Algorithms for modular counting of roots of multivariate polynomials

Abstract

Given a multivariate polynomial P(X1,⋯,Xn) over a finite field double-struck F signq, let N(P) denote the number of roots over double-struck F signqn. The modular root counting problem is given a modulus r, to determine Nr(P) = N(P) mod r. We study the complexity of computing Nr(P), when the polynomial is given as a sum of monomials. We give an efficient algorithm to compute N r(P) when the modulus r is a power of the characteristic of the field. We show that for all other moduli, the problem of computing N r(P) is NP-hard. We present some hardness results which imply that that our algorithm is essentially optimal for prime fields. We show an equivalence between maximum-likelihood decoding for Reed-Solomon codes and a root-finding problem for symmetric polynomials. © Springer-Verlag Berlin Heidelberg 2006.