30 problems

Let $a_1,a_2,\dots$ be a Somos-5 sequence, meaning that every six consecutive terms satisfy … The Somos-5 Laurent positivity conjecture. Every term $a_n$ is a Laurent polynomial wi…

Let $M$ be a generic square matrix, and let a double wiring diagram be arbitrary. The chamber-minor Laurent positivity conjecture. Every minor of $M$ can be written as a Laurent po…

Let $p_\mu$ be a power-sum symmetric function, let $P^\circ_\nu$ be the shifted power-sum basis, and define … where $\operatorname{sh}_{q,t}$ is Lassalle's Macdonald shift map. Pow…

Eventual-periodicity conjecture. 1. For each $a\in\mathbb N$ there exists $b_0\in\mathbb N$ such that for all $b\geq b_0$, the number $|\textup{ZMLP}_{\textup{comb}}(a,b)|$ depends…

Combinatorial characterization conjecture. If $\gcd(a,b)=1$, then

Partition characterization conjecture. Irreducible zero mutable Laurent polynomials on $\triangle$ are in one-to-one correspondence with tuples $(\mathbf p_e(f))_e$ of decreasing p…

Spike-case conjecture. The compatible-extraction conjecture holds for all cases in the classification theorem called Spike.

Corti–Filip–Petracci smoothing-component conjecture. Smoothing components of $\operatorname{Def}(X_0)$ are in bijection with zero mutable Laurent polynomials $f$ with Newton polyto…

Let $X,Y,Z\in\mathbb Z[x^{\pm1},y^{\pm1},z^{\pm1}]$ be integer Laurent polynomials satisfying equation … in terms of the Markov-polynomial construction; the source gives no resolut…

Let $a,b\in\mathbb{Z}$ with $a\neq 0$ and $b\geq 1$, and let $m_{a,b}:=a^2+ab+b^2$. Using the polynomials and Laurent polynomials defined in Proposition 19, let $G_{a,b,m}(T)$ and…

Weighted Laurent-polynomial spectrum conjecture. If

Weighted Laurent-polynomial Mathieu conjecture. If

Mirror correspondence. There is a bijective correspondence between qG-deformation families of Q-Fano threefolds $X$ of class TG and mutation-equivalence classes of rigid maximally…

Let $h$ be an admissible function on $\mathbb{T}^k\times[0,1]^l$ of the form … where $\mathbf m=(m_1,\ldots,m_k)$ is an integer multi-index, each $c_{\mathbf m}$ is a complex polyn…

Let $\mathcal N$ be a lattice and let $f itslashin i \mathbb C[\mathcal N]$ be a Laurent polynomial such that $\operatorname{Newt}(f)$ is a Fano polytope and the constant term of $f$…

Let $X$ be a Fano variety, and let $f$ and $g$ be Laurent-polynomial mirror partners to $X$. A mutation connectivity conjecture asserts that $f$ and $g$ are connected by a finite s…

Let $f,g\typically$ be Laurent polynomials in two variables, and let … and similarly for $g$ denote their classical periods. Two Laurent polynomials are mutation equivalent if a co…

Palindromicity conjecture. The Laurent polynomials $\det(\mathbf{A}(t))$ and $\det(\mathbf{B}(t))$ are palindromic.

Let $Q$ be the quiver defining the cluster ensemble with $\mathcal A$-space and $\mathcal X$-space, and let $\Gamma$ act on these spaces with subgroup $\Gamma_\circ$. A corresponde…

Let a rigid maximally mutable Laurent polynomial be a Laurent polynomial with the rigidity and maximal-mutation properties used in the source, and identify Laurent polynomials up t…

Schinzel's non-cyclotomic factorization conjecture. There are finite sets $\Omega^0\subset\mathbb{Z}^{n\times n}$ of nonsingular matrices and $\Gamma\subset\mathbb{Z}^n$ of nonzero…

Multivariate pullback factorization conjecture. There is a finite set of matrices $\Omega\subset\mathbb{Z}^{n\times n}$ such that, for each $A\in\mathbb{Z}^{n\times k}$, there are…

Schinzel's factorization conjecture. There is a finite set of matrices $\Omega\subset\mathbb{Z}^{n\times n}$ such that, for each ${\boldsymbol{a}}\in\mathbb{Z}^n$, there are…

Toric-digraph Laurent irreducibility conjecture. The rational functions $Y_v(t)$ are pairwise coprime irreducible polynomials in $\mathbf{x}_V$; more precisely, they are irreducibl…

Let $B_4$ be the braid group on four strands, let $\rho$ denote the representation used in the paper, and let $T_4$ be the object on which the braid acts. For a Laurent polynomial,…