58 problems

Consider a chemical reaction network (CRN) whose steady-state ideal satisfies Assumptions 1–3: mass-action kinetics, inclusion of conservation laws, and absence of independent subn…

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

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…

Four-line generation conjecture.

The Gröbner basis characterization. For each $n$, there exists a monomial order such that $\mathcal{C}$ is $0$- or $1$-inductively pierced if and only if the reduced Gröbner basis…

Moreno-Socías's conjecture for generic complete intersections. Then $\operatorname{in}(I)$ is almost reverse lexicographic.

Let $K\{\mathbf{X};P\}$ be the analytic power-series ring over the valued field $K$, with $P$ the parameter set of valuation vectors, and let $I\subset K\{\mathbf{X};P\}$ be an ide…

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…

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 ${\mathcal A}\subset\mathbb Z^d$ be a finite graded, normal configuration, and let $I_{\mathcal A}$ be its toric ideal. Write $\dim({\mathcal A})=d$ for the Krull dimension of…

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.

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…

Let $\mathbf{k}=(2)_{n+1}$ and $\mathbf{r}=(1,r_2,\dots,r_{n-1},1)$. For each $i$, let $G_i^{(r_i+1)}$ denote the set of $(r_i+1)$-minors of the flattening $\psi^{i-1}$, and let…

Let $\mathbf{k}=(k_0,\dots,k_n)$ and $\mathbf{r}=(r_1,\dots,r_n)$, let $\psi$ be a $k_0\times\cdots\times k_n$ tensor of indeterminates, and let $\psi^i$ denote its flattenings. Th…

Let $n$ be a positive integer, let $\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}$ be the space of symmetric $n\times n$ matrices, and let $I^{\mathrm{sym}}_w$ and…

Let $n$ be a positive integer, let $\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}$ be the space of symmetric $n\times n$ matrices, and for…

Initial-ideal decomposition conjecture. The initial ideal decomposes as

Prime-product coefficient conjecture. Each coefficient occurring in the Gröbner basis of $I_{a_1,\ldots,a_{n+1}}$ is a possibly empty product of primes less than…

$k$-ary Gröbner-basis finiteness conjecture. For all $k\in\mathbb{N}$ and $d\equiv1\pmod{k-1}$, the reduced Gröbner basis for the nonsymmetric operad encoding the identity…

Ternary Gröbner-basis finiteness conjecture. For each $d\in\mathbb{N}$, the reduced Gröbner basis for the nonsymmetric operad encoding the identity $w_{2d+1}^{(3)}=0$ is finite.

Quadraticity–quadratic Gröbner basis conjecture. The algebra $\operatorname{A}(X_{n,d}^G)$ is quadratic if and only if $\operatorname{I}(X_{n,d}^G)$ has a quadratic Gröbner basis.

Let $I(z_1,\ldots,z_n)$ be an $L$-loop integral with nonnegative indices, and let $D_1,\ldots,D_n$ be the propagator generators in the associated noncommutative rational double-shi…