The Generalized External Order, and Applications to Zonotopal Algebra by Bryan R. Gillespie A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Olga Holtz, Chair Professor Bernd Sturmfels Professor Lauren Williams Professor David Aldous Summer 2018 The Generalized External Order, and Applications to Zonotopal Algebra Copyright 2018 by Bryan R. Gillespie 1 Abstract The Generalized External Order, and Applications to Zonotopal Algebra by Bryan R. Gillespie Doctor of Philosophy in Mathematics University of California, Berkeley Professor Olga Holtz, Chair Extrapolating from the work of Las Vergnas on the external active order for matroid bases, and inspired by the structure of Lenz's forward exchange matroids in the theory of zonotopal algebra, we develop the combinatorial theory of the generalized external order. This partial ordering on the independent sets of an ordered matroid is a supersolvable join- distributive lattice which is a refinement of the geometric lattice of flats, and is fundamentally derived from the classical notion of matroid activity. We uniquely classify the lattices which occur as the external order of an ordered matroid, and we explore the intricate structure of the lattice's downward covering relations, as well as its behavior under deletion and contraction of the underlying matroid. We then apply this theory to improve our understanding of certain constructions in zonotopal algebra. We first explain the fundamental link between zonotopal algebra and the external order by characterizing Lenz's forward exchange matroids in terms of the external order. Next we describe the behavior of Lenz's zonotopal D-basis polynomials under taking directional derivatives, and we use this understanding to provide a new algebraic construction for these polynomials. The construction in particular provides the first known algorithm for computing these polynomials which is computationally tractible for inputs of moderate size. Finally, we provide an explicit construction for the zonotopal P-basis polynomials for the internal and semi-internal settings. i To my parents, for fanning my passions and giving me a foundation on which to build. To my wife, for supporting me through this and filling my days with love and joy. ii Contents Contents ii 1 Introduction 1 2 Background 8 2.1 Matroids and Antimatroids . 8 2.2 Zonotopal Algebra . 19 3 The Generalized External Order 30 3.1 Feasible and Independent Sets of Join-distributive Lattices . 34 3.2 Definition and Fundamental Properties . 37 3.3 Lattice Theoretic Classification . 45 3.4 Deletion and Contraction . 54 3.5 Passive Exchanges and Downward Covering Relations . 57 4 Applications to Zonotopal Algebra 68 4.1 Forward Exchange Matroids and the External Order . 69 4.2 Canonical Basis Polynomials . 71 4.3 Differential Structure of Central D-Polynomials . 77 4.4 Recursive Construction for the Central D-Basis . 79 4.5 Explicit Construction for the Internal and Semi-internal P-Bases . 85 Bibliography 93 A Software Implementations 96 A.1 Ordered Matroids and the External Order . 96 A.2 Polynomial Vector Spaces . 99 A.3 Zonotopal Spaces . 100 iii Acknowledgments This work could not have been completed without the assistance and support of many individuals, both during graduate school and before it, and I owe a debt of gratitude to everyone who has helped me to reach this point. Thanks first to my parents, whose love and encouragement is responsible for much of who I am today. There is nothing more important in the life of a child than a stable and supportive upbringing, and I can only hope to be as good of a parent someday as you both have been for me. From my earliest years of schooling, I would like to thank two of my grade school math teachers who played a formative role in my education. Thanks to Evan McClintock, my 5th grade teacher, for individual after-school math lessons which put me on track to get the most out of later math curricula. Thanks also to Greg Somers for effectively communicating, in so many small ways, the beauty and elegance of rigorous mathematics. You both have a rare gift in your ability to educate, and I am grateful that my path passed through your classrooms. Among my excellent teachers as an undergraduate at Penn State, thanks especially to Svetlana Katok for encouraging my first awkward steps in theoretical mathematics, and to Omri Sarig for showing me a level of clarity and precision in teaching of which I have yet to see an equal. Thanks to Federico Ardilla, Bernd Sturmfels, Lauren Williams, and Martin Olsson for mathematical and academic advice. Grad school has many ups and downs, and your per- spective and suggestions were invariably helpful in weathering them. Thanks to Matthias Lenz, Sarah Brodsky, Spencer Backman, Jose Samper, and Federico Castillo for helpful conversations and thoughtful insights. Additionally, thanks to