Some New Applications of Toric Geometry

Details Some New Applications of Toric Geometry Some New Applications of Toric Geometry

1997-02-08

arxiv.org/abs/math/9702221v1

Mathematics

Algebraic Geometry

Scientific paper

This paper reexamines univariate reduction from a toric geometric point of view. We begin by constructing a binomial variant of the $u$-resultant and then retailor the generalized characteristic polynomial to fully exploit sparsity in the monomial structure of any given polynomial system. We thus obtain a fast new algorithm for univariate reduction and a better understanding of the underlying projections. As a corollary, we show that a refinement of Hilbert's Tenth Problem is decidable within single-exponential time. We also show how certain multisymmetric functions of the roots of polynomial systems can be calculated with sparse resultants.

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