Cramer-type Formula for the Polynomial Solutions of Coupled Linear Equations with Polynomial Coefficients

Abstract

This paper derives a determinant form formula for the general solution of coupled linear equations with coefficients in $K[x_1,\dots ,x_n]$, where $K$ is a field of numbers, the number of unknowns is greater than the number of equations, and the solutions are in $K(x_1,\dots ,x_{n–1})[x_n]$. The formula represents the general solution by the minimum number of generators, and it is a generalization of Cramer's formula for the solutions in $K(x_1,\dots ,x_n)$. Compared with another formula which is obtained by a method typical in algebra, the generators in our formula are represented by determinants of quite small orders.