110 problems

Let $S=k[X_0,\dots,X_n]$ be a polynomial ring over a field $k$, let $I\subseteq S$ be an ideal, and let $<$ be a monomial order such that the initial ideal $\operatorname{in}_<(I)$…

Let $\mathcal C$ be a neural code labeled so that neuron $i$ is added as a piercing at the $i$th step, and let $T_\mathcal C$ be its toric ideal. Let $\prec$ be the monomial order…

High-accuracy conjecture. High accuracy should still be achievable over the field of rational numbers when

Strong and weak conjectures. The strong conjecture asserts that all essential specializations are singular. The weak conjecture asserts

Initial-ideal decomposition conjecture. The initial ideal decomposes as

Let $e_{i,n}(x)$ denote the $i$th elementary symmetric function in $x_1,\dots,x_n$, and let $h_{i,n-i+1}(x)$ denote the corresponding complete homogeneous symmetric function in the…

Four-line generation conjecture.

Graver-basis conjecture. Every robust toric ideal is minimally generated by its Graver basis ${\mathcal Gr}(A)$. Consequently, robust toric ideals would be precisely the…

Let $P$ and $Q$ be finite posets with $|P|=|Q|=d$. For the toric ideal $I_{\Omega(P,-Q)}$ associated with the twinned order polytope $\Omega(P,-Q)$, quadratic generation conjecture…

Let … be the likelihood function on positive $4 \times 4$ matrices $P=(p_{ij})_{1 \leq i,j \leq 4}$ of rank at most two. Sturmfels's conjecture. The matrix … is a global maximum of…

One-singular-reduction conjecture. If $p\in P_2$ has a unimodular gradient, then one can get

Let $M$ be a matroid and let $I_M$ be its toric ideal, the kernel of the map sending each base variable $y_B$ to the corresponding squarefree monomial. The matroid quadratic Gröbne…

Let $I$ be a toric ideal defining a projectively normal $r$-dimensional toric variety. The normality Gröbner-basis conjecture. The ideal $I$ has a Gröbner basis consisting of binom…

Circular lexicographic delightfulness conjecture. The circular lexicographic term order is $2$-delightful for $I_{p,1}$. More specifically, the reduced Gröbner basis for $I_{p,2}$…

Let $L=(L_{ij})$ be an $m\times n$ matrix whose entries have the form … with $a_{ijk}\in K$ for all $i,j,k$. Assume that, for each $i$, the forms $L_{i1},\dots,L_{in}$ are linearly…

Let $t_{ij}$ be distinct variables over a field $K$ with $1\leq i\leq m$ and $1\leq j\leq n$. Let $L=(L_{ij})$ be an $m\times n$ matrix with … where $a_{ijk}\in K$ for all $i,j,k$.…

Let $w$ be a word in the generators $x,x^*$, and let $w^*$ be its image under the involution exchanging $x$ and $x^*$. Consider the $*$-algebra … A word is unshrinkable if it canno…

Let $k={\mathbb{C}}$, let $m\geq 2$, and consider the generators of the ideal in Theorem Li–Li analogue, namely … Here the ideal defines the set of points in ${\mathbb{C}}^n$ for w…

Let ${\mathbb K}$ be a field and let ${\cal G}^r_{2,\mathbb K}$ be the graph whose vertices are the monomial ideals of colength $r$ in ${\mathbb K}[x,y]$, with two vertices joined…

Sturmfels's conjecture. There exists a Gröbner basis for $I_{S_P}$ consisting of binomials of degree at most $d+1$.

Let $M=(E,\mathcal B)$ be a matroid with matroid base polytope $P(M)$, and let $I_{P(M)}$ be its toric ideal. A triangulation of $P(M)$ is called quadratic when its non-faces have…

Let $U$ be a two-dimensional vector space, let $a,b$ be positive integers, and write $S=\operatorname{Sym}(\operatorname{Sym}^b U)$. Let…

Koszul filtration conjecture. The quotient $S/(G)$ has a Koszul filtration. Moreover, if $(G)$ is toric, then $S/(G)$ has a monomial Koszul filtration.

Ene–Herzog–Hibi conjecture. G-quadraticity need not imply the existence of a Koszul filtration.

Let $r\in\mathbf{Z}_{\geq 2}$, let $\mathcal{S}$ be the ring in which the ideal $\mathfrak{a}$ is defined, and let $J_r$ be the ideal generated by the monomials described above: th…