Classifications, volume bounds and universal Ehrhart inequalities of Gabriele Balletti Academic dissertation for the Degree of Doctor of Philosophy in at Stockholm University to be publicly defended on Monday 22 October 2018 at 13.00 in sal 14, hus 5, Kräftriket, Roslagsvägen 101.

Abstract In this PhD thesis we study relations among invariants of lattice polytopes. Particular emphasis is placed on bounds for the volume of lattice polytopes with interior points, and inequalities for the coefficients of their Ehrhart delta . The major tools used for this investigation are explicit classifications and computer-assisted proofs. In the first paper we give an upper bound on the volume of a which is dual to a d-dimensional lattice polytope with exactly one interior lattice point, in each d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. In the second paper we classify the three-dimensional lattice polytopes with two lattice points in their strict interior. Up to unimodular equivalence there are 22 673 449 such polytopes. This classification allows us to verify, for this case only, the sharp conjectural upper bound for the volume of a lattice polytope with interior points, and provides strong evidence for more general new inequalities on the coefficients of the Ehrhart delta in dimension three. In the third paper we prove the existence of inequalities for the coefficients of the Ehrhart delta polynomial of a lattice polytope P which do not depend on the degree or dimension of P. This proves that the space of all Ehrhart delta polynomials of lattice polytopes have coordinate-projections whose images do not fully cover the codomain. This is done by extending Scott's inequality to lattice polytopes whose Ehrhart delta polynomial has vanishing cubic coefficient. In the fourth paper we associate to any digraph D a simplex P whose vertices are given as the rows of the Laplacian of D, generalizing a work of Braun and Meyer. We show how basic properties of P can be read from D, for example the normalized volume of P equals the complexity of D, and P contains the origin in its relative interior if and only if D is strongly connected. We extend Braun and Meyer's study of cycles, by characterizing properties such as being Gorenstein and IDP. This is used to produce interesting examples of reflexive polytopes with non-unimodal Ehrhart delta vectors. In the fifth paper we describe an algorithm for an explicit enumeration of all equivalence classes of lattice polytopes, once dimension and volume are fixed. The algorithm is then implemented to create a database of small lattice polytopes up to dimension six. The resulting database is then compared with existing ones, used to understand the combinatorics of small smooth polytopes, and to give conjectural inequalities for coefficients of Ehrhart delta polynomials in dimension three. The frequency of some of the most important properties of lattice polytopes can be explicitly studied, and interesting minimal examples are extracted and discussed.

Stockholm 2018 http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-159311

ISBN 978-91-7797-416-1 ISBN 978-91-7797-417-8

Department of Mathematics

Stockholm University, 106 91 Stockholm

CLASSIFICATIONS, VOLUME BOUNDS AND UNIVERSAL EHRHART INEQUALITIES OF LATTICE POLYTOPES

Gabriele Balletti

Classifications, volume bounds and universal Ehrhart inequalities of lattice polytopes

Gabriele Balletti ©Gabriele Balletti, Stockholm University 2018

ISBN print 978-91-7797-416-1 ISBN PDF 978-91-7797-417-8

Printed in Sweden by Universitetsservice US-AB, Stockholm 2018 Distributor: Department of Mathematics, Stockholm University Abstract

In this PhD thesis we study relations among invariants of lattice poly- topes. Particular emphasis is placed on bounds for the volume of lattice polytopes with interior points, and inequalities for the coefficients of their h∗-polynomials. The major tools used for this investigation are explicit classifications and computer-assisted proofs. In the first paper we give an upper bound on the volume vol(P ∗) of a polytope P ∗ dual to a d-dimensional lattice polytope P with exactly one interior lattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. In the second paper we classify the three-dimensional lattice polytopes with two lattice points in their strict interior. Up to unimodular equiv- alence there are 22 673 449 such polytopes. This classification allows us to verify, for this case only, the sharp conjectural upper bound for the volume of a lattice polytope with interior points, and provides strong evidence for more general new inequalities on the coefficients of the h∗- polynomial in dimension three. In the third paper we prove the existence of inequalities for the coeffi- cients of the h∗-polynomial of a lattice polytope P which do not depend on the degree or dimension of P . This proves that the space of all h∗-polynomials of lattice polytopes have coordinate-projections whose images do not fully cover the codomain. This is done by extending Scott’s inequality to lattice polytopes having h∗-polynomial satisfying ∗ h3 = 0. In the fourth paper we associate to any digraph D a simplex PD whose vertices are given as the rows of the Laplacian of D, generalizing a work of Braun and Meyer. We show how basic properties of PD can be read from D, for example the normalized volume of PD equals the complexity of D, and PD contains the origin in its relative interior if and only if D is strongly connected. We extend Braun and Meyer’s study of cycles, by characterizing properties such as being Gorenstein and IDP. This is used to produce interesting examples of reflexive polytopes with non- unimodal h∗-vectors. In the fifth paper we describe an algorithm for an explicit enumeration of all equivalence classes of lattice polytopes, once dimension and volume are fixed. The algorithm is then implemented to create a database of “small” lattice polytopes up to dimension six. The resulting database is then compared with existing ones, used to understand the combinatorics properties of small smooth polytopes, and to give conjectural inequalities for coefficients of h∗-polynomials, in dimension three. The frequency of some of the most important properties of lattice polytopes can be explicitly studied, and interesting minimal examples are extracted and discussed. Sammanfattning

