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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.

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The version of Mason's theorem with arbitrarily many summands (see my paper in Bull. Braz. Math Soc. 1985, or the paper of Brownawell and Masser in Math Proc Cam Phil Soc), gives that a linear dependence among $r$ terms of your set yields $H \le r(r+1)(n+1)/2$, or something like that. From that, you get an estimate on the dimension.

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