The Exact Solution of Systems of Linear Equations with Polynomial Coefficients

An algorithm for computing exactly a general solution to a system of linear equations with coefficients that are polynomials over the integers is presented. The algorithm applies mod-p mappings and then evaluation mappings, eventually solving linear systems of equations with coefficients in GF(p) by a special Gaussian elimination algorithm. Then by applying interpolation and the Chinese Remainder Theorem a general solution is obtained. For a consistent system, the evaluation-interpolation part of the algorithm computes the determinantal RRE form of the mod-p reduced augmented system matrices. The Chinese Remainder Theorem then uses these to construct an RRE matrix with polynomial entries over the integers, from which a general solution is constructed. For an inconsistent system, only one mod-p mapping is needed. The average computing time for the algorithm is obtained and compared to that for the exact division method. The new method is found to be far superior. Also, a mod-p/evaluation mapping algorithm for computing matrix products is discussed briefly.