I denna doktorsavhandling studerar vi relationer mellan invarianter för gitterpolytoper. Särskilt fokus läggs på gränser för volymen av gitter- polytoper med inre punkter och olikheter för koefficienterna till deras h∗-polynom. De huvudsakliga verktygen som används till denna under- sökning är explicita klassifikationer och datorstödda bevis. I den första artikeln ger vi en övre gräns på volymen vol(P ∗) av en polytop P ∗ som är dual till en d-dimensionell gitterpolytop P med exakt en gitterpunkt i dess inre, för alla dimensioner d. Denna gräns, uttryckt i termer av Sylvester-följden, är skarp, och uppnås av dualen till ett särskilt reflexivt simplex. Vårt resultat medför en skarp övre gräns för volymen av d-dimensionella reflexiva polytoper. I den andra artikeln klassificerar vi de tredimensionella gitterpolytoper- na som har två gitterpunkter i deras strikta inre. Upp till unimodulär ekvivalens så finns det 22 673 449 sådana polytoper. Denna klassifikation tillåter oss att verifiera, för enbart detta fall, den skarpa förmodade övre gränsen för volymen av en gitterpolytop med inre punkter, och ger starkt stöd för mer allmänna olikheter på koefficienterna till h∗-polynomet i di- mension tre. I den tredje artikeln bevisar vi existensen av olikheter för koefficienterna till h∗-polynomet av en gitterpolytop P som inte beror på graden eller dimensionen för P Detta visar att rummet av alla h∗-polynom av gitter- polytoper har koordinatprojektioner vars bildmängder inte helt täcker målmängden. Detta görs genom att utvidga Scotts olikhet till gitterpo- ∗ ∗ lytoper med h -polynom som uppfyller h3 = 0. I den fjärde artikeln förknippar vi till alla riktade grafer D ett simplex PD vars hörn ges som raderna i Laplacianen för D, vilket generaliserar ett verk av Braun och Meyer. Vi visar hur grundläggande egenskaper för PD kan avläsas från D, till exempel så är den normaliserade volymen för PD lika med komplexiteten för D och PD innehåller origo i sitt relativa inre om och endast om D är starkt sammanhängande. Vi utvidgar Braun och Meyers studie av cykler genom karakterisering av egenskaper som att vara Gorenstein och IDP. Detta används för att producera intressanta exempel på reflexiva polytoper med ickeunimodala h∗-vektorer. I den femte artikeln beskriver vi en algoritm för en explicit enumeration av alla ekvivalensklasser av gitterpolytoper, när väl dimensionen och vo- lymen är fixerade. Algoritmen är sedan implementerad för att skapa en databas av “små” gitterpolytoper upp till sex dimensioner. Den data- basen jämförs sedan med existerande databaser, används för att förstå kombinatoriska egenskaper för små glätta polytoper och för att ge för- modade olikheter för koefficienterna till h∗-polynom i tre dimensioner. Frekvensen för några av de viktigaste egenskaperna för gitterpolytoper kan studeras uttryckligen och intressanta minimala exempel extraheras och diskuteras. List of Papers

The following papers, referred to in the text by their Roman numerals, are included in this thesis.

PAPER I: On the maximum dual volume of a canonical Fano polytope Balletti, G. , Kasprzyk, A. M. , Nill, B. , submitted (2016).

PAPER II: Three-dimensional lattice polytopes with two in- terior lattice points Balletti, G. , Kasprzyk, A. M. , submitted (2016).

PAPER III: Universal inequalities in Ehrhart Theory Balletti, G. , Higashitani, A. , Israel Journal of Mathe- matics, to appear (2018).

PAPER IV: Laplacian simplices associated to digraphs Balletti, G. , Hibi, T. , Meyer, M. , Tsuchiya, A. , Arkiv för Matematik, to appear (2018).

PAPER V: Enumeration of lattice polytopes by their volume Balletti, G. , preprint (2018).

Reprints were made with permission from the publishers. Papers I–II also appear in the author’s licentiate thesis.

Contents

Abstract i

Sammanfattning iii

List of Papers v

Acknowledgements ix

General Introduction 11 1.1 Convex polytopes ...... 13 1.2 Let there be a lattice! ...... 14 1.3 The Ehrhart Theory ...... 17 1.4 Volume of lattice polytopes ...... 21 1.5 Content of the thesis ...... 22 1.5.1 Paper I: the dual volume of a lattice polytope . . . 22 1.5.2 Paper II: three-dimensional polytopes with two in- terior lattice points ...... 23 1.5.3 Paper III: universal Ehrhart inequalities ...... 24 1.5.4 Paper IV: simplices associated to digraphs . . . . . 25 1.5.5 Paper V: enumeration of lattice polytopes by their volume ...... 26

