98 problems

Linnik's conjecture. There is a constant $c_d>0$ such that

Scholz's degree-comparison conjecture.

Brumer–McGuinness conjecture. The number of elliptic curves with bounded discriminant satisfies

$X_9^2$ component conjecture. These representatives exhaust the components of the complement of the discriminant variety; in total there are 52 such components.

Let $n\geq 2$, let $Q_n$ be the polynomial studied in the paper, and write … For a polynomial with rational coefficients, let $\operatorname{Gal}(Q_n)$ denote its Galois group and…

Let $V$ be the variety under consideration, let its RR discriminant be defined by a polynomial, and call a factor isotropic when it belongs to the isotropic part of that discrimina…

Weak Malle conjecture for pushforward discriminants. For every $\epsilon>0$,

Conjectural tail estimates. For every $\epsilon>0$,

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…

Let $f$ and $g$ be singular functions such that $f$ is adjacent to $g$, so that a versal unfolding of $g$ is also a versal unfolding for $f$. Let $4mu$ and $tilde\mu$ be the corres…

Tropical discriminant conjecture. For any point configuration $A$, the tropical discriminant $\tau(X_A^*)$ is a union of cones in the secondary fan $\Sigma(A)$.

Let $(w_1,\dotsc,w_m)$ be a Schubert problem for $\mathbb{F}\ell(\alpha;n)$, let $\Delta_w$ be its discriminant, and let monotonicity be defined using the descent sets…

Let $(w_1,\dotsc,w_m)$ be a Grassmannian Schubert problem for $\mathbb{F}\ell(\alpha;n)$, let $\delta(w_i)$ be the unique descent of $w_i$, and let $\Delta_w(t_1,\dotsc,t_m)$ be it…

Let $\alpha=\{\alpha_1<\dotsb<\alpha_k\}$ with $\alpha_k<n$, and let $(w_1,\dotsc,w_m)$ be a Grassmannian Schubert problem for $\mathbb{F}\ell(\alpha;n)$. Let…

For a quartic $S_4$-extension of the rationals, let $d$ denote its absolute discriminant, and let the number of such extensions with discriminant $d$ be counted up to isomorphism.…

Let $1\leq m<n$, let $a,b,c$ be integers, and define … Assume that $\gcd(n,mb)=1$ and that $a$ is prime. Jones's monogenicity conjecture. The polynomial $g(x)$ is monogenic if and…

Folklore discriminant-counting conjecture. The number of degree-$n$ number fields with discriminant bounded in absolute value by $X$ is asymptotic to a constant times $X$.

For an integer $n\geq 2$, a positive integer $\Delta\geq 1$, and $\varepsilon>0$, let $K$ range over number fields of degree $n$ satisfying $|\Delta_K|=\Delta$, where $\Delta_K$ is…

A local lacuna is a neighboring component of the complement of a discriminant function corresponding to a wavefront point, in which the associated wave propagation problem has the…

For the products $\mathbb P^1\times\mathbb P^{n-1}$ and $\mathbb P^2\times\mathbb P^{n-1}$, consider the corresponding RR discriminants and their nonisotropic parts. Product RR dis…

Let $C_d\subseteq\mathbb P^d$ be the rational normal curve of degree $d\geq2$, and let its RR discriminant be decomposed into nonisotropic and isotropic parts. Rational normal curv…

Smallest-exponent conjecture. The smallest positive integer $s$ such that

Double-discriminant constant conjecture. The following hold:

Double-discriminant factor irreducibility conjecture. The polynomial $B_{n,0}$ is irreducible; indeed,

Double-discriminant factorization conjecture. For $0\leq k\leq n/2$, there is a factorization in $\mathbb{Z}[a_0,\ldots,\widehat{a_k},\ldots,a_n]$,