|
This article is cited in 13 scientific papers (total in 14 papers)
A note on the Chevalley–Warning theorems
D. R. Heath-Brown Mathematical Institute, University of Oxford, UK
Abstract:
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.
Received: 29.09.2010
Citation:
D. R. Heath-Brown, “A note on the Chevalley–Warning theorems”, Russian Math. Surveys, 66:2 (2011), 427–435
Linking options:
https://www.mathnet.ru/eng/rm9423https://doi.org/10.1070/RM2011v066n02ABEH004744 https://www.mathnet.ru/eng/rm/v66/i2/p223
|
Statistics & downloads: |
Abstract page: | 835 | Russian version PDF: | 369 | English version PDF: | 29 | References: | 65 | First page: | 21 |
|