Recent results on unimodular triangulations of lattice ... · ReeveÕs tetrahedra reeve(q) for integral non-negative q. subdivision has a reÞnement to a full triangulation, for example

Intro Examples Dilations

Recent results on unimodular triangulations oflattice polytopes

Francisco Santos,1 U. Cantabria, Spain.http://personales.unican.es/santosf

DMV Discrete Math Symposium, Frankfurt, May 9 2014

1joint with C. Haase, A. Paffenholz, L. Piechnik, arXiv:1405.1687 (andG. M. Ziegler arXiv:1304.7296)

F. Santos Unimodular triangulations of lattice polytopes

http://personales.unican.es/santosf


What?

What? Who? Why?



What?

What?

Lattice polytope P in Rd := conv(S), S ∈ Zd .

Unimodular simplex := vertices are an affine basis of Zd .(Equivalently, normalized volume equal to 1)

(Lattice) subdivision of P: “face to face” decomposition intolattice subpolytopes.

(Lattice triangulation) of P: same, into simplices.

Unimodular triangulation: triangulation into unimodular simplices.



What?

Dim 2 versus higher dim

Proposition

Every lattice polygon has a unimodular triangulation.UNIMODULAR TRIANGULATIONS 3

Figure 1.1. A non-unimodular triangulation and two unimodular ones.They are all regular, but only the last is quadratic. (See Sec-tion 2.4.2).

Theorem 4.4. If P is a lattice polytope of dimension d and (lattice) volumevol(P ), then the dilation

d!vol(P )!dd2 vol(P )P

has a regular unimodular triangulation.More precisely, if P has a triangulation into N d-simplices, of volumes

V1, . . . , VN , then the dilation

d!PN

i=1 Vi!((d+1)!(d!)d)Vi�1

Thas a regular unimodular refinement.

1.1. What? A lattice polytope in Rd is the convex hull of finitely manypoints in the lattice Zd. We identify two lattice polytopes if they are relatedby a lattice preserving a�ne map. Up to this lattice equivalence, we canalways assume that our polytope is d-dimensional. (For more on convexpolytopes and lattices we refer to [7].)

A unimodular simplex is a lattice polytope which is lattice equivalent tothe standard simplex �d, the convex hull of the origin 0 together with thestandard unit vectors ei (1 i d). Equivalently, unimodular simplicesare characterized as the d-dimensional lattice polytopes of minimal possibleEuclidean volume, 1/d! .

For the purposes of this paper, a (lattice) subdivision of a d-dimensionallattice polytope P is a finite collection of (lattice) polytopes S such that

(1) every face of a member of S is in S,(2) any two elements of S intersect in a common (possibly empty) face,

and(3) the union of the polytopes in S is P .

The maximal (d-dimensional) polytopes in S are called cells of S.A triangulation is a subdivision of a polytope for which each cell of the

subdivision is a simplex. The triangulation is unimodular if every cell is.Figure 1.1 depicts three triangulations of the 9-point square. The first is notunimodular, while the other two are.

A full triangulation is a lattice triangulation which uses all the latticepoints in P . The triangulation on the left in Figure 1.1 is not full. Every



What?

Dim 2 versus higher dim

Remark

In dim≥ 3 there are “empty non-unimodular simplices” ⇒ thereare polytopes without unimodular triangulations.

4 HAASE, PAFFENHOLZ, PIECHNIK, AND SANTOS

reeve(q) := convh

0 0 0 q0 0 1 10 1 0 1

i

Figure 1.2. Reeve’s tetrahedra reeve(q) for integral non-negative q.

subdivision has a refinement to a full triangulation, for example the oneresulting from the strong pulling procedure discussed in Section 2.1. Also,every unimodular triangulation is full, and in dimension at most two the con-verse is true as well. Depending on one’s perspective, this is a consequenceof, or the reason for, Pick’s formula, which says that the area a polygon is oneless than its number of interior lattice points plus half the number of latticepoints on its boundary [92]. The formula yields the following proposition.

Proposition 1.1. Every lattice polygon has a unimodular triangulation.

However, there are 3 dimensional polytopes for which a unimodular tri-angulation does not exist.

Example 1.2 (John Reeve [96]). For q 2 Z>0, the tetrahedron in Figure 1.2contains only four lattice points — the vertices. Its only lattice triangulationis the trivial one. As the Euclidean volume is equal to q/6, this simplex doesnot have a unimodular triangulation for q > 1.

A subdivision is regular if cells are the domains of linearity of a convexpiecewise linear function. (Compare [67, Section 14.3], [34].) Less formally,a regular triangulation can be thought of as a triangulation that can berealized as a “convex folding” of the polytope (Figure 1.3 on the left). Allthree triangulations in Figure 1.1 are regular while the triangulation on theright in Figure 1.3 is not.

A particular method for constructing regular full triangulations for anarbitrary lattice polytope is given in Lemma 2.1. However, in general, toconstruct a regular subdivision of P one specifies weights/heights ! 2 RA,where A = P \Zd is the set of lattice points in P , and defines a subdivisionS! of P as follows. A set F is a face of S! if there is an ⌘F 2 RA and a⇣F 2 R such that

(1.1) !a � h⌘F , ai+ ⇣F for all a 2 A

and

(1.2) F = conv {a 2 A : !a = h⌘F , ai+ ⇣F } .

Geometrically, consider the polyhedron P = conv(a⇥ [!a,1) : a 2 A) in

Rd+1. The bounded/lower faces of P project to the faces of S!. The latterare the domains of linearity of the function

x 7! min�h : (x, h) 2 P

= max

�h⌘F , xi+ ⇣F : F 2 S!

.