References xxix

Acknowledgements

First and foremost I would like to thank my advisor Benjamin Nill for introducing me to the topics of the thesis, and for his constant support during my PhD studies. Under his guidance and through his encourage- ment in these four years I could learn a lot, not only about mathematics. I also thank my friend and master thesis supervisor Matteo Varbaro, a true source of inspiration for me, for everything he taught me and for having always been there all the time I needed advice. Special thanks go to Al Kaprzyk for the time spent working together in London and Nottingham. These visits were extremely valuable for my early career, shaping the way I have been doing mathematics since then. Furthermore, it has been always a pleasure to read his great emails. Many other mathematicians from all over the world played an ex- tremely important role in making this thesis possible. Among them I wish to thank Takayuki Hibi, Akihiro Higashitani, Michael Joswig, Marta Panizzut, Bernd Sturmfels and Akiyoshi Tsuchiya for their invi- tations and support that led to great collaborations. I am also grateful to Bruno Benedetti, for his advice and for stimulating my interest to- ward discrete mathematics, thanks to a cycle of five lectures he held in Genoa back in 2011. Similarly, I thank Aldo Conca and the whole Commutative Algebra group in Genoa, for teaching me to appreciate the algebraic side of every bit of mathematics. For great advice and discus- sions about mathematics and academia I also thank Gennadiy Averkov, Matthias Beck, Christopher Borger, Monica Blanco, Alessio D’Alì, San- dra Di Rocco, Martin Henk, Johannes Hofscheier and Paco Santos. I cannot forget to thank my high school math teacher Francesco Bertolini. My interest in mathematics originated back then, thanks to his great teachings and many embarrassingly low grades. Behind every result that an academic may achieve, there is the work of many people making sure that the workplace, the conditions and the environment are ideal. For this reason I thank all the amazing people working at the Department of Mathematics at Stockholm University, for making my work simple and fun, and for making me feel at home whenever I have been in the math building. I also wish to thank the great group of friends I met here in Stock- holm, many of which have been colleagues, for everything we shared together: the trips, the dinners, the beers, the kräftskivor, the life in the math department and, above all, for having been there when I needed. I am also grateful to the unofficial members of the lattice polytopes com- munity, especially to the youngest ones, for all the fun we had together in Berlin, in Colombia, in Japan and in many other places: with them, any workshop and summer school feels like a vacation. I also thank my friends in Genoa for making me always count the days to my next flight to my hometown and for finding the time to meet me when I am there. I thank my family, in particular my parents Anna and Maurizio, my siblings Francesca and Luca, and their families, for their teachings and unconditional support. I finally thank Chiara: with her I am not afraid to face up all the adventures that life has to offer. The love and support of all these people is, at the end of the day, everything that matters.

A brainstorming session with some colleagues during the “Summer Workshop on Lattice Polytopes” in Osaka, August 2018. General Introduction 12 This introductory chapter is meant to be an introduction the main topics of this PhD thesis. Sections 1.2–1.4 are dedicated to introduce the concept of lattice polytope and to give an idea of the most impor- tant results and open problems concerning lattice polytopes and their invariants. Section 1.5 is an essential introduction to the main results of the papers. Papers I–II and parts of this introduction also appear in the author’s licentiate thesis [2].

1.1 Convex polytopes

A convex polytope P is the of finitely many points v1,...,vn d d in R , i.e. P is the smallest convex set of R containing all the points ◦ v1,...,vn. We denote by P and ∂P the (relative) interior and boundary of P , respectively. The dimension of P is the dimension of the affine space spanned by v1,...,vn. We denote it by dim(P ). If dim(P ) = d, d then we say that P is full-dimensional. Given an hyperplane H ⊂ R , we say that H is supporting P if it does not intersect the relative interior of P , i.e. if P ◦ ∩ H = ∅. Any intersection of P with any supporting hyperplane H defines a face P ∩ H of P . By definition, the faces of a polytope are polytopes. Note that ∅ is a face of any polytope P , and we set dim(∅) = −1. Faces of dimension 0, 1 and dim(P ) − 1 are often called vertices, edges, and facets of P . We also consider P itself as the unique dim(P )-dimensional face of P . We call simplex a polytope having dim(P ) + 1 vertices. Simplices of dimension 1, 2 or 3, usually go under the names of segment, triangle and tetrahedron, respectively.

Figure 1.1: A three-dimensional polytope.

13 1.2 Let there be a lattice!

Convex polytopes are fundamental objects in mathematics, and they are currently an extremely active area of research. It is in our interest to study polytopes endowed by an additional structure. We call lattice a d discrete Z-submodule of R . Here by discrete we mean that there is a constant ε > 0 such that the distance between two distinct elements of the lattice is greater than ε. Anyway, unless otherwise stated, we choose d d Z as the lattice, as any lattice of rank d can be mapped into Z via an invertible linear transformation. In some parts of Papers I and III it will be convenient to work with other lattices, obtained by coarsening of the d original ambient lattice Z . When this will be done, it will be properly specified. We can now define the main objects of investigation of this thesis.

d Definition 1.2.1 A lattice polytope (with respect to the lattice Z ) is d a convex polytope whose vertices are in Z . Due to our choice of lattice, we sometimes refer to lattice polytopes as integer polytopes. Our primary source of inspirations for lattice poly- topes as well as suggested reference for further insights is the book Com- puting the continuous discretely by Beck and Robins [3]. It is clear that, in some contexts, it may make sense to identify a d lattice polytope P with any of its translations P + v, where v ∈ Z . In the same way, rotating, flipping or applying any linear transformation to P preserves most of its properties, provided that the lattice is preserved under such transformations (see Figure 1.2 as an example). We make this concept formal by defining the following equivalence relation on the set of d-dimensional lattice polytopes.

d Definition 1.2.2 Two full-dimensional lattice polytopes P,Q ⊂ R are said to be unimodular equivalent, and we denote this by P =∼ Q, if there d exist an element of the lattice v ∈ Z and a lattice preserving invertible d automorphism φ ∈ GLd(Z) of R such that φ(P + v) = Q.

