Retracts of Laurent polynomial rings

TL;DR

This paper investigates the structure of retracts of localized Laurent polynomial rings over an integral domain, establishing conditions under which these retracts are themselves Laurent polynomial rings or their variants.

Contribution

It characterizes retracts of localized Laurent polynomial rings, showing they are Laurent polynomial rings or similar structures under specific conditions.

Findings

Retracts of $B[1/M]$ with $M=\prod_{i=1}^n x_i$ are Laurent polynomial rings over $R$.

When $R$ is a perfect field and $n=3$, retracts are isomorphic to rings with Laurent variables and additional variables.

The results extend understanding of the algebraic structure of retracts in polynomial and Laurent polynomial rings.

Abstract

Let $R$ be an integral domain and $B=R[x_1,\ldots,x_n]$ be the polynomial ring. In this paper, we consider retracts of $B[1/M]$ for a monomial $M$. We show that (1) if $M=\prod_{i=1}^nx_i$, then every retract is a Laurent polynomial ring over $R$, (2) if $R$ is a perfect field and $n=3$, then every retract is isomorphic to $R[y_1^{\pm1},\ldots,y_s^{\pm1},z_1,\ldots,z_t]$ for some $s,t\geq 0$.