A Combinatorial Algorithm for Computing the Degree of the Determinant of a Generic Partitioned Polynomial Matrix with $2\,\times \,2$ Submatrices

Abstract

In this paper, we consider the problem of computing the degree of the determinant of a block-structured symbolic matrix (a generic partitioned polynomial matrix) $A = (A_{\alpha \beta } x_{\alpha \beta } t^{d_{\alpha \beta }})$, where $A_{\alpha \beta }$ is a $2 \times 2$ matrix over a field $\mathbf {F}$, $x_{\alpha \beta }$ is an indeterminate, and $d_{\alpha \beta }$ is an integer for $\alpha , \beta = 1,2,\dots , n$, and t is an additional indeterminate. This problem can be viewed as an algebraic generalization of the maximum perfect bipartite matching problem.

The main result of this paper is a combinatorial $O(n^5)$-time algorithm for the deg-det computation of a $(2 \times 2)$-type generic partitioned polynomial matrix of size $2n \times 2n$. We also present a min-max theorem between the degree of the determinant and a potential defined on vector spaces. Our results generalize the classical primal-dual algorithm (Hungarian method) and min-max formula (Egerváry’s theorem) for maximum weight perfect bipartite matching.

The author was supported by JSPS KAKENHI Grant Number JP17K00029, 20K23323, 20H05795, Japan.