Denominators for One Variable Poincar\'e Series of Generic Matrices

TL;DR

This paper investigates the rational Poincaré series of trace rings of generic matrices, providing explicit formulas and conjectures for denominators in various matrix sizes.

Contribution

It derives explicit formulas for denominators of Poincaré series for 2x2 matrices and proposes conjectures for higher sizes, advancing understanding of these series.

Findings

Explicit formula for 2x2 matrices in lowest terms

Proposed denominator for 3x3 matrices, unproven in lowest terms

Conjectured denominator for general matrix size

Abstract

We study the Poincar\'e series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of $2\times2$ matrices; a denominator, which we presume but have not been able to prove to be in lowest terms, in the case of $3\times3$ matrices; and a conjectured denominator in the general case.