The Multivariate Resultant is NP-hard in any Characteristic

Abstract : The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system's coefficients which vanishes if and only if the system is satisfiable). In this paper we present several NP-hardness results for testing whether a multivariate resultant vanishes, or equivalently for deciding whether a square system of homogeneous equations is satisfiable. Our main result is that testing the resultant for zero is NP-hard under deterministic reductions in any characteristic, for systems of low-degree polynomials with coefficients in the ground field (rather than in an extension). We also observe that in characteristic zero, this problem is in the Arthur-Merlin class AM if the generalized Riemann hypothesis holds true. In positive characteristic, the best upper bound remains PSPACE.
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Bruno Grenet, Pascal Koiran, Natacha Portier. The Multivariate Resultant is NP-hard in any Characteristic. Mathematical Foundations of Computer Science 2010, Aug 2010, Brno, Czech Republic. pp.477-488, ⟨10.1007/978-3-642-15155-2_42⟩. ⟨ensl-00440842v3⟩

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