On the complexity of the multivariate resultant

  title={On the complexity of the multivariate resultant},
  author={Bruno Grenet and Pascal Koiran and Natacha Portier},

Complexity of constructing Dixon resultant matrix

A detailed analysis of the computational complexity of the recurrence formula for the general multivariate setting is presented and three bivariate polynomials are generalized to thegeneral multivariate case by using the construction of standard Dixon resultant matrix.

Variety Membership Testing, Algebraic Natural Proofs, and Geometric Complexity Theory

The stabilizers of the tensors that generate the orbit closures of the two varieties are determined and proved to be almost characterized by their symmetries, a promising sign that the GCT approach might indeed be successful.

Bit complexity for multi-homogeneous polynomial system solving - Application to polynomial minimization

Optimization Algorithm for Reduction the Size of Dixon Resultant Matrix: A Case Study on Mechanical Application

This paper aims at reducing the number of extraneous factors via optimizing the size of Dixon matrix through an optimization algorithm along with a number of new polynomials to replace the polynomial system.

Sparse resultants and straight-line programs

Processing Succinct Matrices and Vectors

It is shown that accessing an entry, equality checking, matrix multiplication, and other basic matrix operations can be solved in polynomial time for + MTDD-represented matrices.

Quantum Merlin Arthur with Exponentially Small Gap

We study the complexity of QMA proof systems with inverse exponentially small promise gap. We show that this class can be exactly characterized by PSPACE, the class of problems solvable with a

Quantum Merlin Arthur with Exponentially Small Gap Bill Fefferman

We will study the complexity of QMA proof systems with inverse exponentially small promise gap. We will show that this class, QMAexp, can be exactly characterized by PSPACE, the class of problems

Représentations des polynômes, algorithmes et bornes inférieures

La complexite algorithmique est l'etude des ressources necessaires — le temps, la memoire, … — pour resoudre un probleme de maniere algorithmique. Dans ce cadre, la theorie de la complexite



The Multivariate Resultant Is NP-hard in Any Characteristic

The 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).

Solving systems of nonlinear polynomial equations faster

This paper considers projective problems, that is, the polynomials are homogeneous and the solutions are sought in n-dimensional projective space, and shows that the solutions of an affine system are specializations of the solution rays of its homogenized projective version.

Matrices in Elimination Theory

This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bezout and Macaulay, and presents the properties of the different matrix formulations.

Fast parallel matrix inversion algorithms

  • L. Csanky
  • Computer Science, Mathematics
    16th Annual Symposium on Foundations of Computer Science (sfcs 1975)
  • 1975
It will be shown in the sequel that the parallel arithmetic complexity of all these four problems is upper bounded by O(log2n) and the algorithms that establish this bound use a number of processors polynomial in n, disproves I. Munro's conjecture.

On the Intrinsic Complexity of Elimination Theory

The intrinsic complexity of selected algorithmic problems of classical elimination theory in algebraic geometry is considered and reductions of fundamental questions of elimination theory to NP- and P#-complete problems are given and it is confirmed that some of these questions may have exponential size outputs.

On the irreducibility of multivariate subresultants

Algebraic and geometric reasoning using Dixon resultants

Experimental results suggest that the resultant of a set of polynomials which are symmetric in the variables is relatively easier to compute using the extended Dixon's method.

Resolution des Systemes d'Equations Algebriques

Comparison of various multivariate resultant formulations

Compared methods for computing nontrivial projection operators, it is shown that the Dixon matrix is smaller than the sparse resultant matrix which is (by a factor up to O(e’) for a certain class) bigger than the Macaulay matrix.

Sharp estimates for the arithmetic Nullstellensatz

We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring ZZ . The result improves previous work of Philippon, Berenstein-Yger and