Fakultät für Mathematik Lehrstuhl für Angewandte Geometrie und Diskrete Mathematik Switching Components in Discrete Tomography: Characterization, Constructions, and Number-Theoretical Aspects Viviana Ghiglione Vollständiger Abdruck der von der Fakultät für Mathematik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Prof. Dr. Gregor Kemper Prüfer der Dissertation: 1. Prof. Dr. Peter Gritzmann 2. Prof. Dr. Paolo Dulio Die Dissertation wurde am 27.09.2018 bei der Technischen Universität München eingereicht und durch die Fakultät für Mathematik am 24.01.2019 angenommen. The pure and simple truth is rarely pure and never simple. (Oscar Wilde) Abstract We study sets of points that cannot be reconstructed by their X-rays, the so- called switching components: we extend known results to obtain their complete algebraic characterization and we provide two constructions that improve the existing ones by producing examples with few — though still exponentially- many — elements. Furthermore, we extend the connection between switching components and two problems in Number Theory: the first due to Prouhet, Tarry and Escott, and the second involving the so-called pure product polyno- mials. We address complexity and algorithmic aspects of the Prouhet-Tarry- Escott problem. Zusammenfassung Wir betrachten Punktmengen, die durch ihre X-Strahlen nicht rekonstruiert werden können, die sogenannten switching components: Wir erweitern Resul- tate, um eine vollständige algebraische Beschreibung zu erhalten, und geben außerdem zwei Konstruktionen an, die Beispiele mit wenigen — allerdings exponentiell vielen — Elementen produzieren und die bestehenden Konstruk- tionen verbessern. Ferner erweitern wir den Zusammenhang zwischen swit- ching components und zwei Problemen der Zahlentheorie, das erste von Prou- het, Tarry und Escott, und das zweite sogenannte reine Produkt Polynome be- treffend. Wir betrachten das Prouhet-Tarry-Escott Problem auch hinsichtlich Komplexität und Algorithmik. Acknowledgements Over the years, several people have been for me a crucial source of support and inspiration in the completion of my Ph.D. I feel privileged to be able to acknowledge them here. First, I would like to thank my supervisor, Peter Gritzmann, for giving me the opportunity to work in his group, for his broad mathematical interests and for everything he taught me. I am thankful to Andreas Alpers for the countless helpful discussions we shared and that led to parts of this thesis. I deeply enjoyed sharing this journey with so many talented and caring colleagues. Among all, I would like to mention Carolin Bauerhenne, An- dreas Bogner, Jesus De Loera, Andreas Denner, Paolo Dulio, Julia Ehrenmüller, Maximilian Fiedler, Elisabeth Finhold, Richard Gardner, Stephanie Hann, Fe- lix Happach, Peter Heinig, Carl Georg Heise, Raymond Hemmecke, Melanie Herzog, Wei Huang, Thomas Kahle, Stefan Kahler, Gregor Kemper, Florian Lindemann, Dorothee Mannschreck, Nicole Megow, Silvia Pagani, Benedikt Plank, Simon Plazotta, Maurice Rojas, Kathrin Ruf, Katharina Schaar, Kevin Schewior, Tina Schmidt, Andreas Schulz, Anastasia Shakhshneyder, Anusch Taraz, Annika Treyer, Clara Waldmann, Georg Wechslberger, Stefan Weltge, Mel Zürcher. My warmest thanks go to the colleagues with whom I shared the office: Wolfgang Riedl, for his mentoring, Felix Schmiedl, for his invaluable help and advice, Bernardo González Merino, for his constant encouragement, his en- thusiasm and his friendship, Marinus Gottschau, for his positive attitude and kindliness. I am particularly in debt with Steffen Borgwardt, Stefan König and Matthias Silbernagl, for their support during my first years at TUM. I am deeply grateful to René Brandenberg, for his empathy and assistance, and to Michael Ritter, for his never-ending kindness and patience. I owe a huge thank you to Fabian Klemm, for his generosity and for our engaging exchanges, to Anja Kirschbaum, for her thoughtfulness, and to Paul Stursberg, for his wor- thy insights and grace. I wish to thank Klaudia Bachmeier, Margot Wagner, Lydia Weber and Isabella Wiegand, whose benevolence and helpfulness have been of great benefit to me in many situations. I am grateful to TUM-GS and ISAM for the interesting events I had the chance to attend. I would like to express my gratitude to Giuseppe Rosolini and Giuseppe Valla, for their guidance and edorsements. None of this would have been possible without my family and friends, whom I thank from the bottom of my heart, especially Ornella, Lorena, Cristina, Alessandro, Giulia, Laura, Serena, Alice, Michel and Martina. I cannot thank enough my husband Federico, for sharing this experience with me and for being my rock during the critical times. Contents 1 Introduction and Notation 1 1.1 Content Overview and Contributions . .3 1.2 Notation . .4 2 Background Knowledge 13 2.1 Algebraic Background . 13 2.1.1 Gröbner Bases . 18 2.1.2 Toric Ideals . 