Morphism between affine spaces of polynomials by setting equal variables - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T04:03:43Z http://mathoverflow.net/feeds/question/111068 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111068/morphism-between-affine-spaces-of-polynomials-by-setting-equal-variables Morphism between affine spaces of polynomials by setting equal variables Döni 2012-10-30T10:39:49Z 2012-10-30T10:39:49Z <p>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?</p>