Counting curves and their projections
Joachim von zur Gathen, Marek Karpiński, Igor E. Shparlinski · 1993
. Some deterministic and probabilistic methods are presented for counting and estimating the number of points on curves over finite fields, and on their projections. The classical question of estimating the size of the image of a univariate polynomial is a special case. For curves given by sparse polynomials, the counting problem is #P-complete via probabilistic parsimonious Turing reductions. 1. Introduction One of the most celebrated results in algebraic geometry is Weil's theorem on the number of points on algebraic curves over a finite field. In this paper, we address some computational problems related to this question. Our main results are: ffi A "computational Weil estimate" for projections of curves and images of polynomials, in Section 3. ffi #P-completeness of the exact counting problem for sparse curves, in Section 4. We consider a finite field F q with q elements, an algebraic closure K of F q , a polynomial f 2 F q [x; y] of degree n , the plane curve C = ff = 0g = f(a;...