Zero-degree solutions to \(AX+BY=C\) and invariant factors assignment problem (Q1086318)

The paper deals with the polynomial Diophantine matrix equation \(AX+BY=C\) and, for given A, B, C, with the following two problems: 1. Find X, Y such that deg X\(=0\) and deg \(Y\leq \max (\deg A\), deg C) - deg B; 2. Find X, Y such that \(\deg [X^ T\), \(Y^ T]^ T=0\). Necessary and sufficient conditions for the existence of the solutions are respectively proved. For problem 1 and illustrative example is given. Finally it is shown that the invariant factors assignment by state feedback can be reduced to the problem 2.