Who? Why?

Counting lattice points

The Ehrhart series of a lattice polytope P counts how many latticepoints lie in kP, for k ∈ N.It is known that its generating function can be rewritten as

∑

k≥0

#(kP ∩ Zd) tk =h∗P(t)

(1− t)d+1,

for a certain polynomial h∗P of degree (at most) dim(P).

Theorem (Ehrhart 1967)

If T is a unimodular triangulation of P, then h∗P equals theh-polynomial of T .



Who? Why?

Counting lattice points

The Ehrhart series of a lattice polytope P counts how many latticepoints lie in kP, for k ∈ N.It is known that its generating function can be rewritten as

∑

k≥0

#(kP ∩ Zd) tk =h∗P(t)

(1− t)d+1,

for a certain polynomial h∗P of degree (at most) dim(P).

Theorem (Ehrhart 1967)

If T is a unimodular triangulation of P, then h∗P equals theh-polynomial of T .



Who? Why?

Integer programming

Unimodular triangulations arise in two contexts:

If P has a unimodular triangulation then the cone σPgenerated by P × {1} ⊂ Rd+1 is generated in degree one (Weabbreviate this as “P is integrally closed”). In particular,P ∩ Zd is a Hilbert basis for it.

If the normal fan of P = {yA = c, y ≥ 0} has a unimodulartriangulation (using only facet normals as rays) then anyinteger program over P is totally dual integral.



Who? Why?

Integer programming

Unimodular triangulations arise in two contexts:

If P has a unimodular triangulation then the cone σPgenerated by P × {1} ⊂ Rd+1 is generated in degree one (Weabbreviate this as “P is integrally closed”). In particular,P ∩ Zd is a Hilbert basis for it.

If the normal fan of P = {yA = c, y ≥ 0} has a unimodulartriangulation (using only facet normals as rays) then anyinteger program over P is totally dual integral.



Who? Why?

Commutative algebra

Consider again the cone σP generated by P × {1} ⊂ Rd+1, and itsassociated semigroup algebra

RP = K[σP ∩ Zd+1]

Then P is integrally closed ⇔ RP is generated in degree one.(More generally, the subring of RP generated in degree one isnormal if and only if P is integrally closed in the sub latticegenerated by its lattice points).



Who? Why?

Algebraic geometry, I

Unimodular triangulations of P correspond to certain (so-calledcrepant) resolutions of the singular point in the affine toric variety

UP = SpecK[σ∨P ∩ Zd+1].

The Stable Reduction Theorem of Kempf-Knudsen-Mumford-SaintDonat (1973) is based on the following combinatorial result:

Theorem (Knudsen-Mumford-Waterman, 1973)

For every lattice polytope P there is a dilation factor c ∈ N suchthat cP admits a regular unimodular triangulation.



Who? Why?









Who? Why?









Who? Why?

Algebraic geometry, II

Consider the projective toric variety associated to P:

XP = ProjK[σP ∩ Zd+1] = Proj RP .

The following questions are of interest:

Is the ideal IP of XP generated by quadrics?

Is the embedding XP ↪→ Pn−1 projectively normal?

If P has a regular unimodular triangulation the second question hasa positive answer. If the triangulation is, moreover, flag, the firsttoo (in fact, then there is a binomial Grobner basis of degree two).

Conjecture

Every smooth polytope has a regular unimodular triangulation.

Here: P smooth :⇔ P is simple and every vertex normal cone is

unimodular ⇔ XP is smooth.



Who? Why?








Conjecture






Who? Why?








Conjecture






Who? Why?








Conjecture



unimodular ⇔ XP is smooth.F. Santos Unimodular triangulations of lattice polytopes


Who? Why?

Regular triangulations

A triangulation (unimodular or not) is called regular if its simplicesare the domains of linearity of a piece-wise convex function P → R.

UNIMODULAR TRIANGULATIONS 5

Figure 1.3. Regular vs. non-regular subdivisions.

We do not change the function or the subdivision if, for all a 2 A, we drop!a to the value of that function at a, so that

(1.3) a 2 F () !a = h⌘F , ai+ ⇣F .

If these equalities hold, we call the pair (S!,!) tight. If all a 2 A arevertices of S!, then (S!,!) is automatically tight.

A non-face of a triangulation is a set of points whose convex hull doesnot form a face. Particularly important are the minimal non-faces, which donot form faces but for which every proper subset does. The list of minimalnon-faces completely characterizes a triangulation, as does the list of (setsof points that form) cells. A triangulation for which all minimal non-facescontain only two elements is called flag. Putting these properties together, aquadratic triangulation is defined as a regular unimodular flag triangulation.The triangulation on the right in Figure 1.1 is quadratic. However, the threewhite vertices of the triangulation in the middle form a minimal non-face.So that triangulation is not quadratic.

1.2. Why? Who? In this section, we present some applications of uni-modular triangulations and closely related objects (“Why?”), arranged bymathematical discipline (“Who?”).

1.2.1. Enumerative combinatorics. Many counting problems can be phrasedas counting lattice points in (dilates) of polytopes or polyhedral complexes [11,31]. By a fundamental result of Ehrhart, the number of lattice points in pos-itive dilates kP of P is a polynomial function of degree d in k 2 Z>0 [11,36].Consequently, the generating function has the special form:

X

k�0

#(kP \ Zd) tk =h⇤(t)

(1� t)d+1,

where h⇤(t) is a polynomial of degree d. If P has a unimodular triangu-lation, h⇤ equals the combinatorial h-polynomial of that triangulation [15].

A quadratic triangulation is a regular, unimodular, and flagtriangulation (flag:= every clique in the graph spans a simplex).



