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

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

2004-05-14

arxiv.org/abs/math/0405302v1

Mathematics

Number Theory

33 pages

Scientific paper

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.

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