18 problems

Let $q$ be even, let $a,b\in\mathbb F_{q^2}^*$, and define … Assume that $f$ is a permutation polynomial of $\mathbb F_{q^2}$. Tu–Zeng–Li–Helleseth's conjecture. The conditions … a…

Conjecture 2. There exist real or complex numbers $y_1,\ldots,y_n$ such that

Conjecture 1. The system $S$ has a real or complex solution $(x_1,\ldots,x_n)$ satisfying

Equal-sums conjecture. $M_s(f;B)$ is dominated by the $s!B^s$ trivial solutions, namely those for which $x_1,\ldots,x_s$ are a permutation of $x_{s+1},\ldots,x_{2s}$.

Equal-sums conjecture. The quantity $L_s(f;B)$ is dominated by the $s!B^s$ trivial solutions in which $x_1,\ldots,x_s$ is a permutation of $x_{s+1},\ldots,x_{2s}$. This is stated a…

For $n \geq 1$, let $D(n,n)$ be the set of minor specifications described in the paper, and let $M(S)$ be the associated product of minors of a positive $n \times n$ matrix $A$. Le…

Conjectural formula. The number of degree of difficulty is

Finite-supremum conjecture. The supremum of the number of solutions of this equation, as $D$ ranges over integer-coefficient polynomials with $\deg(D)\leq k$, exists; denote it by…

Hilbert's octic conjecture.

Segre's asymptotic resolvent degree conjecture. The difference between $n$ and the resolvent degree $\operatorname{RD}(n)$ tends to infinity as $n$ tends to infinity.

For each $n$, let $\mathcal{P}_n$ be the parameter space of degree-$n$ polynomials and let $\widetilde{\mathcal{P}}_n\to\mathcal{P}_n$ be the associated cover parametrizing a root.…

For each $n$, let $\mathcal{P}_n$ be the parameter space of degree-$n$ polynomials and let $\widetilde{\mathcal{P}}_n\to\mathcal{P}_n$ be the associated cover parametrizing a root.…

Let $P$ be a homogeneous polynomial in $n$ variables, and consider the volume of … as $R\to\infty$. Polynomial volume asymptotics conjecture. In full generality, there are constant…

Joly's conjecture. The number of solutions to the diagonal equation satisfies

Let $\mathbb F_q$ be a finite field, and let $H\in\mathbb F_q[X_1,\dots,X_n]$ be a homogeneous polynomial with $\deg(H)<n$. A zero of $H$ is nontrivial if it is not the zero vector…

Let $p$ be a polynomial, with roots and critical points subject to appropriate assumptions on their distinctness. The Voronoi property conjecture. The Voronoi property is valid und…

Let $\mathbf{f}$ be a set of polynomials over $\mathbb{F}_q$ whose degrees total $d$, and let $V_1,\ldots,V_m$ be the absolutely irreducible varieties over $\mathbb{F}_q$ whose uni…

Euler's conjecture. The roots of a fifth-degree equation should be expressible in the form