Who? Why?

Regular triangulations

A triangulation (unimodular or not) is called regular if its simplicesare the domains of linearity of a piece-wise convex function P → R.

UNIMODULAR TRIANGULATIONS 5

Figure 1.3. Regular vs. non-regular subdivisions.

We do not change the function or the subdivision if, for all a 2 A, we drop!a to the value of that function at a, so that

(1.3) a 2 F () !a = h⌘F , ai+ ⇣F .

If these equalities hold, we call the pair (S!,!) tight. If all a 2 A arevertices of S!, then (S!,!) is automatically tight.

A non-face of a triangulation is a set of points whose convex hull doesnot form a face. Particularly important are the minimal non-faces, which donot form faces but for which every proper subset does. The list of minimalnon-faces completely characterizes a triangulation, as does the list of (setsof points that form) cells. A triangulation for which all minimal non-facescontain only two elements is called flag. Putting these properties together, aquadratic triangulation is defined as a regular unimodular flag triangulation.The triangulation on the right in Figure 1.1 is quadratic. However, the threewhite vertices of the triangulation in the middle form a minimal non-face.So that triangulation is not quadratic.

1.2. Why? Who? In this section, we present some applications of uni-modular triangulations and closely related objects (“Why?”), arranged bymathematical discipline (“Who?”).

1.2.1. Enumerative combinatorics. Many counting problems can be phrasedas counting lattice points in (dilates) of polytopes or polyhedral complexes [11,31]. By a fundamental result of Ehrhart, the number of lattice points in pos-itive dilates kP of P is a polynomial function of degree d in k 2 Z>0 [11,36].Consequently, the generating function has the special form:

X

k�0

#(kP \ Zd) tk =h⇤(t)

(1� t)d+1,

where h⇤(t) is a polynomial of degree d. If P has a unimodular triangu-lation, h⇤ equals the combinatorial h-polynomial of that triangulation [15].

A quadratic triangulation is a regular, unimodular, and flagtriangulation (flag:= every clique in the graph spans a simplex).



Constructions

Some examples



Constructions

Compresed polytopes

A compressed polytope P is a polytope of width one with respectto every facet. That is, for every facet hyperplane H of P, allvertices of P not in H lie in the next lattice translation of H.

All compressed polytopes have regular unimodular triangulations.In fact, all their pulling triangulations are unimodular.

We can use this to show that

Theorem (S. 1996, HPPS 2014+ for flagness)

If a polytope P has a (regular, flag) unimodular triangulation Tthen every integer dilation cP of it has one too.



Constructions

Compresed polytopes

A compressed polytope P is a polytope of width one with respectto every facet. That is, for every facet hyperplane H of P, allvertices of P not in H lie in the next lattice translation of H.All compressed polytopes have regular unimodular triangulations.In fact, all their pulling triangulations are unimodular.






Constructions

Compresed polytopes

A compressed polytope P is a polytope of width one with respectto every facet. That is, for every facet hyperplane H of P, allvertices of P not in H lie in the next lattice translation of H.All compressed polytopes have regular unimodular triangulations.In fact, all their pulling triangulations are unimodular.






Constructions

Compresed polytopes

Sketch of proof.

Consider the dilation cT of T , which subdivides cP into dilatedunimodular simplices.

Slice those simplices by all lattice translates of their facethyperplanes. This produces a subdivision of cT into compressedpolytopes (hypersimplices).

Any pulling refinement of this subdivision is unimodular.



Constructions

Compresed polytopes

Sketch of proof.






Constructions

Compresed polytopes

Sketch of proof.






Constructions

Semidirect product

Join and Cartesian product also preserve existence of unimodulartriangulations.We generalize both (plus dilations) to the following definition.

Definition

Let Q ⊂ Rd and Pi ⊂ Rdi for i = 1, . . . , n be lattice polytopes,and let φ : Zd → Zn be an integer affine map with φ(Q) ⊂ R≥0.The semidirect product of Q and the tuple (P1, . . . ,Pn) along φ is

Q nφ (P1, . . . ,Pn) := conva∈Q({a} ×

∏φi (a)Pi

),

where (φ1, . . . , φn) are the coordinates of φ.



Constructions

Semidirect product


Definition

Let Q ⊂ Rd and Pi ⊂ Rdi for i = 1, . . . , n be lattice polytopes,and let φ : Zd → Zn be an integer affine map with φ(Q) ⊂ R≥0.

The semidirect product of Q and the tuple (P1, . . . ,Pn) along φ is

Q nφ (P1, . . . ,Pn) := conva∈Q({a} ×

∏φi (a)Pi

),




Constructions

Semidirect product


Definition

Let Q ⊂ Rd and Pi ⊂ Rdi for i = 1, . . . , n be lattice polytopes,and let φ : Zd → Zn be an integer affine map with φ(Q) ⊂ R≥0.The semidirect product of Q and the tuple (P1, . . . ,Pn) along φ is

Q nφ (P1, . . . ,Pn) := conva∈Q({a} ×

∏φi (a)Pi

),




Constructions

Semidirect product

This includes:

∆d nId (P0, . . . ,Pd) is the join of P0, . . . ,Pd ,

{pt}n1 (P0, . . . ,Pd) is the product of P0, . . . ,Pd .

{pt}nk (P) is the k-th dilation of P.

The chimney chim(Q, `, u) (Haase-Paffenholz 2007)associated to two functionals ` ≤ u on Q is the semidirectproduct Q nu−` I , where I is a unimodular segment. (This isrelated to the so-called Nakajima polytopes, whose associatedtoric singularities are l.c.i)



Constructions

Semidirect product

This includes:







Constructions

Semidirect product

This includes:







Constructions

Semidirect product

This includes:




The chimney chim(Q, `, u) (Haase-Paffenholz 2007)associated to two functionals ` ≤ u on Q is the semidirectproduct Q nu−` I , where I is a unimodular segment.

