An analytic approach to estimating the solutions of Bézout's polynomial identity

This paper contains sharp bounds on the coefficients of the polynomials $R$ and $S$ which solve the classical one variable Bézout identity $AR+BS=1$, where $A$ and $B$ are polynomials with no common zeros. The bounds are expressed in terms of the separation of the zeros of $A$ and $B$. Our proof involves contour integral representations of these coefficients. We also obtain an estimate on the norm of the inverse of the Sylvester matrix.