On Algebraic Shift Equivalence of Matrices over Polynomial Rings

TL;DR

This paper investigates algebraic shift equivalence of matrices over polynomial rings with up to two variables, establishing key factorizations and equivalences in the one-variable case and providing a counterexample in the two-variable case.

Contribution

It proves that over $D[x]$, all non-zero matrices have full rank factorizations and are shift equivalent to nonsingular matrices, while also presenting a counterexample in the two-variable case.

Findings

All non-zero matrices over $D[x]$ have full rank factorizations.

Non-nilpotent matrices over $D[x]$ are algebraically shift equivalent to nonsingular matrices.

Counterexample in two variables shows some matrices cannot be shift equivalent to nonsingular matrices.

Abstract

The paper studies algebraic strong shift equivalence of matrices over $n$-variable polynomial rings over a principal ideal domain $D$($n\leq 2$). It is proved that in the case $n=1$, every non-zero matrix over $D[x]$ has a full rank factorization and every non-nilpotent matrix over $D[x]$ is algebraically strong shift equivalent to a nonsingular matrix. In the case $n=2$, an example of non-nilpotent matrix over $\mathbb{R}[x,y,z]=\mathbb{R}[x][y,z]$, which can not be algebraically shift equivalent to a nonsingular matrix, is given.