Felix Breuer Ham Sandwiches, Staircases and Counting Polynomials Ham Sandwiches, Staircases and Counting Polynomials Dissertation vorgelegt von Felix Breuer Fachbereich für Mathematik und Informatik Freie Universität Berlin Berlin, 2009 Felix Breuer Freie Universität Berlin Institut für Mathematik Arnimallee 3 D-14195 Berlin Germany [email protected] Betreuer und erster Gutachter: Prof.Dr.Martin Aigner Freie Universität Berlin Institut für Mathematik Arnimallee 2 D-14195 Berlin Germany [email protected] Zweiter Gutachter: Prof.Dr.Volkmar Welker Philipps-Universität Marburg Fachbereich Mathematik und Informatik Hans-Meerwein-Straße D-35032 Marburg Germany [email protected] Datum der Disputation: 17. November 2009 c 2009 Felix Breuer Umschlaggestaltung durch den Autor unter Verwendung von Inkscape und Blender Satz durch den Autor unter Verwendung von LATEX Herstellung: Books on Demand GmbH, Norderstedt Contents Introduction 7 Zusammenfassung 9 Acknowledgments 11 Chapter 1. Uneven Splitting of Ham Sandwiches 13 1.1 Convex Geometry . 15 1.2 The Problem . 17 1.3 Auxiliary functions fi ................................................. 18 1.4 Sandwiches Without Uneven Splittings . 19 1.5 Separability . 20 1.6 The Main Result . 21 1.7 The Topological Tool . 22 1.8 Proof of Main Result . 23 1.9 Existence of the fi ..................................................... 27 1.10 Partitioning one Mass by two Hyperplanes . 29 1.11 Central Spheres . 30 Chapter 2. Staircases in Z2 33 2.1 Linear Algebra and Lattice Points . 34 2.2 Staircases and Related Sequences . 36 2.3 Geometric Observations . 41 2.4 Characterizations of Sturmian Sequences . 53 2.5 Proof of the Characterizations . 58 2.6 Application: Short Representations . 63 2.7 Application: Dedekind-Carlitz Polynomials . 66 2.8 Application: Theorem of White . 68 Chapter 3. Counting Polynomials and Reciprocity Theorems 73 3.1 Graphs and Orientations . 76 3.2 Colorings, Flows and Tensions . 77 3.3 Reciprocity Theorems . 83 3.4 The Linear Algebra Connection . 86 3.5 Total Unimodularity . 89 3.6 A Geometric Proof of Modular Flow Reciprocity . 91 3.7 The Connection to Inside-Out Polytopes . 97 3.8 An Inductive Proof of Modular Flow Reciprocity . 101 3.9 A Geometric View on Modular Tension Reciprocity . 105 3.10 BG and DG ........................................................... 107 3.11 The Tutte Polynomial as a Counting Function . 110 3.12 A Combinatorial Proof of the Tutte Interpretation . 117 6 Contents Chapter 4. Counting Polynomials as Hilbert Functions 119 4.1 Hilbert Functions . 122 4.2 Steingrímsson’s Construction . 124 4.3 The Tension Polynomial as a Hilbert Function (Combinatorially) . 127 4.4 Hilbert equals Ehrhart . 135 4.5 Counting Polynomials as Hilbert Functions (Geometrically) . 140 4.6 The Combinatorial versus the Geometric Approach . 146 4.7 Non-Square-Free Ideals and Non-Standard Gradings . 152 4.8 The Structure of the Tension Polytope and Tension Complex . 154 4.9 Bounds on the Coefficients . 157 Bibliography 160 Introduction This thesis consists of four chapters that are largely independent. A brief outline of each of the four chapters is given below. A more elaborate introduction to each of the four topics is given at the beginning of each chapter. Chapter 1. Uneven Splitting of Ham Sandwiches. Let m1,..., mn be continuous probability n measures on R and a1,..., an 2 [0, 1]. When does there exist an oriented hyperplane H such + + that the positive half-space H has mi(H ) = ai for all i 2 [n]? We call such a hyperplane an a-splitting. It is well known that a-splittings do not exist in general. The famous Ham 1 Sandwich Theorem states that if ai = 2 for all i, then a-splittings exist for any choice of the mi. We give sufficient criteria for the existence of a-splittings for general a 2 [0, 1]n. To better n−1 n keep track of the pairs (mi, ai) we introduce auxiliary functions f1,..., fn : S ! R with the property that for all i the unique hyperplane Hi with normal v that contains the point + fi(v) has mi(Hi ) = ai. Our main result is that if Im f1, . , Im fn are bounded and can be separated by hyperplanes, then an a-splitting exists. Interestingly, the equivariant methods that are classically used to prove the Ham Sandwich Theorem and similar equipartition results cannot be easily applied to show the existence of uneven splittings. We present a novel approach based on the Poincaré-Miranda Theorem. One important property of this result is that it can be applied even if the supports of the mi overlap. This gives a partial answer to a question of Stojmenovi´c. The main result implies several other criteria as corollaries, the weakest of which was also obtained independently by Bárány et al. Also, it allows an easy corollary of the classical Ham Sandwich Theorem to be generalized. Chapter 2. Staircases in Z2. This part is joint