Arbitrariness of the general solution of the partial differential equations and its applications

Abstract

Using the framework of formal theory of partial differential equations, we consider a method of computation of the bi-Hilbert polynomial (i.e. Hilbert polynomial in two variables). Furthermore, present an approach to compute the number of arbitrary functions of positive differential order in the general solution. Then, under the "AC=BD" model for mathematics mechanization developed by Hong-qing ZHANG, we present a method to reduce an overdetermined system to a well-determined one. As applications, the Maxwell equations and weakly overdetermined equations are considered.