal N}(\mathbb{A}^n(\mathbb{F}_q))\ge q^{n+1-d}/(n+2-d)\) for any \(q\) if the polynomials in \(\mathbf{f}\) are homogeneous.
Several applications and some instructive examples are given, too.

Reviewer: Fumio Hazama (Hatoyama)

MSC:

11T06	Polynomials over finite fields
11D79	Congruences in many variables
11G25	Varieties over finite and local fields

Keywords:

Chevalley-Warning theorems; polynomials; finite fields; number of zeros