3 problems

Let $G(t)$ be the rational function … whose extra numerator factor is $(1+t+\cdots+t^{n-1})^2$. Equivalently, if $G(t)=\prod_i(1-t^i)^{-\beta(i)}$, then … Here $\bar{R}(n,k)$ denot…

For positive integers $m,n$, let $X_{m,n}$ be the closure of the image of the map … so that its homogeneous ideal $I(X_{m,n})$ consists of the algebraic relations among the…

Let $S_t(m,n)$ be the algebra generated by the $t$-minors of a generic $m\times n$ matrix, let $J_t(m,n)$ be its ideal of relations, and let ${\bf f}_{u,v}$, ${\bf g}_u$,…