Improved explicit estimates on the number of solutions of equations over   a finite field
  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. 
Antonio Cafure (1), Guillermo Matera (2) ((1)Universidad de Buenos   Aires, Buenos Aires, Argentina, (2)Universidad Nacional...