# (Eran Nevo) The g-Conjecture III: Algebraic Shifting

This is the third in a series of posts by Eran Nevo on the g-conjecture. Eran’s first post was devoted to the combinatorics of the g-conjecture and was followed by a further post by me on the origin of the g-conjecture. Eran’s second post was about the commutative-algebra content of the conjecture. It described the Cohen-Macaulay property (which is largely understood and known to hold for simplicial spheres) and the Lefshetz property which is known for simplicial polytopes and is wide open for simplicial spheres.

## The g-conjecture and algebraic shifting

### Squeezed spheres

Back to the question from last time, Steinitz showed that

any simplicial 2-sphere is the boundary of a convex 3-polytope.

However, in higher dimension

there are many more simplicial spheres than simplicial polytopes,

on a fixed large number of vertices. Continue reading

# Igor Pak’s “Lectures on Discrete and Polyhedral Geometry”

It is a wonderful book, full of gems, contains original look on many important directions, things that cannot be found elsewhere, and great beyond great pictures (which really help to understand the mathematics). Grab it!

# The Polynomial Hirsch Conjecture: Discussion Thread

This post is devoted to the polymath-proposal about the polynomial Hirsch conjecture. My intention is to start here a discussion thread on the problem and related problems. (Perhaps identifying further interesting related problems and research directions.)

First, for general background: Here is a  chapter that I wrote about graphs, skeleta and paths of polytopes. Some papers on polytopes on Gunter Ziegler’s homepage  describe very interesting and possibly relevant current research in this area,  and also a few of the papers under “discrete geometry” (which follow the papers on polytopes) are relevant. Here are again links for the recent very short paper by  Freidrich Eisenbrand, Nicolai Hahnle, and Thomas Rothvoss, the 3-pages paper by Sasha Razborov,  and to Eddie Kim and Francisco Santos’s survey article on the Hirsch Conjecture.

Here are the basic problems and some related problems. When we talk about polytopes we usually mean simple polytopes. (Although looking at general polytopes may be of interest.)

Problem 1: Improve the known upper bounds for the diameter of graphs of polytopes, perhaps finding a polynomial upper bound in term of the dimension $d$ and number of facets $n$.

Strategy 1: Study the problem in the purely combinatorial settings proposed in the EHR paper.

Strategy 2: Explore other avenues.

Problem 2: Improve  the known lower bounds for the problem in the abstract setting.

Strategy 3: Use the argument for upper bounds as some sort of a role model for an example.

Strategy 4:Try to use recursively mesh constructions as those used by EHR.

Problem 3: What is the diameter of a polytopal digraph for a polytope with n facets in dimension d.

A polytopal digraph is obtained by orienting edges according to some generic linear objective function. This problem can be studied also in the abstract setting of shellability. (And even in the context of unique sink orientations.)

Problem 4: Find a (possibly randomized) pivot rule for the simplex algorithm which requires, in the worse case, small number of pivot steps.

A “pivot rule” refers to a rule to walk on the polytopal digraph where each step can be performed efficiently.

Problem 5: Study the diameter of graphs (digraphs) of specific classes of polytopes.

Problem 6: Study these problems in low dimensions.

Here are seven additional relevant problems.

Problem 7: What can be said about expansion properties of graphs of polytopes? Continue reading

# The Polynomial Hirsch Conjecture – How to Improve the Upper Bounds.

I can see three main avenues toward making progress on the Polynomial Hirsch conjecture.

One direction is trying to improve the upper bounds, for example,  by looking at the current proof and trying to see if it is wasteful and if so where it can be pushed further.

Another direction is trying to improve the lower-bound constructions for the abstract setting, perhaps by trying to model an abstract construction on the ideas of the upper bound proof.

The third direction is to talk about entirely different avenues to the problem: new approaches for upper bounds, related problems, special classes of polytopes, expansion properties of graphs of polytopes, the relevance of shellability, can metric properties come to play, is the connection with toric varieties relevant, continuous analogs, and other things I cannot even imagine.

Reading the short  recent paper by  Freidrich Eisenbrand, Nicolai Hahnle, and Thomas Rothvoss will get you started both for the upper bounds and for the lower bounds.

I want to discuss here very briefly how the upper bounds could be improved. (Several people had ideas in this direction and it would be nice to discuss them as well as new ideas.) First, as an appetizer, the very basic argument described for polytopes. Here $\Delta (d,n)$ is the maximum diameter of the graph of a $d$-dimensional polyhedron with $n$ facets.

(Click on the picture to get it readable.)

The main observation here (and also in the abstract versions of the proof) is that

if we walk from a vertex $v$ in all possible directions $\Delta(d,k)$ steps we can reach vertices on at least $k+1$ facets.

But it stands to reason that we can do better.

