3 problems

Let $k[x_1,\dots,x_n]$ be a polynomial ring, and let $I=(f_1,\dots,f_r)$ be a generic ideal whose generators satisfy $\deg f_i=d_i$. For a power series $F(T)$, write $[F(T)]$ for i…

Let $K$ be a field, let $R=K[x_1,\ldots,x_n]$, and let $I=(f_1,\ldots,f_n)$ be a generic homogeneous ideal of type $(n;d_1,\ldots,d_n)$. Let $\operatorname{in}(I)$ be the initial i…

Let $V_n$ be an $n$-dimensional vector space, let $\mathfrak{S}(V_n)$ denote the square-free algebra, and let $I_n$ be a generic ideal with generators of degrees $d_1,\dots,d_r$. W…