11 problems

Saturation conjecture. The terms that appear in the numerator of a Markov polynomial $M_{\rho}$ are precisely those corresponding to the integer lattice points in the Newton polygo…

Let $b/a=[a_1,a_2,\dots,a_n]$ be the continued-fraction expansion of the rational number $b/a$, and let $A_m,B_m$ be the corresponding penultimate-continuant vertices of the Markov…

Markov Sail Duality conjecture. The coefficients of Markov polynomials satisfy these two Markov-sail duality identities. This would determine almost all interior sail coefficients…

Let $\rho=a/b$ be rational, and consider the Markov sail formed by the integer lattice points in the critical triangle of the Newton polygon, together with the coefficients of the…

Let $\rho=a/b$ be rational, and let the critical triangle be the set of lattice points satisfying … Consider the coefficients of the Markov polynomial associated with these lattice…

Let $\alpha\in[0,1]$, let $\mathcal N_\alpha$ be the scaled Newton polygon … and let $\mathcal H_\alpha(\xi,\eta)$ denote the entropy function obtained from the coefficient asympto…

Let $\rho_n=a_n/b_n$ be a sequence of fractions tending to $\alpha\in[0,1]$. Choose lattice points $(i_n,j_n)$ in the Newton polygons … so that $(\xi_n,\eta_n)=(i_n/b_n,j_n/b_n)$ t…

Let $M_\rho$ be a Markov polynomial, and let its coefficients be regarded as weights on the corresponding Newton polygon. A sequence is log-concave when $x_k^2\geq x_{k-1}x_{k+1}$…

Saturation conjecture. The terms appearing in the numerator of $M_\rho$ are precisely those corresponding to the integer lattice points in $\Delta_\rho$. The conjecture is an equiv…

Let $\rho=a/b$ be a rational number, and let $\Delta_{a/b}$ be the Newton polygon of the numerator of the Markov polynomial $M_\rho(x,y,z)$, consisting of the points $i,j\geq0$ sat…

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…