Regularity of $R(X)$ does not pass to finite unions

TL;DR

The paper demonstrates that the regularity property of the algebra of rational functions on compact sets does not necessarily extend to finite unions of such sets, providing specific counterexamples.

Contribution

It introduces counterexamples showing that regularity of $R(X)$ and $R(Y)$ does not imply regularity of $R(X up Y)$, highlighting a limitation in the behavior of rational function algebras.

Findings

Existence of compact sets with regular $R(X)$ and $R(Y)$ but non-regular $R(X up Y)$

Counterexamples in the plane for rational function regularity

Demonstrates non-passivity of regularity property under finite unions

Abstract

We show that there are compact plane sets $X$, $Y$ such that $R(X)$ and $R(Y)$ are regular but $R(X \cup Y)$ is not regular.