9 problems

The $(A_1,A_4)$ Gröbner-basis conjecture. The monomials in $\mathcal{G}(\ell,m)$ constitute a Gröbner basis for $\langle J(x)^2\rangle$, with the recursive description of…

Let $I_{n,k}$ be the determinantal ideal and let $preceq$ be the term order defined in the paper. An initial ideal is called spherical when it satisfies the stated Stanley–Reisner…

Let $R$ be a strongly discrete coherent ring. For $n\a0\geq 1$, let $R[X_1,\ldots,X_n]$ be a polynomial ring, and let $\prec$ be a rational monomial order on its monomials. For a f…

Mascia et al.'s conjecture. $\mathcal{P}$ is prime if and only if $\mathcal{P}$ contains no zig-zag walk.

For the polynomial variables ordered by … use degree-lexicographic ordering, and let $G_n$ be the Gröbner basis of the relevant ideal. For $I_{\ell,m}$, define … with $F_z$ and $F$…

Let $R=\mathbb{Q}[a,\ldots,i]$, and let $I$ be the toric ideal … This ideal defines a projectively normal toric variety. Sturmfels's conjecture. None of the initial ideals of $I$ d…

Let $L$ be a modular lattice, let $K[L]$ be its associated polynomial ring, and let $I_L$ be its ideal generated by the Hibi relations for incomparable pairs in $L$. For a monomial…

Let $v,w$ be permutations with $v\leq w$, and let $I_{v,w}$ denote the Kazhdan–Lusztig ideal. Suppose that $I_{v,w}$ is standardly homogeneous. If $ws_i<w$ but $vs_i>v$, then…

Let $Y(z)=(Y_1(z),\ldots,Y_n(z))$ be an $n$-tuple of invertible $d\times d$-matrices over $\mathbb{Q}(z)$, and let $\mathcal{B}$ be the Bruhat–Tits building. The matrices are not a…