14 problems

Let $\mathcal{A}(X,\mathbf{h},\Lambda,\widetilde{B})$ be a generalized quantum cluster algebra with initial extended cluster $\{X_1,\ldots,X_m\}$. A Laurent polynomial in this clus…

Kontsevich's conjecture. Every component of every such iterate is a noncommutative Laurent polynomial in $x$ and $y$.

Propp's Laurent phenomenon conjecture. Every value $f_v(t)$ of the unbounded cube recurrence is a Laurent polynomial in ${\mathbf x}$.

Let $f_{i,j,k}$ be the Laurent polynomials generated by the cube recurrence from general initial conditions, with coefficients viewed after the recurrence has been expanded as Laur…

Let $1\leq p\leq q\leq r$ and set $k=p+q+r$. Define a sequence by … with initial conditions $a_i=1$ for $0\leq i\leq k-1$. Robinson's integrality conjecture. The resulting sequence…

The sequences Somos-$4$ through Somos-$7$ are defined by the quadratic recurrences … with $n\geq4$ and $a_i=1$ for $0\leq i\leq3$; … with $n\geq5$ and $a_i=1$ for $0\leq i\leq4$; t…

Laurent positivity conjecture. These Laurent polynomials have positive integer coefficients.

Let $L$ be the response matrix of a minimal circular planar network with $m$ edges, and call a minor of $L$ noninterlaced when its row and column index sets are noninterlaced. Base…

Let $P=P(x_1,\ldots,x_{n-1})$ be a symmetric polynomial with positive coefficients that generates a period $1$ seed. Symmetric-polynomial classification conjecture. The only possib…

Let $P=P(x_1,\ldots,x_n)$ be a multilinear polynomial with positive coefficients that generates a period $1$ seed. Reversal-symmetry conjecture. … This predicts reversal symmetry f…

Let $P=P(x_1,cdots,x_{n-1})$ be an irreducible polynomial in the class used to generate the seed $\mathbf{(x,P)}$, and let $P_0$ be the initial exchange polynomial of that seed. A…

Let ${\mathcal A}_\Gamma$ be a graph LP algebra and let $t_{\mathcal S}$ be a seed. Write ${\mathcal L}(t_{\mathcal S})$ for the Laurent polynomial ring associated with this seed.…

Positivity conjecture. Every such Laurent polynomial has non-negative integer coefficients.

Universal TP-basis conjecture. For any cell and any TP-basis, every other Plücker variable in the cell can be expressed as a subtraction-free Laurent polynomial in the variables of…