Some New Applications of Toric Geometry
<|reference_start|>Some New Applications of Toric Geometry: 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.<|reference_end|>
@article{rojas1997some,
    title={Some New Applications of Toric Geometry},
    author={J. Maurice Rojas},
    journal={arXiv preprint arXiv:math/9702221},
    year={1997},
    number={MSRI 1997-016},
    archivePrefix={arXiv},
    eprint={math/9702221},
    primaryClass={math.AG cs.DS}
}