6 problems

A sequence $(h_0,h_1,\dots,h_d)$ is called Gorenstein when it is the $h$-vector of a standard, graded, Gorenstein commutative ring. The conditions of the $g$-theorem are the symmet…

McMullen's conjecture. The face vector should satisfy $g_i\geq 0$ for $1\leq i\leq\lfloor k/2\rfloor$, and, if

Let $P$ be a $d$-dimensional simplicial convex polytope, and let $h(t)=\sum_{i=0}^d h_i t^i$ be its $h$-polynomial, defined by … where $f_i$ is the number of $i$-dimensional faces…

Let $\Sigma$ be a simplicial sphere of dimension $d-1$, with $f$-vector $(f_{-1},f_0,\ldots,f_{d-1})$ and associated $g$-vector defined by $g_k(\Sigma)=h_k-h_{k-1}$ for $k\leq d/2$…

Generalized $g$-conjecture. The characterization given by the $g$-theorem should also apply to the $h$-vectors of simplicial spheres.

A simplicial sphere is a triangulation of a topological sphere, and its ${\mathfrak f}$-vector records the numbers of faces in each dimension. The ${\mathfrak g}$-theorem gives a s…