Sharp estimates of the solutions to B{\'e}zout's polynomial equation and a corona theorem

TL;DR

This paper provides sharp estimates for solutions to B{é}zout's equation, extending corona theorem results to multipliers of certain de Branges--Rovnyak spaces, advancing understanding in complex analysis.

Contribution

It introduces new estimates for B{é}zout's solutions and extends corona theorem results to a broader class of function spaces.

Findings

Derived sharp solution estimates for B{é}zout's equation

Extended corona theorem to multipliers of de Branges--Rovnyak spaces

Connected classical results with modern function space theory

Abstract

In this paper, we obtain estimates for the solutions to the classical B{\'e}zout equation that are analogous to Carleson's solution to the corona theorem for the bounded analytic functions on the open unit disk. As an application, we extend some results of Luo and obtain a corona theorem for the multipliers of a class of de Branges--Rovnyak spaces.