36 problems

Let $R$ be a rational function in $z_1$ with deformation dimension $m=1$, satisfying the conditions of Proposition[necessary condition], and suppose that its first three critical v…

Converse characterization conjecture. If $f$ is an extreme point of $PRP_2$, then $p$ is $\mathbb{T}^2$-saturated and $p-\widetilde{p}$ is irreducible.

Bilu–Luca conjecture. There exists a real number $c>0$ such that, for every sufficiently large integer $N$, at least $cN$ of the number fields…

Tricolour graph construction. For any tricolour graph there exists a function

Hawaiian conjecture. Each chord of $\mathcal{G}_{P}$ contains at least one nonreal zero of $P$.

Let $p>0$, let $p\nmid d$, and let $h_d$ be the reduced separable quotient associated with $H_d^{\rm raw}$ in characteristic $p$. Assume the characteristic-zero case is settled and…

Denominator conjecture. The expression

Let $G$ be a graph whose vertices are colored with $z$ and $w$, and let $i$ be a vertex from which the representation is computed. The contact order is the order describing how the…

Bickel–Hong's contact-order conjecture. The order of contact of $f$ is $2n$. This conjecture concerns the behavior of level curves near boundary singularities; it has been proved i…

Oscillation conjecture. There exists a constant $C>0$ such that, for every positive integer $k$, there is a pair of binary words $(v,w)$ with lengths at most $\exp(Ck)$ for which

Lüroth-type conjecture. There is a unique field extension $L|k$ in $F$ such that

Let $f(x,y)=(y^2-x^3)/x$. For $Q\in\mathbb{N}_{\geq 2}$, let $C_Q$ be the set defined in the paper for this rational function and quotient $Q$. Nonemptiness and infinitude conjectu…

Let $D\subseteq \mathbb{C}(z)$ be a Diophantine set. For each integer $n\geq 0$, write $\mathbb{C}[z]_n=\mathbb{C}^{n+1}$ for the polynomials of degree at most $n$. Kollár's conjec…

Let $K$ be a number field, let $\phi\in K(x)$, and let $\alpha\in K$. The pair $(\phi,\alpha)$ is eventually stable when, writing $\phi^n=p_n/q_n$ with coprime $p_n,q_n\in K[x]$, t…

Disconnected-graph conjecture. If $\Gamma_B\backslash\{B\}$ is disconnected, then $B$ belongs to the Ritt ideal.

Finiteness conjecture. For every positive integer $N$, the number of $N$-admissible rational functions is finite. The conjecture is stated as plausible without a rigorous proof in…

Rational-function primitive normal conjecture. There exists an element $\alpha\in\mathbb{F}_{q^m}$ such that $\alpha$ and $f(\alpha)$ are simultaneously primitive normal over…

Let $n$ be a positive integer and let $\pi\in\mathfrak{S}_n$ be a permutation of $\{1,\ldots,n\}$. Cyclic squared-difference conjecture. If $n>11$, there is a permutation $\pi$ suc…

Let $n$ be a positive integer and let $\pi\in\mathfrak{S}_n$ be a permutation of $\{1,\ldots,n\}$. Sun's squared-difference conjecture. If $n>7$, there is a permutation $\pi$ such…

Let $n$ be a positive integer and let $\pi\in\mathfrak{S}_n$ be a permutation of $\{1,\ldots,n\}$. Sun's reciprocal-sum conjecture. If $n>6$, there is a permutation $\pi$ such that…

Let $Q_{1,1}$ be the set of prime powers $q$ for which every rational function in $R_{1,1}$ admits a primitive pair $(\alpha,f(\alpha))$ with $\alpha\in\mathbb{F}_q$, and let…

Let $m$ be the number of alternatives, let $\binom{m}{2}$ pairwise-comparison intensities be denoted by $\boldsymbol{\lambda}=(\lambda_{ij})$, and let $w_{ij}$ be the weights of a…

Let $q$ be a prime power and let $n\geq 1$. Write … for the least power of $q$ that is at least $n$, and set … Let $T$ be the set of pairs $(g,h)\in\mathbb{F}_q[X]^2$ with…

Let $K\subset\mathbb{C}$ be convex, and let $\epsilon>0$ be given. There exists some constant $C>0$ for which the following holds. Let $g,h\in\mathbb{C}(z)$ be rational functions,…

Let $K\subset\mathbb{C}$ be bounded and convex, and let $\epsilon>0$ be given. There is some constant $C>0$ such that if $k/(n-k)>C$, then $K$ satisfies…