Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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!

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

1 Answer 1

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.

share|improve this answer
    
Thank you! I will take a look at this book soon. –  Erik Friese Apr 9 at 9:19

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.