# Study of convex polytopes via commutative algebra

Let $P \subset \mathbb{R}^d$ be any convex polytope with integral vertices, and let $M$ be the additive submonoid of $\mathbb{R}^{d+1}$ which is generated by $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. Then $M$ is an affine monoid (i.e. finitely generated, torsionfree and cancellative) and all combinatorial data of $P$ are completely determined by the isomorphism class of $M$. By choosing an arbitrary commutative ring $R$, and considering the monoid ring $R[M]$ instead of $M$ we make the whole machinery of commutative algebra available for the study of $P$.

There are many interesting connections between $P$ and $R[M]$. For example, the face lattice of $P$ is anti-isomorphic to the lattice of monomial prime ideals of $R[M]$. So theoretically, we can determine faces of $P$ by studying certain prime ideals of $R[M]$. But does this also work in practice? Are there examples of polytopes $P$ (given by integral vertices) where the algebraic viewpoint makes it easier to study the combinatorics of $P$ (e.g. the face lattice, combinatorial automorphism group, etc.)? Are there famous polytopes which were defined by affine monoids and whose properties were proved by considering monoid rings?

In short: I would really like to see examples where polyhedral geometry benefits from commutative algebra.

There is another theorem of Stanley which says that among all centrally symmetric simple polytopes cube has the minimal h-vector. This immediately implies that in that class of polytopes the cube has also the minimal number of $k$-faces for any $k$. – sva Mar 13 '14 at 10:44