Let $S=\{s_1,\ldots,s_n\}$ be a finite set of complex numbers. Consider the set of polynomials $$U=\{\,(x+s_1)^{k_1}\cdots(x+s_n)^{k_n}:\, 0\leq k_1+\cdots+k_n\leq H\}.$$
I am curious as to what is known on the dimension of the space spanned by $U$ as a function of $H$. My feeling is that one should be able to get pretty good control on this using variations of Mason's theorem.