Sylvester A-Resultants for Bivariate Polynomials with Planar Newton Polygons

摘要

We derive necessary and sufficient conditions which guarantee that a multiplying set of monomials generates exactly a Sylvester A-resultant for three bivariate polynomials with a given planar Newton polygon. We show that valid multiplying sets come in complementary pairs, and any two complementary pairs of multiplying sets can be used to index the rows and columns of a pure Bezoutian A-resultant for the same Newton polygon. The necessary and sufficient conditions include a set of Diophantine equations that can be solved to generate the multiplying sets and therefore the corresponding Sylvester A-resultants. Examples relevant to Geometric Modeling are provided, including a new family of hexagonal examples for which Sylvester formulas were not previously known. These examples not only flesh out the theory, but also demonstrate that none of the conditions are superfluous and that all the conditions are mutually independent. The proof of the main theorem makes use of tools from algebraic geometry, including sheaf cohomology on toric varieties and Weyman's resultant complex.