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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The distribution of solutions to equations over finite fields


Author: Todd Cochrane
Journal: Trans. Amer. Math. Soc. 293 (1986), 819-826
MSC: Primary 11T41; Secondary 11D72
DOI: https://doi.org/10.1090/S0002-9947-1986-0816328-4
MathSciNet review: 816328
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Abstract: Let $ {\mathbb{F}_q}$ be the finite field in $ q = {p^f}$ elements, $ \underline F (\underline x )$ be a $ k$-tuple of polynomials in $ {\mathbb{F}_q}[{x_1}, \ldots ,{x_n}]$, $ V$ be the set of points in $ \mathbb{F}_q^n$ satisfying $ \underline F (\underline x ) = \underline 0 $ and $ S$, $ T$ be any subsets of $ \mathbb{F}_q^n$. Set $ \phi (V,\underline 0 ) = \vert V\vert - {q^{n - k}}$,

$\displaystyle \phi (V,\underline y ) = \sum\limits_{\underline x \in V} {e\left... ...ot \underline y )} \right)\quad {\text{for}}\;\underline y \ne \underline 0 ,} $

and $ \Phi (V) = {\max _{\underline y }}\vert\phi (V,\underline y )\vert$. We use finite Fourier series to show that $ (S + T) \cap V$ is nonempty if $ \vert S\vert\vert T\vert > {\Phi ^2}(V){q^{2k}}$. In case $ q = p$ we deduce from this, for example, that if $ C$ is a convex subset of $ {\mathbb{R}^n}$ symmetric about a point in $ {\mathbb{Z}^n}$, of diameter $ < 2p$ (with respect to the sup norm), and $ \operatorname{Vol} (C) > {2^{2n}}\Phi (V){p^k}$, then $ C$ contains a solution of $ \underline F (\underline x ) \equiv \underline 0 (\bmod p)$.

We also show that if $ B$ is a box of points in $ \mathbb{F}_q^n$ not contained in any $ (n - 1)$-dimensional subspace and $ \vert B\vert > 4 \cdot {2^{nf}}\Phi (V){q^k}$, then $ B \cap V$ contains $ n$ linearly independent points.


References [Enhancements On Off] (What's this?)

  • [Ba] R. C. Baker, Small solutions of congruences, Mathematika 30 (1983), 164-188. MR 737175 (86c:11027)
  • [Car] L. Carlitz, Weighted quadratic partitions over a finite field, Canad. J. Math. 5 (1953), 317-323. MR 0059310 (15:508e)
  • [Cas] J. W. S. Cassels, An introduction to the geometry of numbers, Springer-Verlag, Berlin, 1959.
  • [Ch1] J. H. H. Chalk, The number of solutions of congruences in incomplete residue systems, Canad. J. Math. 15 (1963), 291-296. MR 0146130 (26:3656)
  • [Ch2] -, The Vinogradov-Mordell-Tietäväinen inequalities, Indag. Math. 42 (1980), 367-374. MR 597995 (82d:10053)
  • [CW] J. H. H. Chalk and K. S. Williams, The distribution of solutions of congruences, Mathematika 12 (1965), 176-192. MR 0190112 (32:7526)
  • [De] P. Deligne, La conjecture de Weil. I, Publ. Math. IHES 43 (1974), 273-307. MR 0340258 (49:5013)
  • [Mo1] L. J. Mordell, The number of solutions in incomplete residue sets of quadratic congruences, Arch. Math. 8 (1957), 153-157. MR 0091963 (19:1039a)
  • [Mo2] -, Incomplete exponential sums and incomplete residue systems for congruences, Czechoslovak. Math. J. 14 (1964), 235-242. MR 0170875 (30:1110)
  • [My] G. Myerson, The distribution of rational points on varieties defined over a finite field, Mathematika 28 (1981), 153-159. MR 645095 (83h:10041)
  • [Sm] R. A. Smith, The distribution of rational points on hypersurfaces defined over a finite field, Mathematika 17 (1970), 328-332. MR 0284419 (44:1646)
  • [Sp] K. Spackman, On the number and distribution of simultaneous solutions to diagonal congruences, Canad. J. Math. 33 (1981), 421-436. MR 617633 (83b:10019)
  • [Ti] A. Tietäväinen, On the solvability of equations in incomplete finite fields, Ann. Univ. Turku. Ser. AI 102 (1967), 1-13. MR 0213334 (35:4198)

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DOI: https://doi.org/10.1090/S0002-9947-1986-0816328-4
Article copyright: © Copyright 1986 American Mathematical Society

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