Zero-nonzero patterns for nilpotent matrices over finite fields

TL;DR

This paper explores which zero-nonzero matrix patterns can be potentially nilpotent over finite fields, developing algebraic methods to classify such patterns of small size over Z_p.

Contribution

It introduces algebraic geometry and commutative algebra techniques to identify potentially nilpotent patterns and classifies all irreducible patterns of size two and three over Z_p.

Findings

Classified all irreducible patterns of order two and three over Z_p

Developed algebraic methods to eliminate non-potentially nilpotent patterns

Provided a framework for analyzing zero-nonzero patterns over finite fields

Abstract

Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper we initiate an investigation into which zero-nonzero patterns are potentially nilpotent over F, with a special emphasis on the case that F = Z_p is a finite field. As part of this investigation, we develop methods, using the tools of algebraic geometry and commutative algebra, to eliminate zero-nonzero patterns A as being potentially nilpotent over any field F. We then use these techniques to classify all irreducible zero-nonzero patterns of order two and three that are potentially nilpotent over Z_p for each prime p.