Algebraic Elements in Group Rings

Abstract

In this brief note, we study algebraic elements in the complex group algebra ${\mathbf {C}}[G]$. Specifically, suppose $\xi \in {\mathbf {C}}[G]$ satisfies $f(\xi ) = 0$ for some nonzero polynomial $f(x) \in {\mathbf {C}}[x]$. Then we show that a certain fairly natural function of the coefficients of $\xi$ is bounded in terms of the complex roots of $f(x)$. For $G$ finite, this is a recent observation of [HLP]. Thus the main thrust here concerns infinite groups, where the inequality generalizes results of [K] and [W] on traces of idempotents.