Figure 1.2: Two unimodular equivalent lattice polytopes.

14 Introducing a “grid” structure to a lattice polytope allows to do com- parisons between its classical geometric nature, and the discrete one, carried by the lattice. Therefore we can make a distinction between two different notions of volume for a lattice polytope P : a classical and con- tinuous one, in which the information is distributed homogeneously in the polytope, and a quantized and discrete one, in which the information is focused on the lattice points. We can make this idea explicit by defining the volume of a full- d R dimensional lattice polytope P in R as the P 1 dµ, and by choosing an opportune measure µ depending on the context: if we are inter- ested in the continuous volume of P we choose µ to be the standard d-dimensional Lebesgue measure, if we are interested in its discrete vol-

d d ume we choose µ to be the “counting measure” R ⊃ X 7→ X ∩ Z . To keep things simple (and coherent with the notation used nowadays) we will simply call volume the continuous volume of P and we denote it by

d vol(P ), while its discrete counterpart P ∩ Z will be called size. It is safe (and instructive) to consider the integral vol(P ) in the Rie- mannian sense. Indeed we can think of computing it by approximating 1 1 P by putting boxes having side length k – and therefore volume kd – on 1 d every point of k Z . By taking the limit we get vol(P ):

1 1 d vol(P ) = lim P ∩ Z . (1.1) k→+∞ kd k

k = 1 k = 2 k → +∞

Figure 1.3: From discrete to continuous volume.

Equation (1.1) allows one to find the volume of P by knowing the asymptotic size of integers dilations of a lattice polytope P . Indeed,

1 d d since P ∩ k Z = kP ∩ Z , one can calculate vol(P ) as

1 d vol(P ) = lim kP ∩ Z . (1.2) k→+∞ kd

15 Clearly this is an unhandy way to calculate vol(P ), so it makes sense to investigate some low-dimensional case, looking for stronger relations between the volume of P and its size. The one-dimensional case is straightforward: if P is a line segment having integer endpoints, clearly its volume (or length) satisfies vol(P ) = |P ∩ Z| − 1. In dimension two the situation becomes interesting thanks to Pick’s Theorem. Theorem 1.2.3 (Pick’s Theorem) The volume of a two-dimensional lattice polytope P can be expressed in terms of the number of lattice points in the interior and on the boundary of P as 1 vol(P ) = P ◦ ∩ 2 + ∂P ∩ 2 − 1. Z 2 Z Although elementary proofs of Theorem 1.2.3 are possible, the most suitable way to present this result on this thesis it to see it as a corollary of the Ehrhart-theoretic results presented in Section 1.3. Just by examining the one- and two-dimensional cases, one may feel tempted to conjecture the existence of a formula expressing the volume of a lattice polytope P of arbitrary dimension just by counting the lattice points in P and in its faces. This fails already in dimension three, where it is possible to realize polytopes with a fixed number of lattice points, but with arbitrarily large volume. An example of this is the Reeve’s Tetrahedron of Figure 1.4.

Figure 1.4: The Reeve’s Tetrahedron has no lattice points other than its four vertices (0,0,0), (0,1,0), (0,0,1) and (1,1,a) (for some a ∈ Z≥1).

From the previous example it results clear that in general it is not possible to extract enough information from the distribution of the lat- tice points in a lattice polytope P . The Ehrhart Theory, presented in the following section, finds a compromise between Pick’s Theorem and equality (1.2).

16 1.3 The Ehrhart Theory

Ehrhart Theory studies the distributions of lattice points in lattice poly- topes and in their dilations. It takes name after Eugène Ehrhart, who proved the main theorem (Theorem 1.3.1) in this subject. We define the Ehrhart function, which counts the lattice points in the k-th dilation of a lattice polytope P

d ehrP (k) := kP ∩ Z for k ≥ 1. Ehrhart proved that the Ehrhart function behaves polynomially.

Theorem 1.3.1 (Ehrhart’s Theorem) Let P be a d-dimensional lat- tice polytope. Then the Ehrhart function ehrP agrees with a polyno- mial of degree d called the Ehrhart polynomial of P , i.e. there exist c0,...,cd ∈ Q such that c0 = 1 and d ehrP (k) = c0 + c1k + ... + cdk .

