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Russian Mathematical Surveys, 2011, Volume 66, Issue 2, Pages 427–435
DOI: https://doi.org/10.1070/RM2011v066n02ABEH004744
(Mi rm9423)
 

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
References:
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
Bibliographic databases:
Document Type: Article
UDC: 511
MSC: 11G25
Language: English
Original paper language: Russian
Citation: D. R. Heath-Brown, “A note on the Chevalley–Warning theorems”, Russian Math. Surveys, 66:2 (2011), 427–435
Citation in format AMSBIB
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\by D.~R.~Heath-Brown
\paper A note on the Chevalley--Warning theorems
\jour Russian Math. Surveys
\yr 2011
\vol 66
\issue 2
\pages 427--435
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Linking options:
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  • https://doi.org/10.1070/RM2011v066n02ABEH004744
  • https://www.mathnet.ru/eng/rm/v66/i2/p223
    Erratum
    • Misprints
      Uspekhi Mat. Nauk, 2011, 66:3(399), 207–208
    Remarks
    This publication is cited in the following 14 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Успехи математических наук Russian Mathematical Surveys
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    Abstract page:835
    Russian version PDF:369
    English version PDF:29
    References:65
    First page:21
     
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