19 problems

Let $F$ be a generic homogeneous form, and let $\mathcal{R}_F$ denote the factor of the resultant appearing in Theorem. Irreducibility conjecture. The polynomial $\mathcal{R}_F$ is…

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 ${\mathbb A}$ 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 $c=\pm1$.

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

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…

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

Casas–Alvero conjecture. If $\operatorname{char} K=0$ and $f\in K[x]$ is a Casas–Alvero polynomial of degree $d$ with $a_d=0$, then

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 $F$ 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…

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

Let $f(x_1,\ldots,x_n)$ be a generic form, and let $\Delta(f)$ denote its multivariate discriminant. Let ${\tt Hp}(f,[x_n,\ldots,x_1])$ be the polynomial associated with the iterat…

Let $f_1,f_2$ define two affine plane curves, and let $g$ generate their elimination ideal in the $y$-coordinate, while $R$ denotes their resultant with respect to $x$. Assume that…

Let $f$, $g$, and $h$ be layered polynomials for which the layered resultants below are defined, and let $L$ be the layer structure; write…

Minimum-dimension box-center conjecture. If the data $\boldsymbol l,\boldsymbol d,\boldsymbol s$ are determinantal, then determinantal degree vectors of minimum matrix dimension li…