From now on we identify the Ehrhart function ehrP (k) with the corresponding Ehrhart polynomial. Directly from (1.2) it follows that the leading coefficient of ehrP is the volume of P , indeed

ehrP (k) 1 d cd = lim = lim kP ∩ Z = vol(P ). k→+∞ kd k→+∞ kd It is possible to find an interpretation also for the second leading coefficient cd−1: 1 c = vol(∂P ), d−1 2 P where vol(∂P ) := F facet of P vol(F ). Unfortunately there is still not a canonical way to give a combinatorial interpretation to all the other coefficients. Anyway, for each i it is only known that d!ci ∈ Z. Note that, in case P is two-dimensional, Pick’s Theorem 1.2.3 follows 2 directly from the properties of ehrP . Indeed P ∩ Z = ehrP (1) = 1 + 1 2 vol(∂P ) + vol(P ), so 1 1 vol(P ) = P ∩ 2 − vol(∂P ) − 1 = P ∩ 2 − ∂P ∩ 2 − 1 Z 2 Z 2 Z 1 = P ◦ ∩ 2 + ∂P ∩ 2 − 1. Z 2 Z Once we know that the Ehrhart function can be extended to a poly- nomial, it is possible to evaluate it at any real number. An interesting behavior can be observed when evaluating it at negative integer values.

17 d Theorem 1.3.2 (Ehrhart-Macdonald reciprocity) Let P ⊂ R be a d-dimensional lattice polytope. Then

d ◦ d ehrP (−k) = (−1) kP ∩ Z . Due to the fact that the coefficients of the Ehrhart polynomial do not have a fully understood combinatorial interpretation, it is often con- venient to introduce an equivalent invariant. As always, let P be a d-dimensional lattice polytope. It is well known that the of ehrP can be rewritten in the following form h∗ (t) X ehr (k)tk = P , P (1 − t)d+1 k≥0

∗ ∗ ∗ d where hP (t) = 1 + hP,1t + ··· + hP,dt is a polynomial of degree at most d. We call it the h∗-polynomial of P or Ehrhart δ-polynomial. Note that ∗ hP is nothing but a doppelgänger of ehrP , indeed one can think of it just as ehrP written with a different base of the polynomial ring:

! ! ! ! t + d t + d − 1 t + 1 t ehr (t) = + h∗ + ··· + h∗ + h∗ . P d P,1 d P,d−1 d P,d d

2P × {2}

P × {1}

Figure 1.5: In algebraic language ehrP can be interpreted as the Hilbert func- tion of the semigroup algebra generated by the lattice points in the cone over P × {1}, considered with the grading given by the last coordinate. In this case ∗ hP is the Hilbert polynomial of the semigroup algebra.

The coefficients of the h∗-polynomial have a combinatorial interpre- d tation. Specifically, if P ⊂ R is a d-dimensional simplex with vertices ∗ d+1 v0,...,vd, then hk counts the number of points of Z in the half-open parallelepiped

( d ) X d+1 Π(P ) := λi(vi,1) ⊂ R : 0 ≤ λi < 1 i=0

18 having last coordinate equal to k.

∗ → hP,2 = 1

∗ → hP,1 = 1

∗ → hP,0 = 1

Figure 1.6: The half-open parallelepiped Π(P )of the two-dimensional simplex P = conv{(−1,−1),(1,0),(0,1)}. By counting the lattice points on it, one can ∗ ∗ 2 find the coefficients of the h -polynomial of P . In this case hP (t) = 1 + t + t .

This construction can be extended from simplices to any polytope by the use of half-open triangulations (a good reference for this is [5]). The h∗-polynomial of P naturally defines a new invariant for P : we ∗ ∗ call degree of P the degree of its h -polynomial hP . In the following proposition we summarize some of the properties of the h∗-polynomial.

∗ ∗ ∗ d ∗ Proposition 1.3.3 Let hP (t) = 1+hP,1t+···+hP,dt be the h -polynomial of a d-dimensional lattice polytope. Then

∗ d (i) hP,1 = P ∩ Z − d − 1;

∗ ◦ d (ii) hP,d = P ∩ Z ; ∗ ∗ (iii) 1 + hP,1 + ··· + hP,d = d!vol(P );

◦ d (iv) if s is the degree of P , then for k = 1,...,d−s we have |kP ∩Z | = ◦ d ∗ 0, while |(d − s + 1)P ∩ Z | = hP,s; ∗ (v) hP,i ∈ Z≥0 for each i; ∗ ∗ (vi) if Q ⊆ P is a lattice polytope, then hQ,i ≤ hP,i for each i; While (i)–(iv) follow from Ehrhart’s original approach, (v) and (vi) are two results of Stanley [16; 18]. Proofs of the latter results can be given by using the half-open triangulations method mentioned earlier. One of the most important open problem in Ehrhart Theory is to understand which polynomials can be Ehrhart polynomials for some lattice polytope. This problem can be stated in equivalent terms of h∗- ∗ ∗ polynomials. In the following we identify the h -polynomial hP (t) = 1+ ∗ ∗ d ∗ ∗ hP,1t+···+hP,dt of a lattice polytope P with the vector (1,hP,1,...,hP,d) of its coefficients.

19 Question 1.3.4 In each dimension d, is it possible to characterize all ∗ ∗ ∗ the vectors (1,h1,...,hd), which are h -vectors of some d-dimensional lattice polytope?

The two-dimensional case of Question 1.3.4, has been solved by Scott [15], and can be summarised as follows.

