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[x₁, ..., 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(x₁, ..., 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(x₁, ..., 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.