The Polynomial Part of the Codimension Growth of Affine PI Algebras

TL;DR

This paper provides an algebraic interpretation of the polynomial factor in the codimension growth of affine PI algebras, confirming a conjecture and linking it to the algebra's structure.

Contribution

It establishes a formula for the polynomial part of codimension growth in affine PI algebras using Kemer's theory, especially for basic algebras.

Findings

The polynomial part t is given by (d - q)/2 + s for basic algebras.

Confirmed Giambruno's conjecture on the polynomial factor.

Linked codimension growth to algebraic structure via Kemer's theory.

Abstract

Let $F$ be a field of characteristic zero and $W$ be an associative affine $F$-algebra satisfying a polynomial identity (PI). The codimension sequence associated to $W$, $c_n(W)$, is known to be of the form $\Theta (c n^t d^n)$, where $d$ is the well known (PI) exponent of $W$. In this paper we establish an algebraic interpretation of the polynomial part (the constant $t$) by means of Kemer's theory. In particular, we show that in case $W$ is a basic algebra, then $t = \frac{d-q}{2} + s$, where $q$ is the number of simple component in $W/J(W)$ and $s+1$ is the nilpotency degree of $J(W)$. Thus proving a conjecture of Giambruno.