Theorem 1.3.5 (Scott’s Theorem) The vector with integer entries ∗ ∗ ∗ (1,h1,h2) is the h -vector of a two-dimensional polytope P if and only if one of the following conditions is true

∗ 1. h2 = 0, or

∗ ∗ ∗ 2. 0 < h2 ≤ h1 ≤ 3h2 + 3, or

∗ ∗ 3. (1,h1,h2) = (1,7,1), in this case P is equal to 3∆2, the third dila- tion of the two-dimensional unimodular simplex

∆2 := conv{(0,0),(1,0),(0,1)}.

Henk–Tagami [6] and Treutlein [21] generalised Scott’s Theorem to lat- tice polytopes of any dimension and having degree two. In Paper III this ∗ is generalized further by relaxing the degree two condition to h3 = 0. All the higher-dimensional cases of Question 1.3.4 are widely open. Nevertheless several families of inequalities for the coefficients of the h∗- polynomials have been proven for all . Let P be any lattice polytope of dimension d. In [8] Hibi proved that

∗ ∗ ∗ ∗ hP,d + ··· + hP,d−i ≤ hP,1 + ··· + hP,i+1

j d k for i = 0,1,..., 2 − 1, while Stanley in [17] proved that

∗ ∗ ∗ ∗ hP,0 + ··· + hP,i ≤ hP,s + ··· + hP,s−i

 s  ∗ for i = 0,1,..., 2 , where s is the degree of the h -polynomial. Another result from Hibi [9], in case P has interior points, is the following:

∗ ∗ 1 ≤ hP,1 ≤ hP,i for i = 2,...,d − 1. (1.3)

More recently Stapledon [19; 20] showed the existence of infinite new classes of inequalities and generalised existing ones.

20 1.4 Volume of lattice polytopes

Another way to compare discrete and continuous volume is to study the possible behaviors of the volume of a lattice polytope P when its number of interior points is fixed. This question is not very interesting for hollow polytopes, i.e. for lattice polytope without any lattice point in its relative interior. Indeed once can fit a d-dimensional polytope with an arbitrarily large volume d−1 in the “slab” R × [0,1]. On the other hand, in case P is a d-dimensional lattice polytope with at least one interior points, Hensley [7] proved that the volume of P is bounded by a constant depending on the number of interior lattice points. Refined values for such constant have been proved first by Lagarias–Ziegler [13], later by Pikhurko [14]. It makes sense to define a slightly modified version of the volume of a lattice polytope P . Given a d-dimensional lattice polytope P , we call Vol(P ) := d!vol(P ) the normalized volume of P . The normalized volume of a lattice polytope ∗ P has the advantage of being always an integer, it is equal to hP (1) (see Proposition 1.3.3 iii), and it respects monotonicity, i.e. if Q ⊆ P is a lattice polytope, then Vol(Q) ≤ Vol(P ). Theorem 1.4.1 (Hensley, Lagarias-Ziegler, Pikhurko) Let P be a d-dimensional lattice polytope with k > 0 interior lattice points. Then

2d+1 Vol(P ) ≤ d! (8d)d 15d·2 k. In dimension two a better and sharp bound follows from Scott’s Theorem 1.3.5 together with Proposition 1.3.3 iii. Corollary 1.4.2 Let P be a two-dimensional polytope with k > 0 inte- rior lattice points, then Vol(P ) ≤ 9 if k = 1, or Vol(P ) ≤ 4(k + 1) if k > 1.

Moreover the first inequality is an equality only if P = 3∆2. Also in higher dimension the bound in Theorem 1.4.1 appears to be far from sharp. Indeed the largest known (in terms of volume) d- dimensional lattice polytope having k (with k > 0) interior lattice points is the Zaks–Perles–Wills simplex, or ZPW simplex, from [22]: d Sk := conv(o,s1e1,...,sd−1ed−1,(k + 1)(sd − 1)ed), where k ≥ 1, (1.4)

21 d where o is the origin of the lattice, e1,...,ed is the standard base of Z and (si)i∈Z≥1 is the Sylvester sequence

s1 = 2, si = s1 ···si−1 + 1. (1.5)

It is reasonable to conjecture that, for fixed d ≥ 3 and k ≥ 1, the simplex d Sk maximizes the volume amongst all k-point d-dimensional polytopes. Hints of this conjecture can be tracked back to [22], [7] and [13].

Conjecture 1.4.3 Fix d ≥ 3 and k ≥ 1.A k-point d-dimensional lattice polytope P satisfies

d 2 Vol(P ) ≤ Vol(Sk ) = (k + 1)(sd − 1) . (1.6)

With the exception of the case when d = 3, k = 1, this inequality is an ∼ d equality if and only if P = Sk .

Averkov–Krümpelmann–Nill [1] proved a “simplicial” version of Con- d jecture 1.4.3 when k = 1: S1 is the unique simplex with maximum volume among all 1-point d-dimensional simplices, d ≥ 4.

