Thoughts on Eggert's Conjecture

TL;DR

This paper explores Eggert's Conjecture, a mathematical statement about the dimensions of certain algebraic structures over fields, examining evidence, variants, implications, and related questions.

Contribution

It provides a comprehensive analysis of Eggert's Conjecture, including heuristic evidence, variants beyond positive characteristic, and implications for semigroup theory.

Findings

Heuristic evidence supports the conjecture.

Identifies variants of the conjecture beyond positive characteristic.

Provides examples where the inequality becomes equality.

Abstract

Eggert's Conjecture says that if R is a finite-dimensional nilpotent commutative algebra over a perfect field F of characteristic p, and R^{(p)} is the image of the p-th power map on R, then dim_F R \geq p dim_F R^{(p)}. Whether this very elementary statement is true is not known. We examine heuristic evidence for this conjecture, versions of the conjecture that are not limited to positive characteristic and/or to commutative R, consequences the conjecture would have for semigroups, and examples that give equality in the conjectured inequality. We pose several related questions, and briefly survey the literature on the subject.