Determinants from homomorphisms

TL;DR

This paper presents a new combinatorial approach to understanding the relationship between determinants and traces of matrix powers, leading to efficient algorithms for computing determinants.

Contribution

It introduces a combinatorial explanation that bypasses linear algebra, leveraging subgraph and homomorphism counts to relate determinants and traces.

Findings

Provides polynomial-time algorithms for determinants

Uses combinatorial methods instead of linear algebra

Establishes a classical connection between subgraph counts and homomorphisms

Abstract

We give a new combinatorial explanation for well-known relations between determinants and traces of matrix powers. Such relations can be used to obtain polynomial-time and poly-logarithmic space algorithms for the determinant. Our new explanation avoids linear-algebraic arguments and instead exploits a classical connection between subgraph and homomorphism counts.