m-Hilbert Polynomial and Arbitrariness of the General Solution of Partial Differential Equations

Using the framework of formal theory of partial differential equations, we consider a method of computation of the m-Hilbert polynomial (i.e. Hilbert polynomial with multivariable), which generalizes the Seiler's theorem of Hilbert polynomial with single variable. Next we present an approach to compute the number of arbitrary functions of positive differential order in the general solution, and give a formally well-posed initial problem. Finally,as applications the Maxwell equations and weakly over determined equations are considered.