3 problems

Depth criterion conjecture. One has

Let $r\geq 1$ and $s\geq 1$ be integers. Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring with each $\deg x_i=1$, and let $I\subset S$ be a homogeneous monomial ideal. Write…

Let $S=\mathbb{K}[x_1,\dots,x_n]$, and let $I\subset S$ be a lexsegment ideal generated by monomials of degree $d$. Write $\mu(I)$ for the number of minimal generators of $I$. Lexs…