Generalized typical dimension of a graded module

In this paper, we prove an upper bound for the leading coefficient of the characteristic polynomial of a graded ideal in a ring of generalized polynomials. Examples of such rings are the rings of commutative polynomials (for which the classical Bézout theorem holds), as well as some rings of differential operators. For a system of generalized homogeneous equations in small codimensions we obtain exact estimates that are polynomial in $d$. In the general case, the estimate is double exponential in $\tau$: $O\bigl(d^{2^{\tau-1}}\bigr)$, where $d$ is the maximal degree of generators of a graded ideal and $\tau$ is its codimension. For systems of linear differential equations, bounds of the same asymptotics, but by other methods, were obtained by D. Grigoriev.