Improved explicit estimates on the number of solutions of equations over a finite field

Antonio Cafure; Guillermo Matera

Improved explicit estimates on the number of solutions of equations over a finite field

Number Theory 2007-05-23 v1 Algebraic Geometry

Authors: Antonio Cafure , Guillermo Matera

Abstract

We show explicit estimates on the number of $q$ --rational points of an $F_q$ --definable affine absolutely irreducible variety of the algebraic closure of the finite field $F_q$ of $q$ elements. Our estimates for a hypersurface significantly improve previous estimates of W. Schmidt and M.-D. Huang and Y.-C. Wong, while in the case of a variety our estimates improve those of S. Ghorpade and G. Lachaud in several important cases. Our proofs rely on elementary methods of effective elimination theory and suitable effective versions of the first Bertini theorem.

Keywords

finite fields explicit estimates rational points absolutely irreducible variety effective elimination theory bertini theorem

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@article{arxiv.math/0405302,
  title  = {Improved explicit estimates on the number of solutions of equations over a finite field},
  author = {Antonio Cafure and Guillermo Matera},
  journal= {arXiv preprint arXiv:math/0405302},
  year   = {2007}
}

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33 pages

Improved explicit estimates on the number of solutions of equations over a finite field

Abstract

Keywords

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