Polynomial Systems Solving by Fast Linear Algebra
Résumé
Polynomial system solving is a classical problem in mathematics with a wide range of applications which make its complexity a central study in theoretical computer science. Depending on the context, solving has different meanings. In order to stick to the most general case, we consider a representation of the solutions from which one can easily recover the exact solutions or a certified approximation of them. Under generic assumption, such a representation is given by the lexicographical Gröbner basis of the system and consists of a set of univariate polynomials. The best known algorithm for computing the lexicographical Gröbner basis is in $O(nD^3)$ arithmetic operations where $n$ is the number of variables and $D$ the number of solutions of the system. We show that this complexity can be decreased to $\widetilde{O}(D^\omega)$ where $2 \leq \omega < 2.3727$ is the exponent in the complexity of multiplying two dense matrices and the notation $\widetilde{O}$ means that we neglect logarithmic factors. To achieve this result we propose new algorithms which rely on fast linear algebra. When the degree of the equations are bounded we propose a deterministic algorithm. In the unbounded case we present a Las Vegas algorithm.
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