40 problems

Let $\mathcal{A}$ be the tuple of supports defining a sparse resultant, let $\mathcal{H}_{\mathcal{A},\rho}$ be the Canny–Emiris matrix whose rows encode the coefficients of the po…

Let $A$ be a configuration, let $D_A$ denote its discriminant, and let a rational $A$-hypergeometric function be a rational solution associated with $A$. A resultant…

A rational hypergeometric function is a function associated with a toric configuration whose denominator may involve discriminants of that configuration. A resultant is a special d…

Let $C$ be the trigonal genus-three curve defined by $f(x,y)=0$, with coefficients $\mu_1,\mu_2,\mu_3,\mu_4,\mu_5,\mu_6,\mu_8,\mu_9,\mu_{12}$. For variables $x$ and $y$, write…

Let $L_0(s,\xi)$ be the greatest common divisor of the polynomials $\sigma(L_1)(s,\xi),\ldots,\sigma(L_k)(s,\xi)$, and let the residual resultant be the resultant of…

Let $m$ be fixed, let $f(x)=f_0+\cdots+f_mx^m$, and let $g(x)=g_0+\cdots+g_nx^n$ be a degree $n$ polynomial. Here $H$ denotes the height of an integer polynomial or integer, and…

Let ${\mathbb A}$ be the coefficient ring, let $P_h\in{\mathbb A}$ denote the factor associated with a monomial $h$ of critical degree, and let $\ell$ be the index of the lattice s…

Unit-coefficient conjecture. The constant $c=\pm1$.

Let $f_1,\dots,f_n$ be polynomials such that ${\mathbb M}^0$ is a basis of ${\mathcal A}$. Let $g\in{\mathbb K}[x_1,\dots,x_n]$, and let ${\mathcal B}$ be the matrix of multiplicat…

The resultant formula. If $V_1(A)=\emptyset$ and $V_2(A)\ne\emptyset$, then, up to a nonzero constant,

Let $a,b,c,d$ be the bihomogeneous polynomials considered in the construction of the matrices $M$ and $Q'$, and let $\operatorname{Res}(a,b,c)$ denote their resultant. ZCG determin…

Resultant stability conjecture. For any positive integers $n,m$ such that $n\leq m$, and any $a>0$, the resultant $\operatorname{res}_z(P_n(s,z),P_m(s+a,z))$ is stable.

For positive integers $m,n,H$, let … where $\mathrm{H}(h)$ is the maximum absolute value of the coefficients of $h$ and $L(h)$ is its leading coefficient. For coprime integer polyn…

For positive integers $m,n,H$, let … where $\mathrm{H}(h)$ is the maximum of the absolute values of the coefficients of $h$ and $L(h)$ is its leading coefficient. For coprime integ…

Let $n\geqslant 2$ and let … Landesman's conjecture. There exists $m\geqslant 0$ such that … The conjecture predicts that, for every fixed degree $n\geqslant 2$, the quadratic poly…

Regular-sequence formulation of the Casas–Alvero conjecture. For every $i\in\{1,\ldots,d-2\}$, $R_{i+1}$ is not a zero divisor modulo $(R_1,\ldots,R_i)$; equivalently,…

Nullstellensatz formulation of the Casas–Alvero conjecture. The conjecture is equivalent to

Let $q$ be an odd prime, and let $R_q(y)$ be the resultant polynomial defined in the paper. A polynomial in $\mathbb Z[y]$ is Eisenstein with respect to $q$ when all nonleading coe…

Let $q$ be an odd prime, and let $R_q(y)$ be the resultant polynomial defined in the paper. Norm-form conjecture. There exist polynomials $h_1,h_2\in\mathbb Z[y]$ such that … The p…

Casas–Alvero ideal reformulation. The conjecture is equivalent to $V=\{0\}$, or, equivalently,

Let $P_{m,n}$ be the quadratic multiplier polynomial and let $\Phi_\ell$ be the $\ell$-th cyclotomic polynomial. Write $a\asymp b$ when there are constants $c,C>0$ such that…

Let $G_{d,m,n}^{\zeta}$ be the Misiurewicz polynomial, let $P_{d,m,n}^{\zeta}$ be its multiplier polynomial, and let $\Phi_i$ denote the $i$-th cyclotomic polynomial. For roots…

Let $F$ be the system of sphere equations, and let $\widetilde{F}(\mathbf{t})$ be the system obtained by zero evaluations of the $\delta$-variables. Zero-evaluation criterion. The…

Let a Kokotsakis polyhedron be a polyhedral surface in $\mathbb{R}^3$ consisting of one $n$-gon, quadrilaterals attached to its sides, and triangles between consecutive quadrilater…

Resultant multiplicity conjecture. The following congruences hold modulo $p$: