Bezout equations over bivariate polynomial matrices related by an entire function

Abstract

This study addresses Bezout equations over bivariate polynomial matrices, where the relationship between two variables is described by a real entire function. This paper proposes a solution method that makes optimal use of minor primeness to reduce such Bezout equations to simple equations over univariate scalar polynomials. The proposed solution method requires only matrix calculations, thus making it very useful, especially in the absence of modern computer algebra systems.

Keywords:

• bivariate polynomial matrix
• Bezout equation
• primeness
• entire function

AMS Subject Classifications:

• 15A24
• 13P05
• 13P25

Notes

1 If is not minor right prime, using the algorithm presented in [8], from we can obtain a minor right prime satisfying . Then, it is sufficient to replace in Equation (5.75.7 ) by .

2 It can be proved by mathematical induction that and , , can be represented by bivariate polynomials in and with real coefficients. For example, we have and .

This work was supported by [grant number 25400128], Grant-in-Aid for Scientific Research (C), The Ministry of Education, Culture, Sports, Science and Technology, Japan.