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

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

MSC:

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