(This isrelated to the so-called Nakajima polytopes, whose associatedtoric singularities are l.c.i)



Constructions

Semidirect product

This includes:







Constructions

Semidirect product

Theorem (HPPS 2014+)

If Q,P1, . . . ,and Pn admit unimodular triangulations, then everysemidirect product Q nφ (P1, . . . ,Pn) admits one too.

Remark

Semidirect product is essentially equivalent to nested configurations[Aoki et al. 2008]. Aoki et al. prove the theorem above under theassumption that all factor triangulations are regular.



Constructions

Semidirect product


If Q,P1, . . . ,and Pn admit unimodular triangulations, then everysemidirect product Q nφ (P1, . . . ,Pn) admits one too.

Remark

Semidirect product is essentially equivalent to nested configurations[Aoki et al. 2008]. Aoki et al. prove the theorem above under theassumption that all factor triangulations are regular.



Polytopes related to root systems

Polytopes from root systems

(Crystallographic) root systems give examples of particularly nicelattices. It seems natural to look at lattice polytopes related tothem. We can do this in two ways:

Polytopes cut out by roots: facet normals belong to the rootsystem (alcoved polytopes).

Polytopes with vertex sets contained in the root system.




Alcoved polytopes of type A

Payne (2009) has proved that all alcoved polytopes in the classicaltypes A, B, C and D are integrally closed. This suggests they mayall have unimodular triangulations.

In type A this is easy to show. Remember that.

An = {ei − ej : i , j ∈ [n + 1]} ⊂ Rn+1

An is a totally unimodular vector configuration. In particular, thehyperplane arrangement consisting of all lattice translates of thehyperplanes normal to the roots has all vertices in the lattice.Moreover, all cells in the arrangement are simplices.Hence:





Payne (2009) has proved that all alcoved polytopes in the classicaltypes A, B, C and D are integrally closed. This suggests they mayall have unimodular triangulations.In type A this is easy to show. Remember that.

An = {ei − ej : i , j ∈ [n + 1]} ⊂ Rn+1

An is a totally unimodular vector configuration.

In particular, thehyperplane arrangement consisting of all lattice translates of thehyperplanes normal to the roots has all vertices in the lattice.Moreover, all cells in the arrangement are simplices.Hence:






An = {ei − ej : i , j ∈ [n + 1]} ⊂ Rn+1

An is a totally unimodular vector configuration. In particular, thehyperplane arrangement consisting of all lattice translates of thehyperplanes normal to the roots has all vertices in the lattice.

Moreover, all cells in the arrangement are simplices.Hence:






An = {ei − ej : i , j ∈ [n + 1]} ⊂ Rn+1

An is a totally unimodular vector configuration. In particular, thehyperplane arrangement consisting of all lattice translates of thehyperplanes normal to the roots has all vertices in the lattice.Moreover, all cells in the arrangement are simplices.

Hence:






An = {ei − ej : i , j ∈ [n + 1]} ⊂ Rn+1

An is a totally unimodular vector configuration. In particular, thehyperplane arrangement consisting of all lattice translates of thehyperplanes normal to the roots has all vertices in the lattice.Moreover, all cells in the arrangement are simplices.Hence:





Theorem

Let P be an alcoved polytope of type A. The dicing triangulationobtained slicing P by all lattice hyperplanes normal to the roots isa flag, regular, unimodular (that is, quadratic) triangulation of P.

Remark

If ∆ = conv{v1, . . . , vn} is any lattice simplex with its verticesgiven in a specific order, we can consider the linear map sending itsfacet normals to the (normals of) the simple roots of type An, inthat order. The preimage of the A-dicing gives a canonicaltriangulation of c∆, for every c ∈ N, into simplices of the samevolume as ∆.This canonical triangulation will be important in our proof of theKMW theorem.





Theorem


Remark

If ∆ = conv{v1, . . . , vn} is any lattice simplex with its verticesgiven in a specific order, we can consider the linear map sending itsfacet normals to the (normals of) the simple roots of type An, inthat order. The preimage of the A-dicing gives a canonicaltriangulation of c∆, for every c ∈ N, into simplices of the samevolume as ∆.

This canonical triangulation will be important in our proof of theKMW theorem.





Theorem


Remark

If ∆ = conv{v1, . . . , vn} is any lattice simplex with its verticesgiven in a specific order, we can consider the linear map sending itsfacet normals to the (normals of) the simple roots of type An, inthat order. The preimage of the A-dicing gives a canonicaltriangulation of c∆, for every c ∈ N, into simplices of the samevolume as ∆.This canonical triangulation will be important in our proof of theKMW theorem.




Alcoved polytopes in other types

Theorem (HPPS-2014+)

Every alcoved polytope P of type B has a regular unimodulartriangulation

Sketch of proof.

First slice P by the hyperplanes corresponding to the “short roots”of type B. This gives a regular subdivision into compressed cells.Any pulling refinement of this is unimodular.

The triangulation in the theorem may need to use simplices thatare not alcoved.For other types:

In F4 and E8 we have explicit examples of polytopes withoutr.u.t.’sIn Cn, Dn, E6 and E7 we do not know.







Sketch of proof.


The triangulation in the theorem may need to use simplices thatare not alcoved.

For other types:








Sketch of proof.


The triangulation in the theorem may need to use simplices thatare not alcoved.For other types:





Polytopes spanned by roots

Polytopes spanned by the roots in An are arc-polytopes of digraphs.

