Combinatorial relaxation algorithm for mixed polynomial matrices

Abstract.The notion of mixed polynomial matrices is a mathematical tool for faithful description of discrete physical/engineering dynamical systems. It distinguishes two kinds of numbers: fixed constants that account for conservation laws and independent parameters that represent physical characteristics.This paper presents an algorithm for computing the maximum degree of subdeterminants of a fixed order in a polynomial matrix that is obtained by substituting specific numerical values to the independent physical parameters of a mixed polynomial matrix. The algorithm first computes the generic solution by a combinatorial method, i.e., by solving a valuated matroid intersection problem associated with the mixed polynomial matrix, and then finds the exact solution for the specified parameter values with a minimum amount of numerical computation.This is intended to be a nontrivial application of mathematical programming techniques to computer algebra and systems analysis.

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