# 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 $k$. – MKO Mar 13 '14 at 10:44