Geodesic
A note on the Chevalley–Warning theorems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 66 (2011) no. 2, pp. 427-435 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $f_1,\dots,f_r$ be polynomials in $n$ variables, over the field $\mathbb{F}_q$, and suppose that their degrees are $d_1,\dots,d_r$. It was shown by Warning in 1935 that if $\mathscr N$ is the number of common zeros of the polynomials $f_i$, then $\mathscr N\geqslant q^{n-d}$. It is the main aim of the present paper to improve on this bound. When the set of common zeros does not form an affine linear subspace in $\mathbb{F}_q^n$, it is shown for example that $\mathscr N\geqslant2q^{n-d}$ if $q\geqslant4$, and that $\mathscr N\geqslant q^{n+1-d}/(n+2-d)$ if the $f_i$ are all homogeneous. Bibliography: 5 titles.
Keywords: Chevalley–Warning theorems, polynomials, finite fields, zeros, lower bound, number of zeros, affine linear space. 
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     title = {A note on the {Chevalley{\textendash}Warning} theorems},
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D. R. Heath-Brown. A note on the Chevalley–Warning theorems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 66 (2011) no. 2, pp. 427-435. http://geodesic.mathdoc.fr/item/RM_2011_66_2_a5/

[1] C. Chevalley, “Démonstration d'une hypothèse de M. Artin”, Abh. Math. Sem. Univ. Hamburg, 11 (1935), 73–75 | Zbl

[2] E. Warning, “Bemerkung zur vorstehenden Arbeit von Herrn Chevalley”, Abh. Math. Semin. Hamb. Univ., 11 (1935), 76–83 | Zbl

[3] J. Ax, “Zeros of polynomials over finite fields”, Amer. J. Math., 86:2 (1964), 255–261 | DOI | MR | Zbl

[4] N. M. Katz, “On a theorem of Ax”, Amer. J. Math., 93:2 (1971), 485–499 | DOI | MR | Zbl

[5] S. Lang, A. Weil, “Number of points of varieties in finite fields”, Amer. J. Math., 76:4 (1954), 819–827 | DOI | MR | Zbl