Bounds on Determinantal Complexity of Two Types of Generalized Permanents

TL;DR

This paper introduces two new polynomial families generalizing permanents and establishes bounds on their determinantal complexities, advancing understanding of algebraic complexity related to these generalized structures.

Contribution

It defines two novel polynomial families based on signed permutations and surjective functions, providing bounds on their determinantal complexities.

Findings

Established upper and lower bounds for the new polynomial families.

Extended known complexity bounds from permanents to their generalizations.

Enhanced understanding of algebraic complexity in generalized permanent-like polynomials.

Abstract

We define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permanents. One of these families is obtained by replacing permutations by signed permutations, and the other by replacing permutations by surjective functions with preimages of prescribed sizes.