It is easy to show that:

Theorem

Let S be a set of roots of type An. then conv(S ∪ {O}) always hasa regular unimodular triangulation (pulling from the origin).

Remark

When O 6∈ conv(S), conv(S) may not have any unimodulartriangulation (e.g. S = {12, 23, 34, 45, 14, 35}). Or it may haveunimodular triangulations but none of them flag (e.g.S = {12, 23, 13, 34, 45, 35})

In other types much less is known (Ohsugi-Hibi 2001 have somepartial results).





Polytopes spanned by the roots in An are arc-polytopes of digraphs.It is easy to show that:

Theorem


Remark








Theorem


Remark








Theorem


Remark






Edge polytopes

For a given graph G , we denote PG the convex hull of itsedge-vertex incidence vectors. (Put differently, edge-polytopes ofgraphs are the subpolytopes of the second hyper simplex{ei + ej : i , j ∈ [n]}).

PG has a unimodular cover iff no induced subgraph consists oftwo odd cycles (Ohsugi-Hibi, Simis-Vasconcelos-Villarreal,1998).

Even if PG is integrally closed, it may not have a r.u.t.(Hibi-Ohsugi 1999).

The Hibi-Ohsugi example has a non-regular unimodulartriangulation (Firla-Ziegler 1999).




Edge polytopes








Edge polytopes








Edge polytopes







Some smooth polytopes

Smooth empty polytopes

It is open whether every smooth (⇔ simple and with unimodularvertex-cones) has a unimodular triangulation (“Oda’s conjecture”).We can prove it if P is empty (no other lattice point than itsvertices).


Every empty smooth polytope is lattice equivalent to a product ofunimodular simplices.

Our proof is inspired by Kaibel-Wolff’s 2000 proof of: “any simple0/1-polytope is a product of 0/1-simplices”.




Smooth empty polytopes

It is open whether every smooth (⇔ simple and with unimodularvertex-cones) has a unimodular triangulation (“Oda’s conjecture”).We can prove it if P is empty (no other lattice point than itsvertices).


Every empty smooth polytope is lattice equivalent to a product ofunimodular simplices.

Our proof is inspired by Kaibel-Wolff’s 2000 proof of: “any simple0/1-polytope is a product of 0/1-simplices”.




Smooth reflexive polytopes

A reflexive polytope is a lattice polytope P with O ∈∫

(P) andsuch that its polar is also a lattice polytope. (Equivalently, alattice polytope that can be defined as P = {Ax ≤ 1} for aninteger matrix A).

There are finitely many reflexive polytopes in each dimension, andsmooth ones have been enumerated up to d = 9. The following isknown [Haase-Paffenholz 2007]:

All smooth reflexive d-polytopes for d ≤ 8 are integrallyclosed.

All smooth reflexive polytopes for d ≤ 6 admit a quadratictriangulation.

All but perhaps one (out of 72256) of the 7-dimensionalsmooth reflexive polytopes have a r. u. t.






(P) andsuch that its polar is also a lattice polytope. (Equivalently, alattice polytope that can be defined as P = {Ax ≤ 1} for aninteger matrix A).There are finitely many reflexive polytopes in each dimension, andsmooth ones have been enumerated up to d = 9. The following isknown [Haase-Paffenholz 2007]:









(P) andsuch that its polar is also a lattice polytope. (Equivalently, alattice polytope that can be defined as P = {Ax ≤ 1} for aninteger matrix A).There are finitely many reflexive polytopes in each dimension, andsmooth ones have been enumerated up to d = 9. The following isknown [Haase-Paffenholz 2007]:






The KMW Theorem

The KMW Theorem

There is the following classical theorem of Knudsen, Mumford, andWaterman (1973):

Theorem

Given a polytope P, there is a factor c = c(P) ∈ N such that thedilation c · P admits a regular unimodular triangulation.

This raises some questions:

Is there a c(d) common to every d-polytope?

What is a (good?) bound on c(P) for a given P?

What is the structure of the set of valid c(P)’s of a given P?Is it additively closed? (It at least contains all its multiples).

There are examples where cP has a r.u.t. but (c + 1)P is notintegrally closed [Cox-Haase-Hibi-Higashitani 2012].



The KMW Theorem

The KMW Theorem


Theorem









The KMW Theorem

The KMW Theorem


Theorem









The KMW Theorem

The KMW Theorem

If we relax the requirement from r.u.t. to weaker properties, thefollowing is known:

cP is integrally closed for every c ≥ d − 1 [Bruns-Gubeladze-Trung, 1997].

There is a c0(d) ∈ O(d6) such that cP has a unimodular coverfor every c ≥ c0 [Bruns-Gubeladze 2002, von Thaden 2008].



The KMW Theorem

The KMW Theorem

In contrast:

Neither the original KMW proof nor the reworking of it byBruns and Gubeladze (2009) contains any explicit bound onthe c needed for a given P.

Working out a bound from those proofs is not easy, and wouldcertainly lead to a tower of exponentials of length related tothe volume of P.

The regularity part of the proof is not totally clear (it isomitted in [Bruns-Gubeladze 2009]).



The KMW Theorem

The KMW Theorem

In contrast:






The KMW Theorem

The KMW Theorem

In contrast:






The KMW Theorem

An effective KMW Theroem

We prove the following:


If P is a lattice polytope of dimension d and (lattice) volumeVol(P), then the dilation

d!Vol(P)!dd2 Vol(P)P

has a regular unimodular triangulation.

More precisely, if P has a triangulation T into N d-simplices, ofvolumes V1, . . . ,VN , then the dilation

d!∑N

i=1 Vi !((d+1)!(d!)d)Vi−1

T

has a regular unimodular refinement.



