I wonder if something like the following is true: let $V$ be a finite-dimensional vector space over a field of characteristic zero and $S \subset V$ a Zariski-dense subset. Does $\{ v^d \ | \ v \in S \}$ span $\text{Sym}^d(V)$? If $S$ is a full lattice this is classical, but perhaps the proof of that result could be simplified by arguing in this way?
Powers of Zariski-dense subset span symmetric power
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linear-algebra
algebraic-geometry