On the Bezout equation in the ring of periodic distributions

TL;DR

This paper establishes a corona theorem for the ring of periodic distributions with polynomially growing Fourier coefficients and determines that both the Bass and topological stable ranks of this ring are equal to 1.

Contribution

It provides a corona theorem for the ring of periodic distributions and computes its stable ranks, advancing the understanding of its algebraic and topological structure.

Findings

Corona theorem for the ring of periodic distributions

Bass stable rank of the ring is 1

Topological stable rank of the ring is 1

Abstract

A corona type theorem is given for the ring R of periodic distributions in R^d in terms of the sequence of Fourier coefficients of these distributions, which have at most polynomial growth. It is also shown that the Bass stable rank and the topological stable rank of R are both equal to 1.