9 problems

Let $n\ge3$ and let $I_n=\langle p_n,q_n\rangle\subset\mathbf Z[z,t]$, with $p_n$ and $q_n$ as defined in the paper. The intersection $I_n\cap\mathbf Z[t]$ is an ideal of…

Let $n\ge3$ be an integer, write $n=2s-1$ when $n$ is odd and $n=2s$ when $n$ is even, and let $p_n,q_n\in\mathbf Z[z,t]$ and $r_n\in\mathbf Z[t]$ be the polynomials defined in the…

Let $I$ be an ideal of polynomials in several variables, viewed with the relative topology induced by the Hardy space of the unit polydisk. Let $V(I)$ denote the variety of $I$, an…

Let $q$ be a prime power, let $n>1$, and let $S\subset\F_q[X_1,\dots,X_n]$ be a minimal generating set such that $\deg f\leq d$ for every $f\in S$. Here $\Theta_{n,q}$ denotes a po…

Let $\mathcal{I}_{h,n}$ be a family of polynomial ideals in the variables $X_1,\ldots,X_n$, and let $e_i(\mathbf{X}_n)$ denote the elementary symmetric polynomials. The family is r…

Generator-growth conjecture. The number of minimal generating elements of the ideal will increase dramatically as the dimension increases, thereby making Buchberger's algorithm com…

Let $m$ be a positive integer. For $0\leq i\leq m-1$ and $0\leq j\leq m-i$, let $H_{ij}(m)$ and $D(m,k)$ be the polynomials defined in the paper, and let…

Completely continuous polynomial ideal conjecture. The class $\mathcal{CC}\circ\mathcal{P}$ is not a polynomial two-sided ideal.

Let $\mathcal M$ be a Hilbert module of holomorphic functions on a bounded open connected subset $\Omega$ of $\mathbb C^m$, and let $\mathcal I\subseteq\mathbb C[\underline{z}]$ be…