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.