Anne Schilling and Nicolas Thi´eryfor a whirlwind initiation into the world of SageMath coding. Thanks especially to my advisor, Olga Holtz, for providing the starting point for my work and much invaluable guidance, and also for allowing me the freedom to explore the mathematics that excited me the most. This dissertation simply would not have been possible without your input and assistance. Thanks also to Bernd Sturmfels, Lauren Williams, and David Aldous for serving on my dissertation committee. Finally, thank you to my wife Maria, for your unwavering support and encouragement throughout graduate school. Your energy, enthusiasm, and determination are an inspiration to me. I look forward to sharing many more years of mathematical adventures with you. The material in this dissertation is based in part upon work supported by the National Science Foundation under Grant No. DMS-1303298. Thanks to the NSF for the time and freedom this support afforded me over several semesters, and thanks also to the Berkeley Graduate Division for support provided by the Graduate Division Summer Grant. 1 Chapter 1 Introduction The theory of zonotopal algebra studies the characteristics and applications of certain finite- dimensional polynomial vector spaces, and related annihilating ideals, which are derived from the structure of particular combinatorial and geometric objects formed from a finite ordered list of vectors. The theory lies at an interface between several starkly contrasting mathemat- ical disciplines, especially approximation theory, commutative algebra, and matroid theory, and wide-ranging connections have been found with topics in enumerative combinatorics, representation theory, and discrete geometry. Zonotopal algebra has at its foundation a collection of ideas in numerical analysis and approximation theory that were developed in the 1980s and early 90s, in works such as [1, 9, 10, 11, 17, 19, 16]. In these works, the central zonotopal spaces were discovered in relation to the approximation of multivariate functions by so-called (exponential) box splines, and as the solutions of certain classes of multivariate differential and difference equations. The central zonotopal spaces lie in the real polynomial ring Π = R[x1; : : : ; xd], and consist of two finite-dimensional polynomial vector spaces D(X) and P(X), and two related ideals I(X) and J (X) which are constructed from the columns of a d × n matrix X. The D-space, also known as the Dahmen-Michelli space, is the space of polynomials spanned by the local polynomial pieces of the box spline associated with the matrix X. It can be realized as the differential kernel or Macaulay inverse system of the J -ideal, an ideal generated by products of linear forms corresponding with the cocircuits of the linear independence matroid of the columns of X. The P-space is defined using the notion of matroid activity of bases of X, and acts as the dual vector space of the D-space under the identification p 7! hp; ·i, where h·; ·i :Π×Π ! R : denotes the differential bilinear form given by hp; qi = p(@)q x=0, and where p(@) is the differential operator obtained from p by replacing each variable xi with the operator @=@xi. This space is realized as the differential kernel of the I-ideal, a power ideal generated by certain powers of linear forms corresponding to orthogonal vectors of the hyperplanes spanned by column vectors of X. The D- and P-spaces in particular each contain a unique interpolating polynomial for any function defined on the vertex set of certain hyperplane arrangements associated with CHAPTER 1. INTRODUCTION 2 the matrix X, a property known as correctness. In addition, the Dahmen-Michelli space has a simple construction in terms of the least map, an operator introduced in [11] which plays an important role in interpolation theory. Furthermore, relations dual to these can be developed with respect to generic vertex sets for the zonotope of X, which is formed by taking the Minkowski sum of the segments [0; x] = ftx : t 2 [0; 1]g for the column vectors x of X. It is from this geometric connection (and related combinatorial structure) that the name zonotopal algebra is derived. 1 0 1 X = 0 1 1 Figure 1.1: A hyperplane arrangement and corresponding dual zonotope of the matrix X. Particularly in the last decade, the area of zonotopal algebra has witnessed a resurgence of interest, with applications and generalizations of the theory emerging in a variety of new directions. In [38], Sturmfels and Xu study sagbi bases of Cox-Nagata rings, and discuss an algebraic generalization of the zonotopal spaces called zonotopal Cox rings which can be described as subalgebras of particular Cox-Nagata rings. In [33], Luca Moci generalizes some of the results of zonotopal algebra regarding hyperplane arrangements to a discrete setting related to the geometry of toric arrangements.