It is known that for affine and projective varities $V$ that it is possible to construct a polynomial such that for all $s$ sufficiently large, the polynomial is equal to the Hilbert function of $V$. The polynomial is called the Hilbert polynomial and it has the property that the degree of the polynomial is equal to the dimension of the variety, and that $(\dim V)!$ times the leading coefficient of the Hilbert polynomial is equal to the degree of the variety.

I am wondering what the analogous result would be for weighted projective varieties. A weighted projective variety is a variety in a weighted projective space, say with weight vector $(w_1, \cdots, w_n) \in \mathbb{Z}_{>0}^n$. Is it still true that the correct notion of degree for a variety $V$ in such a space be the product of $(\dim V)!$ times the leading coefficient of the Hilbert polynomial, or should the weight vector be factored in some how?

Thanks for any references or insights.