Given a multivariate polynomial P(X(1),..., X) over a finite field F(q), let N(P) denote the number of roots over F(q)(n). 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,(P), when the polynomial is given as a sum of monomials. We give an efficient algorithm to compute N,(P) when the modulus r is a power of the ... [Show full abstract] characteristic of the field. We show that for all other moduli, the problem of computing N,(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.