28 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 $f$ 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 $cA$ 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…

Let $S_n$ be the symmetric group on $n$ letters. For permutations $u,w\in S_n$, let $D_u^w$ denote the dual Schubert polynomial, and let SCNP denote the property that its Newton po…

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…

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…

Saturated Newton polytope conjecture. Every double Schubert polynomial ${\mathfrak{S}}_\pi({\normalfont\mathbf{t}}, {\normalfont\mathbf{s}})$ has the Saturated Newton Polytope prop…

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 $\mathcal{A}(\Sigma)$ 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 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…

Non-SAGE separation conjecture. If every $\bm{a}_i$ lies in either $\operatorname{ext}\mathcal{P}(\bm{A})$ or $\operatorname{int}\mathcal{P}(\bm{A})$, but $\bm{A}$ does not satisfy…

Let $\mathcal L_\alpha$ be the Lascoux atom obtained by replacing the Demazure operator in the definition of the key polynomial with $\widehat\tau_i=\tau_i-\operatorname{id}$, wher…

Let $\alpha\in\mathbb Z_{\geq0}^{\infty}$, and let $\Omega_\alpha$ be the inhomogeneous analogue of the key polynomial obtained by replacing the Demazure operators with…

For $w\in S_\infty$, let ${\mathfrak G}_w$ be the Grothendieck polynomial obtained recursively using $\overline\pi_i(f)=\partial_i((1-x_{i+1})f)$. A polynomial has SNP when every l…

Let $X=\{x_1,x_2,\ldots\}$ and $Y=\{y_1,y_2,\ldots\}$, and let ${\mathfrak S}_w(X;Y)$ be the double Schubert polynomial defined recursively from the product for the longest permuta…

For $w\in S_\infty$, let ${\mathfrak S}_w$ be the Schubert polynomial defined recursively from the longest permutation using divided-difference operators. A polynomial has SNP when…

Let $\alpha$ be a weak composition, and let $A_\alpha$ be the Demazure atom obtained from $x^\alpha$ by applying the operators $\widehat\pi_i=\pi_i-\operatorname{id}$ according to…

For compositions $\gamma$ and $\alpha$, write $\gamma\preceq\alpha$ when $\lambda(\gamma)=\lambda(\alpha)$ and $w(\gamma)\leq w(\alpha)$ in Bruhat order, where $\lambda(\gamma)$ is…

Let $\alpha=(\alpha_1,\alpha_2,\ldots)\in\mathbb Z_{\geq0}^{\infty}$ have finite size, and let $\kappa_\alpha$ be the key polynomial defined using Demazure operators. A polynomial…

Let $\alpha\in\mathbb Z_{\geq0}^n$, and let $E_\alpha(X;q,t)$ be the nonsymmetric Macdonald polynomial. Write … where $<_S$ is the ordering generated by the stated transposition an…