A Property of Euclid’s Algorithm and an Application to Padé Approximation

Abstract

If a and b are fixed polynomials with $\deg ( a ) > \deg ( b )$, we show that all solutions to the congruence $qb \equiv p( {\bmod a} )$ with $\deg ( q ) + \deg ( p ) < \deg ( a )$ can be obtained via Euclid’s algorithm. Using this result, we show that the Padé approximants to a given power series can also be obtained from Euclid’s algorithm.