Algorithms for Modular Counting of Roots of Multivariate Polynomials

Given a multivariate polynomial P(X 1 ,..., X n ) over a finite field F q , let N(P) denote the number of roots over F n q . The modular root counting problem is given a modulus r, to determine N r (P) = N(P) mod r. We study the complexity of computing N r (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.

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