1.5 Content of the thesis

1.5.1 Paper I: the dual volume of a lattice polytope In Paper I we prove a “dual” version of Conjecture 1.4.3 for polytopes with one interior lattice point. This translates in a sharp bound for the volume of reflexive polytopes. d Let P ⊂ R be a full-dimensional lattice polytope having one interior lattice point (suppose that such point is the origin 0 of the lattice), and d ∼ d d let HomZ(Z ,Z) = Z be the lattice dual to Z . The dual polytope of P is the polytope

∗ d ∗ P := {y ∈ (R ) : hy,xi ≥ −1 for every x ∈ P }.

Note that P ∗ has one interior lattice points (which is the origin of the dual lattice), but may not be a lattice polytope, as its vertices may have rational coordinates (in the dual lattice). This naturally select a privileged family of lattice polytopes having one interior lattice point.

Definition 1.5.1 If the dual polytope of a lattice polytope P is a lattice polytope, then we say that P is a reflexive polytope.

22 Reflexive polytopes are an extremely interesting family of lattice polytopes, and they are object of studies in many different fields (see d [12] for a brief survey). The ZPW simplex S1 (see (1.4)) is an example of reflexive simplex, provided that it has been translated in such a way that its only interior lattice point is the origin. The main result of Paper I is the following sharp upper bound for the normalized volume of duals of lattice polytopes having one interior lattice point. The bound is expressed via the Sylvester sequence (1.5).

Theorem 1.5.2 Let d ≥ 3 and P be a d-dimensional lattice polytope with one interior lattice point. Then

∗ 2 Vol(P ) ≤ 2(sd − 1) .

With the exception of the case d = 3, this inequality is an equality if and ∼ d ∗ only if P = (S1 ) .

Corollary 1.5.3 Let P be a d-dimensional reflexive polytope, where d ≥ 3. Then 2 Vol(P ) ≤ 2(sd − 1) . With the exception of the case d = 3, this inequality is an equality if and ∼ d only if P = S1 .

The strategy to prove Theorem 1.5.2 is to note that it suffices to prove it for polytopes satisfying some notion of minimality. Such polytopes admit a special decomposition in simplices, for which the result was already known [1], and from which we derive the desired result. This strategy leaves some gaps in low dimension. In order to fill them it is necessary to perform some specific classifications of low-dimensional minimal polytopes.

1.5.2 Paper II: three-dimensional polytopes with two interior lat- tice points A well-known result by Lagarias–Ziegler [13, Theorem 2] states that, up to unimodular equivalence, there are finitely many d-dimensional lattice polytopes having volume lower than any constant V . From this and from Theorem 1.4.1 it is possible to deduce the following corollary.

Corollary 1.5.4 Let k and d be positive integers. Up to unimodular equivalence, there are finitely many d-dimensional lattice polytopes with k interior lattice points.

23 Despite this, obtaining explicit lists of lattice polytopes (when the dimension and the number of interior lattice points are fixed) is a hard computational task. The first significative step in this direction has been taken by Kasprzyk [11], who classified the 674,688 three-dimensional lattice polytopes with one interior lattice point. In Paper II we con- tinue with the classification of the 22,673,449 three-dimensional lattice polytopes having two interior lattice points. In this special case Con- jecture 1.4.3 holds. Moreover we approach Question 1.3.4 by studying the distribution of the coefficients of the h∗-polynomials of the classi- fied polytopes. We conjecture two inequalities for the coefficients of the h∗-polynomials of three-dimensional polytopes that cannot be deduced from the known ones listed in Section 1.3.

∗ ∗ ∗ ∗ Conjecture 1.5.5 For any h -vector (1,hP,1,hP,2,hP,3) of a non-hollow three-dimensional polytope P , the inequalities

∗ ∗ ∗ ∗ hP,1 ≤ 16hP,3 + 19 and hP,2 ≤ 19hP,3 + 16

∗ hold. Moreover, if h3 > 1, the first inequality is an equality if and only 3 if P = S ∗ . h3 This conjecture would imply Conjecture 1.4.3 in dimension three.

1.5.3 Paper III: universal Ehrhart inequalities A way to approach Question 1.3.4 is to study coordinate projections of the space of all h∗-vectors. It makes sense to ask the following question.

∗ ∗ Question 1.5.6 Let (1,h1,...,hk) be a fixed vector of nonnegative in- ∗ ∗ ∗ ∗ ∗ tegers. Can it be extended to an h -vector (1,h1,...,hk,hk+1,...,hd) of some d-dimensional lattice polytope, for some d ≥ k and integers ∗ ∗ hk+1,...,hd?

In this paper we answer negatively to this question, but we do so with a positive result. We indeed prove the following generalization of Scott’s Theorem (Theorem 1.3.5).

∗ ∗ ∗ Theorem 1.5.7 Let P be a lattice polytope whose h -vector (1,h1,...,hd) ∗ satisfies h3 = 0. Then one of the following conditions is satisfied:

∗ (1) h2 = 0;

∗ ∗ (2) h1 ≤ 3h2 + 3;

24 ∗ ∗ (3) h1 = 7 and h2 = 1. Moreover, the case (3) is satisfied only by the h∗-vector of lattice sim- plices obtained via multiple lattice pyramid constructions over the tri- angle conv({0,3e1,3e2}) and then considered as lattice polytopes with respect to a refined lattice.

