The multivariate resultant is NP-hard in any characteristic

  • Authors:

  • Bruno Grenet;Pascal Koiran;Natacha Portier

  • Affiliations:

  • LIP, UMR, École Normale Supérieure de Lyon, CNRS, UCBL, INRIA, École Normale Supérieure de Lyon, Université de Lyon and Department of Computer Science, University of Toron ...;LIP, UMR, École Normale Supérieure de Lyon, CNRS, UCBL, INRIA, École Normale Supérieure de Lyon, Université de Lyon and Department of Computer Science, University of Toron ...;LIP, UMR, École Normale Supérieure de Lyon, CNRS, UCBL, INRIA, École Normale Supérieure de Lyon, Université de Lyon and Department of Computer Science, University of Toron ...

  • Venue:
  • MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
  • Year:
  • 2010

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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.