Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
Rectangular corner cutting and Sylvester A-resultants
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
The n-sided toric patches and A-resultants
Computer Aided Geometric Design
Implicitization by Dixon A-Resultants
GMP '00 Proceedings of the Geometric Modeling and Processing 2000
The resultant of an unmixed bivariate system
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Intersection and self-intersection of surfaces by means of Bezoutian matrices
Computer Aided Geometric Design
Expressing a fraction of two determinants as a determinant
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Single-lifting Macaulay-type formulae of generalized unmixed sparse resultants
Journal of Symbolic Computation
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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.