The KMW Theorem

An effective KMW Theroem

We prove the following:


If P is a lattice polytope of dimension d and (lattice) volumeVol(P), then the dilation

d!Vol(P)!dd2 Vol(P)P

has a regular unimodular triangulation.More precisely, if P has a triangulation T into N d-simplices, ofvolumes V1, . . . ,VN , then the dilation

d!∑N

i=1 Vi !((d+1)!(d!)d)Vi−1

T




Canonical refinement of a dilated simplex

Canonical triangulation

Our proof is not substantially different from the previous ones, butuses a better “book-keeping” based on the canonical triangulationof dilations of an ordered simplex:

Definition

An ordered simplex ∆ is a simplex with its vertices given in aspecified order.

The canonical triangulation of c∆ is the inverse image of thedicing triangulation of type A, under the natural affine mapsending ∆ to an alcoved simplex of type A.





Our proof is not substantially different from the previous ones, butuses a better “book-keeping” based on the canonical triangulationof dilations of an ordered simplex:

Definition

An ordered simplex ∆ is a simplex with its vertices given in aspecified order.The canonical triangulation of c∆ is the inverse image of thedicing triangulation of type A, under the natural affine mapsending ∆ to an alcoved simplex of type A.





Canonical triangulations glue together nicely; if F is a face of P,the c.t. of P restricted to F is the c.t. of F . In particular:

Lemma

If T is a triangulation of P, canonically refining each simplex ofcT produces a triangulation of cP in which:

Volume of simplices is preserved. (Each simplex in the finaltriangulation has the volume of the simplex of of T that itrefines).

Regularity and flagness are preserved.

(Remark: as a corollary we recover that “if P has a quadratictriangulation then cP has one, for every c ∈ N”)






Lemma










Lemma










Lemma







Volume reduction

Reducing the volume of a single dilated simplex

Let ∆ be a non-unimodular simplex. let Λ∆ be the lattice spannedby its vertices (rather, the linear lattice parallel to it...), so thatVol(∆) = |Z2/Λ∆|.

A box point is a non-zero element of thisquotient.A box point for a simplex ∆ is non-degenerate if it is not in the(affine span) of any proper face of ∆.


Figure 4.3. The points in c� \ (m + L�). Here c = 5 and c0 = 2

a multiple of the number of non-zero coordinates in m. However, since wewill later apply the procedure to several box points at the same time, havingc be a multiple of (d + 1)! is convenient).

The structure of c� \ (m + L�) is straightforward: it is a translation ofthe points of L� in (c� c0)� (see Figure 4.3).

Assuming � is an ordered simplex allows us to speak of the canonicaltriangulation of c�, and of cF for every face F of �.

Let F be a face in a triangulation T and let m be a box point of F . Wesay that m is non-degenerate for F if it is not a box point of a proper faceof F . Equivalently, m is non-degenerate if all the F -coordinates of m arefractional. Then, the full-dimensional simplices of T for which m is a boxpoint are precisely the star of F . Recall that the (closed) star of a face Fin a simplicial complex T is the set star(F ; T ) of all simplices of T thatcontain F , together with all their faces. Equivalently:

star(F ; T ) := {F 0 2 T : F [ F 0 2 T }.

By @ star(F ; T ) we mean the faces of star(F ; T ) that do not contain F . If Tis a triangulation of a polytope and F is not contained in the boundary ofP then @ star(F ; T ) coincides with the topological boundary of star(F ; T ).

As before, a global ordering is assumed on the vertices of T , so that wecan speak of the canonical triangulation of each dilated face in cT .

Lemma 4.14. If T is a lattice triangulation on an ordered set of verticeslying in a lattice ⇤, and F = {v0, . . . , vk} is a non-unimodular face witha non-degenerate box point m = (m0, . . . , mk) 2 LF \ ⇤F , then for everyinteger c 2 kN, c · star(F ; T ) has a refinement Tm such that:

(1) The volume of every full-dimensional simplex �0 in Tm is strictlyless than the volume of simplex � for which �0 ⇢ c�.

(2) Tm induces the canonical triangulation on the boundary c·@ star(F ; T ).



Volume reduction


Let ∆ be a non-unimodular simplex. let Λ∆ be the lattice spannedby its vertices (rather, the linear lattice parallel to it...), so thatVol(∆) = |Z2/Λ∆|. A box point is a non-zero element of thisquotient.

A box point for a simplex ∆ is non-degenerate if it is not in the(affine span) of any proper face of ∆.







star(F ; T ) := {F 0 2 T : F [ F 0 2 T }.








Volume reduction


Let ∆ be a non-unimodular simplex. let Λ∆ be the lattice spannedby its vertices (rather, the linear lattice parallel to it...), so thatVol(∆) = |Z2/Λ∆|. A box point is a non-zero element of thisquotient.A box point for a simplex ∆ is non-degenerate if it is not in the(affine span) of any proper face of ∆.







star(F ; T ) := {F 0 2 T : F [ F 0 2 T }.








