Growth for the central polynomials

TL;DR

This paper investigates the growth behavior of central polynomials in specific algebraic structures, revealing that matrix algebras satisfy numerous proper central polynomials, which enhances understanding of their polynomial identities.

Contribution

It provides new insights into the growth rates of central polynomials for the Grassmann algebra and matrix algebras, highlighting their abundance of proper central polynomials.

Findings

Matrix algebras satisfy many proper central polynomials

Growth rates of central polynomials are characterized for specific algebras

Enhanced understanding of polynomial identities in algebraic structures

Abstract

We study the growth of the central polynomials for the algebras $G$ and $M_k(F)$, the infinite dimensional Grassmann algebra and the $k\times k$ matrices over a field $F$ of characteristic zero. In particular it follows that $M_k(F)$ satisfy many proper central polynomials.