An approximation algorithm for the number of zeros of arbitrary polynomials over GF[q]
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Approximating the number of zeroes of a GF[2] polynomial
Journal of Algorithms
Computational Complexity - Special issue on circuit complexity
Finite fields
Counting curves and their projections
Computational Complexity
Solvability of systems of polynomial congruences modulo a large prime
Computational Complexity
Counting Rational Points on Curves and Abelian Varieties over Finite Fields
ANTS-II Proceedings of the Second International Symposium on Algorithmic Number Theory
The Computational Complexity of ({\it XOR, AND\/})-Counting Problems
The Computational Complexity of ({\'it XOR, AND\'/})-Counting Problems
Maximum-likelihood decoding of Reed-Solomon codes is NP-hard
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
Counting rational points on curves over finite fields
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
From randomizing polynomials to parallel algorithms
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
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Given a multivariate polynomial P(X1, ⋯, Xn) over a finite field ${\mathbb F}_{q}$, let N(P) denote the number of roots over ${\mathbb F}^{n}_{q}$. 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 Nr(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 Nr(P) is Nr(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.