Cramer-type formula for the polynomial solutions of coupled linear equations with polynomial coefficients (Q1071067)

This paper is a contribution to the study of the systems of linear equations with coefficients in a domain which is not a field. We mention that there are general results on this problem [\textit{H. Matsumura}, Honam Math. J. 4, 13-21 (1982)] in the case of Dedekind domains. The paper under review presents effective methods in some important particular cases. The author studies the solutions of a linear system of r equations in s indeterminates \((r<s)\) with coefficients in a ring \(K[x_ 1,...,x_ n]\), where K is a field of numbers and \(x_ 1,...,x_ n\) are indeterminates over K. He obtains effective algorithms for determining the solutions of the above system that are in the ring \(K(x_ 1,...,x_{n-1})[x_ n]\). The solutions are represented by determinant formulas. The results obtained are compared with a result of \textit{K. Hentzelt} presented in \textit{G. Hermann}'s paper on effective algorithms in polynomial rings [Math. Ann. 95, 736-788 (1926)].