work with Frederik von Heymann. Motivated by the study of lattice points inside polytopes, we seek to understand the set of lattice points “close” to a rational line in the plane. To this end, we define a staircase in the plane to be the set of lattice point in the plane below a rational line that have Manhattan distance less than 1 to the line. This set of lattice points is closely related to the Beatty and Sturmian j b k j b k sequences defined in number theory, i.e. to sequences of the form ( a (n − 1) − a n )n2N for a, b 2 N with gcd(a, b) = 1. We present three characterizations of these sequences from a geometric point of view. One of these characterizations is known, two are new. The most important one is recursive and closely related to the Euclidean Algorithm. In particular, we obtain recursive descriptions of staircases, as well as of the sets of lattice points inside the fundamental parallelepipeds of rational cones and inside certain triangles in the plane. We then present several applications of our geometric observations. 1) We give a new proof of Barvinok’s Theorem in dimension 2. Barvinok’s Theorem states that the generating function of the lattice points inside a rational simplicial cone can be written as a short rational function. While Barvinok uses a signed decomposition of the cone into unimodular cones, we use a partition of the cone into sets that have a short representation. 2) We give a recursion formula b a−1 k−1 b a kc for Dedekind-Carlitz polynomials, i.e. polynomials of the form ∑k=1 x y . This answers 8 Introduction a question of Beck, Haase and Matthews. 3) We simplify Scarf’s proof of White’s Theorem, which characterizes lattice simplices that contain no lattice points except their vertices. Chapter 3. Counting Functions and Reciprocity Theorems. This part is joint work with Raman Sanyal. Results that give the values of counting polynomials at negative integers a combi- natorial interpretation are called reciprocity theorems. We consider five natural counting polynomials determined by graphs. The chromatic polynomial, the modular flow and ten- sion polynomials and the integral flow and tension polynomials. Reciprocity theorems for all of these are known, except for the modular flow polynomial. We fill in this missing result, thus answering a question of Beck and Zaslavsky. The key ingredient in our proof of the Modular Flow Reciprocity Theorem is our construction of a disjoint union of open polytopes whose Ehrhart function is the modular flow polynomial. Such a construction was not known before, and thus the method of Beck and Zaslavsky, who used inside-out Ehrhart-Macdonald Reciprocity to prove the Integral Flow Reciprocity Theorem, could not be applied. We use classical Ehrhart-Macdonald Reciprocity on each of the polytopes in the union to obtain our reciprocity result. In their unpublished manuscript [BB] Babson and Beck give a similar reciprocity result, independently from our work. In the remainder of the chapter we relate our construction to the theory of inside-out poly- topes and apply these methods to give a new proof of the Modular Tension Reciprocity Theorem. It turns out that these two reciprocity results give rise to a very nice interpretation of the Tutte polynomial as a counting polynomial which is already implicit in the work of Reiner [Rei99]. Moreover this approach provides a unified framework in which the value of the Tutte polynomial at every lattice point in the plane can be interpreted. The chapter is rounded off by direct induction proofs for both the reciprocity result and the interpretation of the Tutte polynomial. Chapter 4. Counting Functions as Hilbert Functions. This part is joint work with Aaron Dall. Steingrímsson showed that the chromatic polynomial of a graph, shifted by one, is the Hilbert function of a relative Stanley-Reisner ideal. More precisely, given a graph G, Steingrímsson defines two square-free monomial ideals I1 ⊂ I2 ⊂ K[x1,..., xn] such that the k + 1-colorings of G are in bijection with the monomials of degree k inside I2 but outside I1. Going beyond the shifted chromatic polynomial, the question arises, which of the five count- ing polynomials introduced in Chapter 3 are Hilbert functions of this type? All five are! Even the chromatic polynomial itself, without the shift by one. We show this by giving a general theorem, that provides a sufficient criterion for when the Ehrhart function of a rela- tive polytopal complex is a Hilbert function of Steingrímsson’s type. We give two proofs of this theorem, one from a more combinatorial and one from a more geometric point of view. Also we present variations of this result, where weaker hypotheses lead to weaker conclu- sions. When applying our criterion to the five counting polynomials we make use of some of the polytopal complexes constructed in Chapter 3.