43 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…

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

Let $A(n,p,q)$ be the perturbed Gale–Robinson Laurent polynomial defined by … with initial variables $A(n,p,q)=x_{n,p,q}$ for $0\leq n<m$. Thus each $A(n,p,q)$ is a Laurent polynom…

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 $x_{i,j,k}$ denote the initial variables in a polynomial generated by the cube recurrence. Propp's exponent-bound conjecture. In each term of any polynomial generated by the cu…

The paper considers instances of the Laurent phenomenon, in which the quantities generated by the relevant recurrences are Laurent polynomials in the initial variables with integer…

For integers $k,l\geq0$, let … be an admissible function, written as … Define its torus spectrum by … The $xz$-conjecture asserts that … where each torus is equipped with normalize…

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…

Let $\Lambda_{[q^{\pm1},t^{\pm1}]}$ be the Laurent lattice of Macdonald symmetric functions, let $L_\mu$ be the Macdonald multiplication-derived operator, and let $L_\mu^\dagger$ d…

Laurent-integrality conjecture for shifted Macdonald skew functions. For all $\mu\subseteq\lambda$,

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

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 $m$ be a natural number, and let $n$ and $ell$ be parameters of the twisted hybrid weaving knot, with $Omega((m,m,n,\ell);q)$ the symmetric Laurent-polynomial trace term define…

Let $n$ be a positive integer. A classification conjecture for rigid maximally mutable Laurent polynomials asserts that rigid MMLPs in $n$ variables, up to mutation, are in one-to-…

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…