We derive necessary and sufficient conditions which
guarantee that a multiplying set of monomials generates exactly
a Sylvester Aresultant 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 Aresultant 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 Aresultants.
Examples relevant to Geometric Modeling are provided, including a new family of
hexagonal supports 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.
