Eggert's Conjecture for 2-Generated Nilpotent Algebras

Abstract

Let A be a commutative nilpotent finitely-dimensional algebra over a field $F$ of characteristic $p \gt 0$. A conjecture of Eggert says that $p^. \operatorname{dim} A^{(p)} \operatorname{dim} A$, where $A^{(p)}$ is the subalgebra of $A$ generated by elements $a^p , a ∈ A$. We show that the conjecture holds if $A^{(p)}$ is at most 2-generated.