The Canonical Diophantine Equations with Applications

A new fundamental relationship involving polynomial matrices, called the canonical Diophantine equations, is introduced and then employed to constructively resolve some "standard problems" involving polynomial matrix pairs. In particular, one such "standard problem" involves solving the general Diophantine equation, H(s)R(s) + K(s)PR(s) = F(s), for an appropriate polynomial matrix pair, H(s), K(s), given any arbitrary F(s). The Bezout equation, when F(s) = I, would represent a special case of the general Diophantine equation. The question of obtaining a dual, prime factorization of T(s) from any given (not necessarily prime) factorization could also be considered another "standard problem" and finally, the determination and possible extraction of a greatest common right or left divisor of a given pair of polynomial matrices could be considered the third and final "standard problem."