Proofs for Nondivisibility of Sparse Polynomials underthe Extended Riemann

Symbolic manipulation of sparse polynomials, given as lists of exponents and nonzero coeecients, appears to be much more complicated than dealing with polyno-mials in dense encoding (see e.g. GKS 90, KT 88, P 77a, P 77b]). The rst results in this direction are due to Plaisted P 77a, P 77b], who proved, in particular, the NP-completeness of divisibility of a polynomial x n ?1 by a product of sparse polynomials. On the other hand, essentially nothing nontrivial is known about the complexity of the divisibility problem of two sparse integer polynomials. (One can easily prove that it is in PSPACE with the help of M 86].) Here we prove that nondivisibility of two sparse multivariable polynomials is in NP, provided that the Extended Riemann Hypothesis (ERH) holds (see e.g. LO 77]). The divisibility problem is closely related to the rational interpolation problem (whose decidability and complexity bound are determined in GKS 90]). In this setting we assume that a rational function is given by a black box for evaluating it. We prove also that the problem of deciding whether a rational function given by a black box equals a polynomial belongs to the parallel class NC ((KR 88]), provided the ERH holds and moreover, that we know the degree of some sparse rational representation of it.