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.
Highlights
The size of the resultant matrix is determined by the inputThe problem of solving a system of nonlinear polytopes, named Newton polytopes, denoted polynomial equations f1 = f2 = ... = fm = 0 over a field X such that the system has a solution, arise in many application domains
Based on the ideas that have been presented in the analysis of the Dixon case, the research is conducted with the aim of inferring and characterizing sparseness conditions on the structure of the Newton polytopes and support interiors of the class of polynomial systems that produce exact resultants for sparse systems
An optimal matrix formula is expected from the class of multigraded systems
Summary
In algebra, mechanics, kinematics, robotics, structural molecular biology, logic, geometric and solid modelling, image understanding and vision have been well-documented [1, 6, 12, 19, 21] It is for this reason that techniques of resultant for solving such applications as above have received considerable research attention due to its efficientness and robustness to certain problems Based on the ideas that have been presented in the analysis of the Dixon case, the research is conducted with the aim of inferring and characterizing sparseness conditions on the structure of the Newton polytopes and support interiors of the class of polynomial systems that produce exact resultants for sparse systems In this preliminary work, Maple multires package is used to compute the resultant matrices of some multivariate polynomial systems. An optimal matrix formula is expected from the class of multigraded systems
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