5 problems

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 $S=K[x_1,\ldots,x_n]$ and let $f_1,f_2,\ldots,f_n\in S$ be a generic sequence of homogeneous polynomials of degrees $d_1,d_2,\ldots,d_n$. Set $I=(f_1,\ldots,f_n)$ and $A=S/I$.…

Let $V_n$ be an $n$-dimensional vector space, let $I_n=(f_1,\dots,f_r)$ be a generic ideal in the exterior algebra on $V_n$, and suppose that $f_i$ has degree $d_i$. Let $q_n(t)$ b…

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…

Let $K$ be a field, let $V_n$ be an $n$-dimensional vector space, and let $I$ be a generic homogeneous ideal in the exterior algebra on $n$ generators, with numerical character…