25 3 Switching Components: Algebraic Interpretation and the Hunt for Small Sizes 29 3.1 Algebraic Interpretation of Switching Components . 29 3.2 Switching Components as Solutions of a Diophantine Equation 36 3.2.1 Enforcing Distinct Points in B and W in a Fixed Grid . 37 3.2.2 Bounding n and l ....................... 39 3.3 Switching Components as Cube Projections . 40 3.3.1 Switching Components as Union of Zonotopes . 43 3.3.2 Switching Components in a Bounded Grid . 49 3.4 On the Minimal Size of a Switching Component . 51 3.5 Small Polynomials . 58 3.6 Selecting Good Directions . 61 3.6.1 Directions leading to Few Terms . 61 3.6.2 Directions leading to Low Coefficients . 63 3.7 Large Size Pure Product leading to a Small Switching Component 64 3.7.1 Number of Copies of the Pure Product . 67 3.8 Switching Components of Size in O(1.38m) ............ 68 3.9 Switching Components from Special Polytopes . 76 3.9.1 The Permutahedron . 76 3.9.2 The Truncated Cuboctahedron . 80 3.10 Bounds Comparison . 89 4 The Prouhet-Tarry-Escott Problem in Discrete Tomography 91 4.1 The Prouhet-Tarry-Escott Problem . 91 4.2 Characterization of PTEd Solutions . 98 4.3 Switching Components as Solutions to PTE . 107 vii Contents 4.3.1 Vectors in Generic and Uniform Position . 107 4.3.2 Hyperplane Switching Components and the PTE problem 115 4.3.3 Line Switching Components and the PTE Problem . 118 4.3.4 Algebraic Proof of the Relation between Switching Com- ponents and PTE Problem . 120 4.4 PTE-Solutions are Projections of Switching Components . 122 4.4.1 Summary on the Relation between Switching Compo- nents and PTE . 125 4.5 Small PTE-Solutions . 126 4.6 The PTE-Problem as an Integer Linear Program . 129 4.6.1 Search for Ideal Solutions of Small Magnitude . 132 4.7 Complexity Aspects . 135 5 Pure Product Polynomials and Switching Components 139 5.1 Bounds on A¥(m) and A1(m) .................... 140 5.2 Multivariate Pure Products Polynomials . 142 5.3 Pure Product Polynomials and the Size of Switching Components146 Appendix 149 Bibliography 153 Index 171 viii List of Symbols Symbol Meaning N f0, 1, 2, . g N∗ Nnf0g [l] f1, . , lg [l]0 f0g [ [l] A [· B Disjoint union of the (multi)sets A and B Sk Set of all permutations of order k K A field containing Z X Vector of variables (X1,..., Xd) K[X] Ring of polynomials in the variables X and coefficients in K ( f1,..., fs) Ideal generated by the polynomials f1,..., fs f jg f divides g gcd greatest common divisor det(M) Determinant of the matrix M rank (M) Rank of the matrix M d u1,..., ud Unit vectors in R lin(C) Linear hull of C aff (C) Affine hull of C conv(C) Convex hull of C bd (C) Boundary of C int (C) Interior of C relint (C) Relative interior of C vol(C) Volume of C diam (C) Diameter of C log Natural logarithm exp Exponential function O, W, o Landau symbols _, ^ Logical “or”, “and” ix Chapter 1 Introduction and Notation Tomography is concerned with the problem of recovering information on a geometric object from its X-rays or projections. Its mathematical aspects were first investigated by Radon [150]. Our interest focuses on its discrete version, where the objects to be reconstructed are finite sets. According to Herman and Kuba [99], the term discrete tomography was first introduced by Lawrence Shepp [162] in 1994. Since then, the topic has developed in several directions: studying the X-rays needed in order to allow unique reconstruction under some restrictions on the sets [39, 44, 58, 63, 78], addressing upper and lower bounds on the size of sets that cannot be reconstructed from the information on their X-rays [16, 127], introducing point X-rays [64], investigating complexity aspects [65, 79, 80, 158], algebraic aspects [18, 94], reconstruction algorithms [24, 39, 50], approximation algorithms [91] and discussing the effect of pertur- bations on the reconstruction [9, 12, 15, 57]. Applications of discrete tomogra- phy include image processing [110], crystallography and material science [8, 17, 21, 156], and plasma physics [13]. In this thesis, we study sets of points that cannot be reconstructed by their X-rays. A pair of (multi-) sets that have the same X-rays with respect to a given subspace is called switching component. Hajdu and Tijdeman [94] showed that every switching component with re- spect to a fixed number of reduced lattice directions corresponds to a polyno- mial divisible by certain binomials. A few years earlier Wiegelmann [175] had shown a characterization in terms of toric ideals of the points x∗ 2 Nd that are unique solutions to Ax = b, with A and b respectively a matrix and a vector with integer non-negative entries. We extend the two approaches, and obtain a unified algebraic characterization of switching components with respect to subspaces of every dimension. A crucial problem in discrete tomography is understanding how many di- rections are needed in order to be able to reconstruct any finite subset of Zd.