
Universität Bonn > Institut für Informatik > Abteilung V  
CSReports 1991  Copyright 1991 Universität Bonn, Institut für Informatik, Abt. V  
8562

Nondivisibility of Sparse Polynomials is in NP Under the Extended Riemann Hypothesis
Dima Grigoriev, Marek Karpinski, Andrew M. Odlyzko [Download PostScript] [Download PDF] Symbolic manipulation of sparse polynomials, given as lists of exponents and nonzero coefficients, appears to be much more complicated than dealing with polynomials in dense encoding (see e.g. \cite{GKS,KT,Pla,Pla1}). The first results in this direction are due to Plaisted \cite{Pla,Pla1}, who proved, in particular, the NPcompleteness of divisibility of a polynomial $x^n1$ 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 \cite{Mul}.) Here we prove that nondivisibility of two sparse multivariable polynomials is in NP, provided that the Extended Riemann Hypothesis (ERH) holds (see e.g. \cite{LO}). The divisibility problem is closely related to the rational interpolation problem (whose decidability and complexity bound are determined in \cite{GKS}). 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 (\cite{KR88}), provided the ERH holds and moreover, that we know the degree of some sparse rational representation of it. 

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Universität Bonn > Institut für Informatik > Abteilung V 