Zero-nonzero patterns for nilpotent matrices 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 an investigation is initiated 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. A necessary condition on F is observed for a pattern to be potentiallynilpotent when the associated digraph has m loops but no small k-cycles, 2 ≤ k ≤ m − 1. As part of this investigation, methods are developed, using the tools of algebraic geometry and commutative algebra, to eliminate zero-nonzero patterns A as being potentially nilpotent over any field F. These techniques are then used to classify all irreducible zero-nonzero patterns of order two and three that are potentially nilpotent over Z_p for each prime p.