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A note on succinct representations of graphs
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On the intrinsic complexity of elimination theory
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Comparison of various multivariate resultant formulations
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Artificial Intelligence
Hilbert's Nullstellensatz is in the polynomial hierarchy
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Counting curves and their projections
Computational Complexity
Matrices in elimination theory
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Complexity of Problems on Graphs Represented as OBDDs (Extended Abstract)
STACS '98 Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
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On the complexity of factoring bivariate supersparse (Lacunary) polynomials
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Computing multihomogeneous resultants using straight-line programs
Journal of Symbolic Computation
Expressing a fraction of two determinants as a determinant
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Computational Complexity: A Modern Approach
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VPSPACE and a Transfer Theorem over the Reals
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The multivariate resultant is NP-hard in any characteristic
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Succinct algebraic branching programs characterizing non-uniform complexity classes
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
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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 investigate the complexity of computing the multivariate resultant. First, we study the complexity of testing the multivariate resultant for zero. Our main result is that this problem 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). In null characteristic, we observe that this problem is in the Arthur-Merlin class AM if the generalized Riemann hypothesis holds true, while the best known upper bound in positive characteristic remains PSPACE. Second, we study the classical algorithms to compute the resultant. They usually rely on the computation of the determinant of an exponential-size matrix, known as Macaulay matrix. We show that this matrix belongs to a class of succinctly representable matrices, for which testing the determinant for zero is proved PSPACE-complete. This means that improving Canny's PSPACE upper bound requires either to look at the fine structure of the Macaulay matrix to find an ad hoc algorithm for computing its determinant, or to use altogether different techniques.