This result generalizes [21], which extends Scott’s Theorem to lattice polytopes of degree two. This theorem gives the first relation among the coefficients of the h∗-polynomial of a lattice polytope P that is valid independently of both the dimension and the degree of P (hence the term “universal”). In most of the cases, this result can be proven using the recent theory of spanning polytopes developed in [10], but a technical proof is necessary whenever a lattice polytope P , if considered with respect to the lattice affinely spanned by its lattice points, is a Lawrence prism (definition in Section 3, Paper III). ∗ Finally, via examples, it is shown that the condition h3 = 0 cannot be relaxed, and it is not too strict, as there are families of polytopes satisfying it that cannot be obtained with standard constructions.

1.5.4 Paper IV: simplices associated to digraphs Here, we generalize a recent work by Braun and Meyer [4] on simplices associated to graphs. The idea behind this is very simple: given a digraph D we associate to it the simplex PD whose vertices are the rows of the Laplacian of D. We prove the following structural results.

Theorem 1.5.8 Let D be a digraph on n vertices. Then PD is a (n−1)- simplex if and only if D has positive complexity, i.e. if D has at least a spanning converging tree. In this case: 1. the numbers of spanning trees converging to each of the vertices of D encode the barycentric coordinates of the origin with respect to the vertices of PD;

2. PD contains the origin 0, which is in the strict relative interior of PD if and only if D is strongly connected;

3. the normalized volume of PD equals the total complexity of D, i.e. the total number of spanning converging trees.

The study of cycle graphs of Braun–Meyer [4] is extended to cy- cle digraphs, i.e. strongly connected simple digraphs whose underlying

25 graph is a cycle. In this setting, properties as being terminal Fano, being Gorenstein, and being IDP are characterized. Those characterizations have been made thanks to tools developed in the study of weighted pro- jective spaces, and was used to build interesting examples of Gorenstein simplices having a symmetric but non-unimodular h∗-vector.

1.5.5 Paper V: enumeration of lattice polytopes by their volume It is a well known result by Lagarias and Ziegler [13] that, up to equiva- lence, there are finitely many d-dimensional lattice polytopes of normal- ized volume K, for fixed values of d and K. Using basic theory of point configurations, we describe an algorithm for the explicit enumeration of all lattice polytopes having such parameters. This in turns, gives an- other proof of the finiteness result. We perform computations for small values of K up to dimension six. The resulting database is immediately applied to the study Ehrhart Theory in dimension three and to broaden the understanding of properties such as smoothness and being IDP, with focus on their frequency. The numbers of the database are the following.

Theorem 1.5.9 Up to unimodular equivalence there are

• 408788 2-dimensional lattice polytopes having volume at most 50.

• 6064034 3-dimensional lattice polytopes having volume at most 36.

• 989694 4-dimensional lattice polytopes having volume at most 24.

• 433273 5-dimensional lattice polytopes having volume at most 20.

• 117084 6-dimensional lattice polytopes having volume at most 16.

Their distribution according to their volume, can be read from the tables in Appendix C, Paper V.

Smooth polytopes in the database are analyzed. In particular, sev- eral classifications of smooth polytopes having few lattice points are included in our lists. We show that smooth polytopes having at most 3d − 4 lattice points have a simple structure, and we use this result, to classify smooth polytopes with small volume (regardless of the dimen- sion). Using a bound proven by Pikhurko [14] for the volume of simplices with interior lattice points, we classify all three dimensional lattice sim- plices having up to 11 interior lattice points. We use these data to give

26 a conjectural set of inequalities for the coefficients of h∗-polynomials of three dimensional lattice polytope. This suggests a possible answer to the three-dimensional case of Question 1.3.4.

Conjecture 1.5.10 Let P be a three-dimensional lattice polytope hav- ing at least one interior lattice point. Then, except for a few exceptional ∗ ∗ ∗ ∗ cases (see Conjecture 8.7, Paper V), its h -vector (1,h1,h2,h3) satisfies the following inequalities.

∗ ∗ 1. h3 ≤ h1, ∗ ∗ 2. h1 ≤ h2, ∗ ∗ 3. h2 ≤ 19h3 + 16, ∗ ∗ ∗ 4. h2 − h1 ≤ 9h3 + 7, ∗ ∗ ∗ ∗2 ∗ ∗ ∗ 5. 5h3h1 + 4h1 + 4 ≤ 4h3 + 4h3h2 + 5h2.

This conjecture, unifies several existing conjectures (Conjectures 1.4.3 and 1.5.5, among the others), and can be seen as a three-dimensional version of Scott’s Theorem (Theorem 1.3.5). In the final part of the paper, we focus on properties such as being spanning, being very ample, being IDP, having a unimodular cover and having a unimodular triangulation. It is well known that they satisfy the hierarchy

unimodular unimodular ⇒ ⇒ IDP ⇒ very ample ⇒ spanning, triangulation cover while counterexamples to the opposite implications are mostly nontriv- ial. Using our database we estimate the frequency of such properties and look for minimal counterexamples to the opposite implications. Although there are counterexamples in higher dimensions, no three- dimensional IPD polytope without a unimodular cover has been found.

Question 1.5.11 Is there a three-dimensional IDP polytope that does not have a unimodular cover?

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