On the Complexity of Counting Irreducible Components and Computing Betti Numbers of Algebraic Varieties Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urElektrotechnik, Informatik und Mathematik der Universit¨atPaderborn vorgelegt von Peter Scheiblechner Paderborn, den 11. Juli 2007 ii Gutachter: Prof. Dr. Peter B¨urgisser Prof. Dr. Joachim von zur Gathen Prof. Dr. Felipe Cucker Abstract This thesis is a continuation of the study of counting problems in algebraic geometry within an algebraic framework of computation started by B¨urgisser, Cucker, and Lotz in a series of papers [BC03, BC06, BCL05]. In its first part we give a uniform method for the two problems #CCC and #ICC of counting the connected and irreducible components of complex algebraic varieties, respectively. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work in parallel polynomial time, i.e., they can be implemented by algebraic circuits of polynomial depth. The design of our algorithms relies on the concept of algebraic differential forms. A further important building block is an algorithm of Sz´ant´o[Sz´a97]computing a variant of characteristic sets. The second part contains lower bounds in terms of hardness results for topo- logical problems dealing with complex algebraic varieties. In particular, we show that the problem of deciding connectedness of a complex affine or projective va- riety given over the rationals is PSPACE-hard. We further extend this result to higher Betti numbers. More precisely, we prove that it is also PSPACE-hard to decide whether a Betti number of fixed order of a complex affine or projective variety is less then some given integer. In the third part we study the dependency of the complexity of #ICC on its combinatorial parameters. The crucial complexity parameter for the problem turns out to be the number of equations. This fact is illustrated by our result about counting the absolutely irreducible factors of a multivariate polynomial, the restriction of the general problem to the case of a single equation. We show that one can solve this problem in parallel polylogarithmic time. Furthermore, we describe a generic parsimonious reduction of the prob- lem #ICC for a fixed number of equations to a fixed number of variables. The consequences are that one can solve #ICC for a fixed number of equations in the BSS-model in polynomial time, and in the Turing model in randomised parallel polylogarithmic time. These results hold also for polynomials given by straight-line programs using their length and the degree as input parameters. iii iv Danksagungen Mein allergr¨oßterDank gilt meinem Doktorvater Peter B¨urgisser f¨ursein Ver- trauen und daf¨ur, dass er mir die M¨oglichkeit der Promotion gegeben hat, seine sehr gute und immer freundliche Betreuung und Unterst¨utzung in vielerlei Hin- sicht, und alles, was ich von ihm (auch außermathematisch) gelernt habe. Weit- erhin herzlich bedanken m¨ochte ich mich bei Thilo Pruschke f¨urdas Finden eines Fehlers und hilfreichen Diskussionen ¨uber das Hilbertpolynom. Außer- dem haben mir meine Arbeitsgruppenkollegen Martin Lotz und Martin Ziegler mit einer angenehmen Arbeitsatmosph¨areund vielen wertvollen Gespr¨achen geholfen. Nicht zuletzt bedanke ich mich ganz herzlich bei meiner Familie, insbeson- dere bei Pia f¨urihre endlose Geduld und Toleranz. Diese Arbeit wurde durch die DFG Sachbeihilfe BU 1371 unterst¨utzt. Contents 0 Introduction 1 0.1 General Upper Bounds . 2 0.2 Lower Complexity Bounds . 7 0.3 Fixing Parameters . 9 0.4 Outline . 13 0.5 Credits . 14 1 Preliminaries 17 1.1 Algebraic Geometry . 17 1.2 Differential Forms . 23 1.3 Models of Computation . 27 1.4 Structural Complexity . 32 1.5 Efficient Parallel Algorithms . 35 1.6 Squarefree Regular Chains . 39 I Upper Bounds 43 2 Transfer Results 45 2.1 Transfer Results for Complexity Classes . 45 2.2 Generic and Randomised Reductions . 49 3 Counting Connected Components 55 3.1 The Zeroth de Rham Cohomology . 56 3.2 Modified Pseudo Remainders . 59 3.3 Computing Differentials . 62 3.4 Proof of Theorem 3.1 . 65 4 Counting Irreducible Components 67 4.1 Affine vs. Projective Case . 68 4.2 Locally Constant Rational Functions . 69 4.3 Proof of Theorem 4.1 . 71 5 Hilbert Polynomial 75 5.1 Bound for the Index of Regularity . 76 5.2 Computing the Hilbert Polynomial . 78 v vi CONTENTS II Lower Bounds 81 6 Connectedness 83 6.1 Basic Notations . 83 6.2 Obtaining an Acyclic Configuration Graph . 84 6.3 Embedding the Configuration Graph . 86 6.4 Equations for the Embedded Graph . 89 6.5 Proof of Theorem 6.1 . 94 6.6 Appendix. The Real Reachability Problem . 95 7 Betti Numbers 97 7.1 The Affine Case . 97 7.2 The Projective Case . 98 III Fixing Parameters 103 8 Counting Irreducible Factors 105 8.1 Cohomology of a Hypersurface Complement . 106 8.2 Structure Theorem for Closed 1-Forms . 