Let $V=\mathbb{C}[x_1^{1}, \ldots , x_1^{m_1}, \ldots , x_n^{1}, \ldots , x_n^{m_n}]_d$ be the vector space of polynomials of degree $d$. Let $W \subseteq V$ be a Zariski closed subset. Consider the map $\Phi : V \to \mathbb{C}[x_1, \ldots , x_n]_d, f \mapsto f(x_1,\ldots, x_1, \ldots, x_n, \ldots ,x_n)$ obtained by equating $x_i^{1}, \ldots , x_i^{m_i}$ for all $i$. (This map is linear in the coefficients of $f$.) Are there some results about the restriction map $\Phi|_W$? If not in general, perhaps for some non-trivial special cases? Perhaps realized as some quotient map of a group action?
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