68 problems

A rational discrete analytic function is a discrete analytic function represented as a rational function. Isaacs' conjecture. Every rational discrete analytic function is a polynom…

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 $K$ be a number field, let $d$ be a positive integer, and let … be a rational function such that each $a_i$ is the $d$-th power of an element of $K$. Pisot's d-th root conjectu…

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

For each $m\geq3$, let $P^{(m)}$ be the higher-point generating series, expressed using $w_1,\ldots,w_m$. Rationality and denominator conjecture. For each $m\geq3$, $P^{(m)}$ is a…

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

Let $F$ be a finitely generated extension of the complex field, and let theorem 1 denote the result asserting that, under the stated hypotheses on a smooth projective surface $S$,…

Rational-summation reduction conjecture. If

Let $S:{\mathbb S}^2\to \operatorname{U}(N)$ be a uniton, and regard it as a map into $\operatorname{GL}(N)$ via the inclusion…

Let $\operatorname{Par}_n$ denote the set of partitions with at most $n$ parts, let $\lambda'\!$ be the conjugate partition, and let $N^Z_{\lambda}$ be the dual coefficient associa…

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

Parity conjecture. The parity of the minimal number $N_v$ of variables determines the character of the differential Galois group: it is symplectic, contained in $Sp$, when $N_v$ is…

Minimal-variable conjecture. The diagonal of $R_1$ should coincide with the diagonal of a rational function $R$ depending on a minimal number $N_v$ of variables, with

Rationality classification conjecture. The integral

Nonexistence conjecture. The Wronskian $W(p_1,\ldots,p_n)$ cannot be a nonzero constant.

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

A branch datum records the prescribed branching information of a branched covering of the Riemann sphere, and its degree is denoted by $n$. Prime-degree conjecture. Any branch datu…

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

Let $f^h(\mathrm{\textcircled{s}};a_1,\dots,a_k)$ denote the number of rational functions produced by a strategy $h$ for choosing valid multipliers in the decomposition of a consta…

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…

B. and M. Shapiro conjecture. If all critical points of $R$ are real, then $R$ can be transformed, by post-composition with a linear fractional transformation, into a quotient of t…

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…