9 problems

Let $S$ be a polynomial ring over a field, and let $I\subseteq S$ be an ideal generated by $n$ quadrics and of height $h$. Ananyan–Hochster's quadratic bound conjecture. One has ……

Eliahou–Villarreal conjecture. If the edge ideal of a connected graph $G$ has a 2-linear resolution, then

Let $n_1$ be a positive integer, and let $R_{n_1,1}$ be the ring and $I_{L_2(n_1,1)}$ the ideal appearing in the paper, with $\mathrm{in}_{\prec}(I_{L_2(n_1,1)})$ denoting its init…

Monotonicity conjecture. Let $r,n\geq 2$ be arbitrary integers. If $(p,r)\in\operatorname{pdreg}(n)$, then $(p,r-1)\in\operatorname{pdreg}(n)$. The preceding results show that the…

Polynomial Stillman-bound conjecture. For all characteristics, the projective dimension of $R/I$ is at worst

Let $d_1,\dots,d_r$ be positive integers. For homogeneous polynomials $f_1,\dots,f_r$ of degrees $\deg(f_i)=d_i$ in a polynomial ring … where $n$ is arbitrary, consider the quotien…

Let $M$ be a monomial ideal minimally generated by more than one monomial. Suppose that some minimal generator divides the least common multiple of every pair of generators. Projec…

Converse characterization conjecture. Then $\operatorname{pd}(\mathcal{H})=|V(\mathcal{H})|=2$ if and only if $\mathcal{H}$ satisfies $(\sharp)$. The authors are asking whether thi…

Let $K$ be a field and let $R=K[x_1,\ldots,x_N]$ be a polynomial ring. For fixed positive integers $n$ and $d_1,\ldots,d_n$, let $I\subseteq R$ be an ideal generated by $n$ homogen…