50 problems

Let $C_f$ be the hypersurface defined by a polynomial $f$, let $H_f$ be its Newton polytope, let $G(H)$ be the greatest common divisor of all coordinates of all elements of $H$, an…

Let a polynomial have saturated Newton polytope (SNP) if its support is M-convex and its Newton polytope contains no lattice points other than those in its support. Monical–Tokcan–…

For a symmetric function $f$, let $f(x_1,\ldots,x_k)$ denote its specialization obtained by setting $x_m=0$ for all $m\geq k+1$. A polynomial…

Let $p(z)$) be a maximally sparse polynomial, meaning that the support of $p$ is equal to the set of vertices of its Newton polytope. The Passare conjecture. The amoeba of $p(z)$ i…

Let $u=( u1, u2,ldots, ut)$ be a partition, and let $uvdash n$. For a skew partition $lambda/mu$ of length $n$, let $H(lambda,mu)$ denote its Jacobi–Trudi matrix, and let…

Let $f$ be a Newton non-degenerate homogeneous polynomial of degree $d\geq 3$ defining a homaloidal hypersurface. Let $D_0,\dots,D_n$ be the polytopes associated with $f$ in Propos…

Let $C$ be a cone and let $mathbf P\subset C\subset\mathbb{R}^n$ be a finite set whose convex hull $P=\operatorname{Conv}(\mathbf P)$ is convenient in $C$. Let $C_{\min}$ be the mi…

Let $d_{\operatorname{\mathbf{x}}}$ and $D_{\operatorname{\mathbf{x}}}$ denote the parameters governing the reported support bound, and let its accuracy be the percentage of the ac…

Theorem gives an inequality relating the analytic spread of a monomial ideal to the dimensions determined by its Newton polytope and supporting hyperplanes. Equality conjecture. Th…

Let $cA$ be the parameter space of maps associated with a triple $cbmA\inccIthth$ of integer polytopes, and let $cA_1$, $cA_2$, and $cA_3$ denote the indicated singularity types. L…

For $w\in S_n$, write … where $\theta_w$ is the number of supports of $\mathfrak{S}_w(x)$ and $p_u(w)$ counts occurrences of the permutation pattern $u$ in $w$. The coefficients ar…

For $n\geq 1$, let $\theta_w$ denote the number of supports of the Schubert polynomial $\mathfrak{S}_w(x)$, equivalently the number of lattice points in its Newton polytope, and de…

Maximal mixed-covolume conjecture. The maximal mixed covolume is

Generalized permutahedron conjecture. The Newton polytope $\mathbf{N}[\mathcal{F}]$ is a generalized permutahedron in the Euclidean regime for all kinematics, including exceptional…

Let ${\mathcal A}$ be a TSSS cluster algebra with principal coefficients. For a coefficient-free cluster monomial, let its Newton polytope be the convex polytope determined by its…

Landau-polytope facet-labeling conjecture. There are $1:1$ correspondences

Let $(S,M)$ be a marked surface and let $\mathcal{A}^{\mathrm{bd}}$ be the cluster algebra from $(S,M)$ with boundary frozen variables. For a triangulation $T$ and an arc $\gamma$,…

Let $(S,M)$ be a marked surface and let $\mathcal{A}^{\mathrm{bd}}$ be the cluster algebra from $(S,M)$ with boundary frozen variables. For a triangulation $T$ and an arc $\gamma$,…

Let $P$ be the Newton polytope of a cluster variable. Kalman's relative-interior conjecture. The polytope $P$ has no lattice points in its relative interior. The source attributes…

Let $\mathcal{A}(\Sigma)$ be a cluster algebra with principal coefficients at the seed $\Sigma$. A Newton polytope is the convex hull of the exponent vectors of the monomials occur…

Let an irregular GKZ system be associated with a Newton polytope, and let a facet of that polytope be a codimension-one face that does not contain the origin. Fang–Feng conjecture.…

Let a finite-type cluster algebra be given, and let an $F$-polynomial be one of its cluster-algebra $F$-polynomials. Its Newton polytope is the convex hull of the exponent vectors…

Yong's SNP conjecture. Every Schubert polynomial ${\mathfrak{S}}_\pi$ has the SNP property; moreover, the Newton polytope of each Schubert polynomial is defined by the inequalities…

Let $Q$ be a 2-acyclic quiver with a finite-dimensional Jacobian algebra, let ${\check{\delta}}$ be a $\delta$-vector, and suppose the theta and triangular basis elements have coef…

Let $f$ be a simplicial convenient non-degenerate function in three variables. Let $(\alpha_i,w_i)$, $i=1,\ldots,\mu_f$, be its spectral pairs; let ${\rm ICF}(f)$ and…