The Canny-Emiris conjecture for the sparse resultant
Abstract
We present a product formula for the initial parts of the sparse resultant associated to an arbitrary family of supports, generalising a previous result by Sturmfels. This allows to compute the homogeneities and degrees of the sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain a similar product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated to a mixed subdivision of a polytope. Applying these results, we prove that the sparse resultant can be computed as the quotient of the determinant of such a square matrix by a certain principal minor, under suitable hypothesis. This generalises the classical Macaulay formula for the homogeneous resultant, and confirms a conjecture of Canny and Emiris.
Keywords
Cite
@article{arxiv.2004.14622,
title = {The Canny-Emiris conjecture for the sparse resultant},
author = {Carlos D'Andrea and Gabriela Jeronimo and Martin Sombra},
journal= {arXiv preprint arXiv:2004.14622},
year = {2021}
}
Comments
52 pages, latex, uses kbordermatrix.sty, revised version accepted for publication at Journal of Foundation of Computational Mathematics