74 problems

Let $G$ be a graph of even order, and let $k$ be an integer. A graph is 1-factorable if its edges can be partitioned into 1-factors, equivalently if its chromatic index equals its…

Weak Factorization Conjecture. Every birational map of smooth projective varieties admits such a factorization.

Strong factorization conjecture. There exists a diagram

Weak factorization conjecture. Every such birational map admits this factorization, with the stated preservation of the open set and normal-crossings boundary.

Let $\sigma$ and $\tau$ be cones in the local toric factorization setting, and let players $A$ and $B$ play the following game: $A$ subdivides one of the cones $\tau$ or $\sigma$,…

Let $X$ and $Y$ be smooth algebraic varieties over a field of characteristic zero, and let $f\colon X\dashrightarrow Y$ be a proper birational map. Strong factorization conjecture.…

Algorithmic completeness conjecture. The new procedure should provide an algorithm for generalized factorization and a closed-form complete solution, precisely in the sense of Tsar…

Let $\mathcal R_{\mathbb Q}=\bigcup_N\mathcal R_{1/N}$ be the rational Robba ring over $\mathbb C_p$, completed under the Gauss norm. Rational Robba-ring admissibility conjecture.…

Let $G$ be a connected Lie group, let $E$ be a Fréchet space, and let $E^{\omega}$ be the space of analytic vectors associated with a representation of exponential type of $G$ on…

Let $\mathcal{L}$ be a rational Landau diagram, and let $V_G$ be the associated variety with irreducible components $V_{G,\sigma}$. Let $\Delta_{\mathcal{L}}$ denote its LS discrim…

Let $\mathbb{F}$ be a field of characteristic zero, and let $f\in\mathbb{F}[x_1,\dots,x_n]$ be computed by an algebraic circuit of size $s$. A polynomial $g$ is a factor of $f$ if…

Let $n$ be a positive integer, and let $\mathbb{Z}^{\widetilde{O}(n^{n^{1/3}})}$ denote the integers in the indicated size range. The residue-class formulation. There exists a set…

Let $n$ be a positive integer, and let divisibility of a product mean that each integer in $\{1,\ldots,n\}$ divides that product. The most general divisor-set conjecture. For every…

Let $n$ be a positive integer. The $n$-divisor property means that every integer from $1$ through $n$ divides at least one member of a specified product. The generalized arithmetic…

Let $n$ be a positive integer. An integer set has the $n$-divisor property if every $j\in\{1,\ldots,n\}$ divides at least one member. The arithmetic-progression divisor conjecture.…

Let $S,T\subseteq\mathbb{Z}^{+}$ satisfy all the requirements of the strong $(\alpha,\beta)$-Divisor Conjecture, including the $n$-divisor property for their pairwise differences.…

Let $n$ be a positive integer, let $S,T\subseteq\mathbb{Z}^{+}$, and define … The set $A$ satisfies the $n$-divisor property when every $i\in\{1,\ldots,n\}$ divides some element of…

The $(\alpha,\beta)$-Divisor Conjecture. For infinitely many $n$, there exist subsets $S,T\subseteq\mathbb{Z}^{+}$ such that $A$ satisfies the $n$-divisor property, all elements of…

Prime-perturbation irreducibility conjecture. The Dirichlet polynomial

Let $H_3$ be the Siegel upper half-space of degree $3$, and let $H_1\times H_2\subset H_3$ be the block-diagonal locus … Let $A:H_3\to W$ be the map in the paper, let $P$, $I$, and…

Irreducible common-zero conjecture. If $h_1$ and $h_2$ have at least one common zero in $\mathbb C$, then

Ritt-type factorization conjecture. For every non-zero $E$-function $f$, there exist a unit $u\in\mathcal E^\times$, simple normalized $E$-functions $s_1,\ldots,s_p$ with pairwise…

Simple-factor obstruction conjecture. The failure of factoriality in $\mathcal E$ occurs only with simple functions.

Bessel irreducibility conjecture. The $E$-function $\Gamma(\alpha+1)x^{-\alpha}J_\alpha(x)$ is irreducible in the ring of $E$-functions.

Jossen's factorization conjecture. Every non-zero $E$-function $f$ can be written as a finite product of powers of $E$-functions with simple zeros, with distinct factors having no…