Suppose that $n$ is not too small (say $n=d^2$.). Suppose that we start from a vertex $v$ and walk in all possible directions $t$ steps for

$t=\Delta (d,10d)+\Delta (d-1,10d)+\Delta(d-2,10d)+\dots +\Delta(2,10d)$.  (We can simply take the larget quantity $t = d \Delta (d,11d)$.)

The main observation we just mentioned implies that with paths of this length starting with the vertex $v$ we can reach vertices on $10d$ facets  and on every facet we reach we can reach vertices on $10d$ facets and in every facet of a facet we can reach vertices on $10d$ facets and so on. It seems that following all these paths we will be able to reach vertices on many many more than $10d$ facets. (Maybe a power greater than one of $d$ or more.)  Unless, unless something very peculiar happens that perhaps we can analyze as well.

# The Polynomial Hirsch Conjecture, a Proposal for Polymath3 (Cont.)

## The Abstract Polynomial Hirsch Conjecture

A convex polytope $P$ is the convex hull of a finite set of points in a real vector space. A polytope can be described as the intersection of a finite number of closed halfspaces. Polytopes have a facial structure: A (proper) face of a polytope $P$ is the intersection of  $P$ with a supporting hyperplane. (A hyperplane $H$ is a supporting hyperplane of $P$ if $P$ is contained in a closed halfspace bounded by $H$, and the intersection of $H$ and $P$ is not empty.) We regard the empty face and the entire polytope as trivial faces. The extreme points of a polytope $P$ are called its vertices. The one-dimensional faces of $P$ are called edges. The edges are line intervals connecting a pair of vertices. The graph $G(P)$ of a polytope $P$  is a graph whose vertices are the vertices of $P$ and two vertices are adjacent in $G(P)$ if there is an edge of $P$ connecting them. The $(d-1)$-dimensional faces of a polytop are called facets.

The Hirsch conjecture: The graph of a d-polytope with n  facets has diameter at most n-d.

A weaker conjecture which is also open is:

Polynomial Hirsch Conjecture: Let G be the graph of a d-polytope with n facets. Then the diameter of G is bounded above by a polynomial in d and n.

The avenue which I consider most promising (but I may be wrong) is to replace “graphs of polytopes” by a larger class of graphs. Most known upper bound on the diameter of graphs of polytopes apply in much larger generality. Recently, interesting lower bounds were discovered and we can wonder what they mean for the geometric problem.

Here is the (most recent) abstract setting:

Consider the collection ${\cal G}(d,n)$ of graphs $G$ whose vertices are labeled by $d$-subsets of an $n$ element set.

The only condition we will require is that if  $v$ is a vertex labeled by $S$ and $u$ is a vertex labeled by the set $T$, then there is a path between $u$ and $v$ so that all labels of its vertices are sets containing $S \cap T$.

Abstract Polynomial Hirsch Conjecture (APHC): Let $G \in {\cal G}(d,n)$  then the diameter of $G$ is bounded above by a polynomial in $d$ and $n$.

Everything that is known about the APHC can be described in a few pages. It requires only rather elementary combinatorics; No knowledge about convex polytopes is needed.

A positive answer to APHC (and some friends of mine believe that $n^2$ is the right upper bound) will apply automatically to convex polytopes.

A negative answer to APHC will be (in my opinion) extremely interesting as well, Continue reading

# The Polynomial Hirsch Conjecture: A proposal for Polymath3

This post is continued here.

Eddie Kim and Francisco Santos have just uploaded a survey article on the Hirsch Conjecture.

The Hirsch conjecture: The graph of a d-polytope with n vertices  facets has diameter at most n-d.

We devoted several posts (the two most recent ones were part 6 and part  7) to the Hirsch conjecture and related combinatorial problems.

A weaker conjecture which is also open is:

Polynomial Diameter Conjecture: Let G be the graph of a d-polytope with n facets. Then the diameter of G is bounded above by a polynomial of d and n.

One remarkable result that I learned from the survey paper is in a recent paper by  Freidrich Eisenbrand, Nicolai Hahnle, and Thomas Rothvoss who proved that:

Eisenbrand, Hahnle, and Rothvoss’s theorem: There is an abstract example of graphs for which the known upper bounds on the diameter of polytopes apply, where the actual diameter is $n^{3/2}$.

Update (July 20) An improved lower bound of $\Omega(n^2/\log n)$ can be found in this 3-page note by Rasborov. A merged paper by Eisenbrand, Hahnle, Razborov, and Rothvoss is coming soon. The short paper of Eisenbrand,  Hahnle, and Rothvoss contains also short proofs in the most abstract setting of the known upper bounds for the diameter.

This is something I tried to prove (with no success) for a long time and it looks impressive. I will describe the abstract setting of Eisenbrand,  Hahnle, and Rothvoss (which is also new) below the dividing line.

