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

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