A note on the Chevalley--Warning theorems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 66 (2011) no. 2, pp. 427-435
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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.
@article{RM_2011_66_2_a5,
author = {D. R. Heath-Brown},
title = {A note on the {Chevalley--Warning} theorems},
journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
pages = {427--435},
publisher = {mathdoc},
volume = {66},
number = {2},
year = {2011},
language = {en},
url = {http://geodesic.mathdoc.fr/item/RM_2011_66_2_a5/}
}
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/