I was playing with the idea of attempting a “polymath”-style  open collaboration (see here, here and here) aiming to have some progress for these conjectures. (The Hirsch conjecture and the polynomial diameter conjecture for graphs of polytopes as well as for more abstract settings.) Would you be interested in such an endeavor? If yes, add a comment here or email me privately. (Also let me know if you think this is a bad idea.) If there will be some interest, I propose to get matters started around mid-August.

Here is the abstract setting of Eisenbrand, Hahnle, and Rothvoss: Continue reading

# (Eran Nevo) The g-Conjecture II: The Commutative Algebra Connection

Richard Stanley

This post is authored by Eran Nevo. (It is the second in a series of five posts.)

## The g-conjecture: the commutative algebra connection

Let $K$ be a triangulation of a $(d-1)$-dimensional sphere. Stanley’s idea was to associate with $K$ a ring $R$, and study the relations between algebraic properties of $R$ and combinatorial properties of $K$.

### Face ring

Fix a field $k$. The face ring (Stanley-Reisner ring) of $K$ over $k$ is $k[K]=k[x_{1},..,x_{n}]/I_{K}$ where $I_{K}$ is the homogenous ideal generated by the monomials whose support is not in $K$, $\{\prod_{i\in S}x_i:\ S\notin K\}$. For example, if $K$ is the boundary of a triangle, then $k[K]=k[x,y,z]/(xyz)$. $k[K]$ is graded by degree (variables have degree one, $1$ has degree zero), and let’s denote the degree $i$ part by $k[K]_i$. This part is a finite dimensional $k$-vector space and we can collect all these dimensions in a sequence, or a series, called the Hilbert series of $k[K]$, which carries the same information as $f(K)$. More precisely,

$hilb(k[K]):=\sum_{i\geq 0}\dim_k k[K]_i t^i = \frac{h_0(K)+h_1(K)t+...+h_d(K)t^d}{(1-t)^d}$

(recall that $K$ is $(d-1)$-dimensional).

### Cohen-Macaulay (CM) ring

The ring $k[K]$ is called Cohen Macaulay (CM) if there are $d$ elements $\Theta=\{\theta_1,..,\theta_d\}$ in $k[K]_1$ such that $k[K]$ is a free $k[\Theta]$-module. As $hilb(k[\Theta])=\frac{1}{(1-t)^d}$, the numerical consequence is that $hilb(k[K]/(\Theta))=h(K)$ (we use $h$ both as a vector and as a polynomial, with the obvious identification).

Macaulay (revisited) showed that the Hilbert series of standard rings (=quatient of the polynomial ring by a homogenous ideal) are exactly the $M$-vectors (sequences).

A theorem of Riesner characterizes the simplicial complexes $K$ with a CM face ring over a fixed field $k$ in terms of the homology of $K$ and its face links (with $k$-coefficients). It follows that if $K$ is a simplicial sphere then $k[K]$ is CM, hence $h(K)$ is an $M$ vector! This gives more inequalities on $f(K)$. This is also how Stanley proved the Upper Bound Conjecture, for face number of spheres: It follows that if $K$ is a $(d-1)$-sphere with $n$ vertices, and $C(d,n)$ is the boundary of the cyclic $d$-polytope with $n$ vertices, then for every $i$, $f_i(K)\leq f_i(C(d,n))$. This is as $h_i(K)\leq \binom{n-d+i-1}{i}=h_i(C(d,n))$.

### Hard Lefschetz

Let $K$ be the boundary of a simplicial $d$-polytope. Stanley observed that the hard Lefschetz theorem for toric varieties, an important theorem in algebraic geometry, translates in the language of face rings as follows: there exists $\Theta$ as above and $\omega\in k[K]_1$ such that the maps

$w^{d-2i}: (k[K]/(\Theta))_i\rightarrow (k[K]/(\Theta))_{d-i}$

are isomorphisms between those vector spaces for any integer $0\leq i\leq \frac{d}{2}$. In particular, $w: (k[K]/(\Theta))_{i-1}\rightarrow (k[K]/(\Theta))_{i}$ is injective for $1\leq i\leq \frac{d}{2}$. Thus, the quotient ring $k[K]/(\Theta, \omega)$ has Hilbert series starting with $g(K)$. This means, again by Macaulay theorem, that $g(K)$ is an $M$-vector!

Later, in 1993, McMullen gave a different proof of this part of his conjectured $g$-theorem. His proof actually proves hard Lefschetz for this case. See McMullen’s survey paper `Polyhedra and polytopes: algebra and combinatorics’.

### Problems

Does hard Lefschetz theorem hold for non polytopal spheres?

Can you think of examples of simplicial spheres which cannot be realized as the boundary of convex polytopes?