Sylvester-type matrices for sparse resultants

Abstract

The resultant matrix of a polynomial system depends on the geometry of its input Newton polytopes. Therefore for sparse inputs, the matrix is lower in dimension. The aim of the study is to infer conditions on the class of polynomial systems that can give a resultant matrix whose size is minimized, that is an optimal or Sylvester-type sparse resultant matrix. From the work of Emiris, the ‘incremental algorithm’ has been claimed to produce optimal matrices for the class of multi-homogeneous (or multigraded) systems of special structure. Cyclic polynomial systems for n-root problems also fall under this classification. We have applied the Maple multires package to obtain Sylvester-type matrices for some examples. The ultimate aim of the study is to verify whether the multigraded systems constitute to the only class of polynomial systems that can give sparse resultant optimal matrix; hence giving a necessary and sufficient condition for producing exact sparse resultants.