Volume reduction

Reducing the volume in a single simplex

Lemma (Elementary volume reduction)

If T is a lattice triangulation on an ordered set of vertices andF = {v0, . . . , vk} is a non-unimodular face with a non-degeneratebox point m = (m0, . . . ,mk) ∈ Zd \ ΛF , then for every integerc ∈ (k + 1)N, c · Star(F ; T ) has a refinement Tm such that:

1 The volume of every full-dimensional simplex ∆′ in Tm isstrictly less than the volume of simplex ∆ for which ∆′ ⊂ c∆.

2 Tm induces the canonical triangulation on the boundaryc · ∂ Star(F ; T ).

3 Tm is a regular refinement of T , so if T is regular then Tm isregular.



Volume reduction


Lemma (Elementary volume reduction)

If T is a lattice triangulation on an ordered set of vertices andF = {v0, . . . , vk} is a non-unimodular face with a non-degeneratebox point m = (m0, . . . ,mk) ∈ Zd \ ΛF , then for every integerc ∈ (k + 1)N, c · Star(F ; T ) has a refinement Tm such that:

1 The volume of every full-dimensional simplex ∆′ in Tm isstrictly less than the volume of simplex ∆ for which ∆′ ⊂ c∆.

2 Tm induces the canonical triangulation on the boundaryc · ∂ Star(F ; T ).

3 Tm is a regular refinement of T , so if T is regular then Tm isregular.



Volume reduction




Volume reduction



Figure 4.4. star(F ; T ) (left) and the subdivision of c · star(F ; T ) intoCayley polytopes via concentric dilated copies of @ star(F ; T )(right)

particular, to compute the number of simplices there is no loss of generalityin assuming that all the non-zero {mi}’s are equal to one another, henceequal to c0/d0. In this case, all the simplices in Tm have volume exactlyc0/d0 times the volume of �, so the number of them needed to fill c� equals

d0cd

c0 (d + 1)cd.

In the last inequality we use that d0 d + 1 and c0 � 1. ⇤

4.4. A proof of the KMW Theorem. Lemma 4.14 can be applied simul-taneously to several faces F1, . . . , FN each with a non-degenerate box pointm1, . . . , mN , as long as the stars of the Fi’s intersect only on their bound-aries. Equivalently, as long as there is not an i, j pair for which conv(Fi[Fj)is a simplex, or as long as the open stars of the Fi’s are disjoint. Here theopen star of a face in a simplicial complex is defined as:

star(F ; T ) = {F 0 2 T : F ⇢ F 0}.

(Observe that open stars are not simplicial complexes.)In the following statement, we assume that c is a multiple of d! to guar-

antee that it is a multiple of the dimension of every Fi.

Corollary 4.16. Let T be a lattice triangulation (with an ordering of itsvertices). Suppose there is a family F1, . . . , FN of faces of T with respectivenon-degenerate box points m1, . . . , mN such that conv(Fi [ Fj) 62 T , forevery pair i, j 2 [N ].

Then, for every integer c 2 d!N, the dilation cT can be refined into atriangulation T 0 with the following properties:



Volume reduction

Reducing the volume in several simplices at a time

Remarks:

If we have non-degenerate box-points m1, . . . ,mN for a familyof simplices F1, . . . ,FN with disjoint stars, the reductionlemma can be applied simultaneously to all of them, to reducethe volumes in all stars simultaneously.

This happens, for example, for all simplices of prime volume:

Corollary

Let T be a triangulation of a lattice polytope P and assume thatthe maximal volume V among all simplices in T is a prime. Then(d + 1)!T can be refined to a triangulation with all simplices ofvolume < V .

. . . so, if every number was a prime, . . . (d + 1)!V T would have aunimodular refinement . . . .



Volume reduction


Remarks:



Corollary


. . . so, if every number was a prime, . . .

(d + 1)!V T would have aunimodular refinement . . . .



Volume reduction


Remarks:



Corollary


. . . so, if every number was a prime, . . . (d + 1)!V T would have aunimodular refinement . . . .



Volume reduction

Reducing the volume in all simplices iteratively

What we can still do is apply the redcution lemma over and over,hoping that eventually we get rid off all simplices of maximalvolume V , then go to those of volume V − 1, etc.

Problem

If we do not process all simplcies of volume V at the same time, inthe unprocessed ones what we get is the canonical refinement ofthe dilation, which creates a lot of new simplices of volume V .The number of simplices of volume V will actually increase, notdecrease.

Knudsen-Mumford-Waterman and Bruns-Gubeladze solve this viathe use of “rational structures” or “local lattices”.. . . which leadsto a tower of exponentials.We solve it by taking advantage of some properties of canonicaltriangulations.



Volume reduction



Problem

If we do not process all simplcies of volume V at the same time, inthe unprocessed ones what we get is the canonical refinement ofthe dilation, which creates a lot of new simplices of volume V .

The number of simplices of volume V will actually increase, notdecrease.




Volume reduction



Problem





Volume reduction



Problem


Knudsen-Mumford-Waterman and Bruns-Gubeladze solve this viathe use of “rational structures” or “local lattices”.

. . . which leadsto a tower of exponentials.We solve it by taking advantage of some properties of canonicaltriangulations.



Volume reduction



Problem


Knudsen-Mumford-Waterman and Bruns-Gubeladze solve this viathe use of “rational structures” or “local lattices”.. . . which leadsto a tower of exponentials.

We solve it by taking advantage of some properties of canonicaltriangulations.



Volume reduction



Problem





An effective KMW Theorem

Canonical refinement, revisited

Definition

An ordered k-simplex is a simplex with a specified order in itsvertices. Two ordered simplices ∆ = conv{p0, . . . , pk} and∆′ = conv{p′o , . . . , p′k} are called A-equivalent if

{pi − pi−1 : i = 1 . . . k} = {p′i − p′i−1 : i = 1 . . . k}

Lemma (A-equivalence)

1 All the simplices in the canonical triangulation of c∆ areA-equivalent to ∆.

2 If two simplices ∆ and ∆′ are A-equivalent then the A-dicingdefined by ∆ and by ∆′ are the same, modulo a translation.





Definition

An ordered k-simplex is a simplex with a specified order in itsvertices. Two ordered simplices ∆ = conv{p0, . . . , pk} and∆′ = conv{p′o , . . . , p′k} are called A-equivalent if

{pi − pi−1 : i = 1 . . . k} = {p′i − p′i−1 : i = 1 . . . k}

Lemma (A-equivalence)

1 All the simplices in the canonical triangulation of c∆ areA-equivalent to ∆.

2 If two simplices ∆ and ∆′ are A-equivalent then the A-dicingdefined by ∆ and by ∆′ are the same, modulo a translation.





Part (2) of the previous lemma allows us to consider a box pointfor a simplex ∆ as a box point for any other A-equivalent simplex∆′ (by the unique, modulo L∆ translation sending one A-dicing tothe other).

