113 problems

Let $I(a_{1}+m,dots,a_{n}+m)$ denote the toric ideal associated to the sequence $(a_{1}+m,dots,a_{n}+m)$, where $m\geq 0$ is an integer. HerzogSrinivasan conjecture. The Betti numb…

Diaconis–Eriksson conjecture. The ideals of the Birkhoff polytopes are generated in degree $3$.

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…

Bøgvad's conjecture. The toric ideal $I_P$ is quadratic.

Let $A$ be a $d \times n$ matrix with $n-d=3$, and let $\mathcal{G}_A$ denote its Gröbner fan. Sturmfels–Thomas conjecture. Every maximal cone in $\mathcal{G}_A$ has three or four…

Let $A$ be an integer matrix, let $I_A$ be its toric ideal, and define the $A$-degree of a binomial $x^u-x^v$ with $\deg_A(x^u)=\deg_A(x^v)$ by $\deg_A(x^u-x^v)=\deg_A(x^u)$. A pri…

Let $G$ be a graph and let $I_G$ be its cut ideal. A graph is $K_5$-minor-free when it has no minor isomorphic to $K_5$. Quartic generation conjecture. … This extends the proposed…

Let $G$ be a simple graph, let $A_G$ be its incidence matrix, let $I_G$ be the corresponding toric ideal, and let $M$ be a minimal Markov basis for $I_G$. A Markov basis is distanc…

Let $I_A$ be a toric ideal, and let the true degree of a circuit of $A$ mean the degree computed without dividing by the common factor $c$ in the circuit formula. Sturmfels's conje…

Distance-irreducible basis conjecture. If $M$ is not distance reducing, then

Let $M$ be a matroid with ground set $E$. A basis sequence is a finite sequence of bases of $M$. Two basis sequences are equivalent if one can be obtained from the other by a compo…

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…

White's $TE$ conjecture. All matroids satisfy each of the three properties:

Klein four-group phylogenetic complexity conjecture. The phylogenetic complexity of the group $G=\mathbb{Z}_2\times\mathbb{Z}_2$ is four:

Finite abelian group bound. For any finite abelian group $G$ and any positive integer $n$,

Corank-three four-neighbor conjecture. Every vertex in the state polytope of $I_A$ has at most four neighbors.

State-polytope valency conjecture. The maximum valency of a vertex in the state polytope of $I_A$ is bounded above by a function depending only on the corank of $A$.

Let $G$ be a graph, let $I_G$ be its cut ideal, and let $\mu(I_G)$ denote the maximal degree of a binomial in a minimal generating set of $I_G$. A graph $H$ is a minor of $G$ if it…

Let $M$ be a matroid, with toric ideal $I_M$. For bases $B_1,B_2,D_1,D_2$, call $y_{B_1}y_{B_2}-y_{D_1}y_{D_2}$ a double-swap binomial when the pair $D_1,D_2$ is obtained from…

Let $M$ be a matroid and let $I_M$ be its toric ideal. A quadratic binomial in $I_M$ is a binomial homogeneous of degree two in the base variables. The quadratic-generation conject…

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 $M$ be a matroid on the ground set $\{1,2,\ldots,n\}$. Fix a field $k$, let $S_M=k[y_B:B\text{ a base of }M]$, and let $I_M$ be the kernel of the homomorphism…

Let $S_n$ be the symmetric group, and let $I_{S_n}$ denote the toric ideal associated with the Fourier model for ranked data on $S_n$. Degree-three generation conjecture. The ideal…