Computing Puiseux-Series Solutions to Determinantal Equations via Combinatorial Relaxation

Abstract

Let $A(t,x) = (A_{ij} (t,x))$ be a square matrix with $A_{ij}$ being a polynomial in t and x. This paper proposes an algorithm for computing the Puiseux (= fractional power) series solutions $x = x(t)$ to the equation ${\operatorname{det}}A(t, x) = 0$. The algorithm is based on an observation which links the Newton diagram (polygon) for det ${\operatorname{det}}A(t, x)$ with the perfect matchings of a bipartite graph associated with A. The algorithm is efficient, making full use of available fast network-type algorithms.