6 problems

Lecture hall position conjecture. If $\pi=\varphi(\lambda)$, then

Let $S_n:=\mathbb{Q}[y_1,\dots,y_n]$ with the degree-lexicographic term order $\prec$ satisfying $y_1\succ y_2\succ\dots\succ y_n$. Assume the polynomiality conjecture, so that eve…

Let $\ell_i$ denote the $i$-th Lecture Hall polynomial, constructed in the paper as a Laurent polynomial depending on $y_1,\dots,y_i$. Polynomiality conjecture for Lecture Hall pol…

Let $n\in\mathbb{N}$. There exist a $\mathbb{Z}^2$-graded polynomial ring $\Bbbk[x_1,\dots,x_m]$ for some $m\in\mathbb{N}$, a graded subalgebra $A_n\subseteq\Bbbk[x_1,\dots,x_m]$,…

Let $\mathbf{L}_n$ be the Lecture Hall cone, and let its Ehrhart ring be the graded algebra generated by lattice points of the cone. A toric deformation is described via initial su…

Bousquet–Mélou and Eriksson's conjecture. If a non-decreasing sequence $\mathbf{s}=(s_1,\ldots,s_n)$ of positive integers is polynomic, then there is an integer $m$ with…