A local decision test for sparse polynomials

An ℓ-sparse (multivariate) polynomial is a polynomial containing at most ℓ-monomials in its explicit description. We assume that a polynomial is implicitly represented as a black-box: on an input query from the domain, the black-box replies with the evaluation of the polynomial at that input. We provide an efficient, randomized algorithm, that can decide whether a polynomial f:Fqn→Fq given as a black-box is ℓ-sparse or not, provided that q is large compared to the polynomial's total degree. The algorithm makes only O(ℓ) queries, which is independent of the domain size. The running time of our algorithm (in the bit-complexity model) is poly(n,logd,ℓ), where d is an upper bound on the degree of each variable. Existing interpolation algorithms for polynomials in the same model run in time poly(n,d,ℓ). We provide a similar test for polynomials with integer coefficients.