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.

Thank you in advance!

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    $\begingroup$ What about Stanley's proof of the upper bound conjecture? $\endgroup$ – Benjamin Steinberg Mar 12 '14 at 14:31
  • $\begingroup$ @BenjaminSteinberg: Thank you for this hint! I was not aware of Stanley's proof, but it is indeed a beautiful example. $\endgroup$ – Erik Friese Mar 12 '14 at 15:41
  • $\begingroup$ 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 $k$. $\endgroup$ – MKO Mar 13 '14 at 10:44
  • $\begingroup$ @semyonalesker: Thank you, this theorem is also a great example! $\endgroup$ – Erik Friese Mar 13 '14 at 12:16

The Sturmfels correspondence is my favorite example. It relates the toric ideal of a point configuration (which could be the vertices of your polytope) to a regular triangulation of the point configuration. It is beautifully described, e.g., in Rekha Thomas' book Lectures in Geometric Combinatorics.

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