107 8.3 Proof of Theorem 8.3 . 110 8.4 Characterising Exact Forms . 112 8.5 Proof of Theorem 8.1 . 114 8.6 Counting Irreducible Components Revisited . 115 9 Fixed Number of Equations 117 9.1 Proof of the Main Results . 117 9.2 Transversality . 119 9.3 Explicit Genericity Condition for Bertini . 122 9.4 Expressing the Genericity Condition . 129 Chapter 0 Introduction A common principle in many mathematical areas is the possibility to construct complicated objects out of simpler ones. Ideally there exists some set of “sim- plest” objects in the sense that they cannot be built up by even simpler ones; these are called prime or irreducible. An elementary and well-known example of this principle is the factorisation of integers into a product of prime numbers. For example, 156 = 22 · 3 · 13 is the factorisation of 156 into prime numbers. As one sees in this example, it is easy to combine (multiply) the prime numbers to obtain the resulting number. Conversely, the “inverse” problem of constructing the factorisation from the result seems to be a much harder (hence much more interesting) task. This thesis adresses such inverse problems in the realm of complex algebraic geometry. Structurally very similar to factorisation of integers is the factori- sation of polynomials into irreducible ones. This seemingly algebraic problem is a special case of the decomposition problem of an algebraic variety into ir- reducible components. This also has a geometric flavour as the decomposition problem of a topological space (e.g., an algebraic variety) into connected com- ponents. More specifically, these problems are studied from a computational complexity point of view, i.e., we try to figure out how hard they are to solve algorithmically. One aim of this research is to identify complexity classes such as P or PSPACE, for which the problems are complete. Here P and PSPACE denote the class of decision problems decidable in polynomial time and space, respectively [Joh90, Pap94]. That a problem A is complete for the class C means that it is among the hardest problems in C. This is a twofold statement: • A is as most as hard as the problems in C, i.e., it lies in C; • A is as least as hard as any problem B from C in the sense that an efficient algorithm solving A also solves B efficiently. The first of these statements is also called an upper bound and the second a lower bound for A. In complexity theory it is convenient to restrict oneself to problems asking for the existence (decision problems) or the number of solutions (counting problems) of some question. We will focus on the counting versions of our problems. 1 2 CHAPTER 0. INTRODUCTION 0.1 General Upper Bounds The problems we prove upper bounds for are specified as follows. #CCC (Counting connected components) Given finitely many complex poly- nomials, compute the number of connected components of their affine zero set. #ICC (Counting irreducible components) Given finitely many complex poly- nomials, compute the number of irreducible components of their affine zero set. To be specific we use for the problem #CCC the Euclidean topology and for #ICC the Zariski topology. We discuss these problems in two models of computation. The first one is the model of algebraic circuits, which are capable of doing complex arithmetic exactly with unit cost. This is a standard model in algebraic complexity theory. Adding uniformity conditions it is equivalent to the BSS model named after Blum, Shub, and Smale [BSS89]. The second model is the discrete model of Boolean circuits. In order to study our problems in the discrete model, we restrict their inputs to rational polynomials whose coefficients are thought of as pairs of binary numbers. These restricted versions are denoted by #CCQ and #ICQ, respectively. Both algebraic and Boolean circuits serve as models for parallel computation, since the depth of a circuit can be interpreted as the running time when there are enough processors evaluating the gates of the circuit in parallel (ignoring communication and synchronisation cost). According to this observation we refer to the depth of a circuit as the parallel and to its size as the sequential running time of the algorithm modelled by the circuit. Using this terminology we can state our general upper bound results as follows. Theorem 0.1. The problems #CCC and #ICC can be solved in parallel poly- nomial and sequential exponential time. The same is true for #CCQ and #ICQ in the discrete setting. 0.1.1 Counting Connected Components The basic mathematical ideas behind the algorithms proving Theorem 0.1 are very classic. We first focus on the problem #CCC. It is well-known from point-set topology that the connected components of a topological space X can be characterised by the locally constant functions on X.