The crucial property that we need is:

Lemma

Let ∆ and ∆′ be two A-equivalent simplices in a triangulation T ,and let m be a box point for both (in the above sense). Let F andF ′ be the faces of ∆ and ∆′ for which m is non-degenerate.Then, either F = F ′ or they have disjoint stars.





Part (2) of the previous lemma allows us to consider a box pointfor a simplex ∆ as a box point for any other A-equivalent simplex∆′ (by the unique, modulo L∆ translation sending one A-dicing tothe other).The crucial property that we need is:

Lemma

Let ∆ and ∆′ be two A-equivalent simplices in a triangulation T ,and let m be a box point for both (in the above sense). Let F andF ′ be the faces of ∆ and ∆′ for which m is non-degenerate.Then, either F = F ′ or they have disjoint stars.





Thus:

Corollary

The elementary volume reduction can be applied simultaneously toall simplices of a given A-equivalence class.

Corollary

Let T be a triangulation of a lattice polytope P and let V be themaximal volume V . Let N be the number of A-equivalence classesof maximal simplices of volume V in T .Then, (d + 1)!NT can be refined to a triangulation with allsimplices of volume < V .





Thus:

Corollary

The elementary volume reduction can be applied simultaneously toall simplices of a given A-equivalence class.

Corollary

Let T be a triangulation of a lattice polytope P and let V be themaximal volume V . Let N be the number of A-equivalence classesof maximal simplices of volume V in T .Then, (d + 1)!NT can be refined to a triangulation with allsimplices of volume < V .




An algorithm

To get a unimodular refinement of cP for some constant c :

1 Construct any lattice triangulation T of P. Let V be themaximal volume among its simplices and N the number ofA-equivalence classes of them.

2 While N > 0, apply the reduction lemma to all the simplicesin one of the A-equivalence classes of volume V . This reducesby (at least) one the number of them.

3 At the end of step 2 all simplices have volume bounded by aV ′ < V . Iterate.

Remark: in all steps regularity of the triangulation can bepreserved.




An algorithm

To get a unimodular refinement of cP for some constant c :

1 Construct any lattice triangulation T of P. Let V be themaximal volume among its simplices and N the number ofA-equivalence classes of them.

2 While N > 0, apply the reduction lemma to all the simplicesin one of the A-equivalence classes of volume V . This reducesby (at least) one the number of them.

3 At the end of step 2 all simplices have volume bounded by aV ′ < V . Iterate.

Remark: in all steps regularity of the triangulation can bepreserved.




Analysis of the algorithm

Each elementary reduction substitutes an equivalence class ofA simplices of a certain volume V to at most (d + 1)d!d

classes of simplices of volume at most V − 1.

Setting up the corresponding recursion, to unimodularly refinea dilation of a particular A-class of volume V , at most

V !((d + 1)!cd

)V−1A-equivalence classes of intermediate

simplices are used.Theorem (HPPS 2014+)

If a lattice polytope P of dimension d has a triangulation into Nd-simplices, of volumes V1, . . . ,VN , then the dilation of factor

d!

(∑Ni=1 Vi !((d+1)!(d!)d)

Vi−1)T







classes of simplices of volume at most V − 1.Setting up the corresponding recursion, to unimodularly refinea dilation of a particular A-class of volume V , at most

V !((d + 1)!cd


simplices are used.



d!

(∑Ni=1 Vi !((d+1)!(d!)d)

Vi−1)T







classes of simplices of volume at most V − 1.Setting up the corresponding recursion, to unimodularly refinea dilation of a particular A-class of volume V , at most

V !((d + 1)!cd


simplices are used.Theorem (HPPS 2014+)


d!

(∑Ni=1 Vi !((d+1)!(d!)d)

Vi−1)T




KMW in 3d

Dimension 3

In dimension three the following is known:

For every lattice 3-polytope P, 2P has a unimodular cover(Ziegler 1997, Kantor-Sarkaria 2003).

Not for every lattice 3-simplex ∆, 2∆ has a unimodulartriangulation (Ziegler 1997, Kantor-Sarkaria 2003).

For every lattice 3-polytope P and every c ≥ 4, 4P has aunimodular triangulation (Ziegler 1997).

For every lattice 3-polytope P, 4P has a unimodulartriangulation (Kantor-Sarkaria 2003).

Theorem (SZ 2013)

For every c ≥ 4 except perhaps c = 5 and for every lattice3-polytope P, cP has a unimodular triangulation.

Remark: We cannot guarantee regularity.



KMW in 3d

Dimension 3






Theorem (SZ 2013)





KMW in 3d

Dimension 3






Theorem (SZ 2013)





KMW in 3d

Dimension 3






Theorem (SZ 2013)





KMW in 3d

Dimension 3






Theorem (SZ 2013)





KMW in 3d

Dimension 3






Theorem (SZ 2013)


Remark: We cannot guarantee regularity.F. Santos Unimodular triangulations of lattice polytopes


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