DOCSLIB.ORG

Zeta Functions in Algebraic Geometry Mircea Mustat˘A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Zeta functions in algebraic geometry Mircea Mustat¸˘a Contents Foreword vii Chapter 1. Introduction: An overview of zeta functions1 1.1. The Hasse-Weil zeta function1 1.2. The zeta function of an arithmetic variety2 1.3. The Igusa zeta function3 1.4. Motivic versions of the above (local) zeta functions4 1.5. Zeta functions in group theory5 Chapter 2. Basics of Hasse-Weil zeta functions7 2.1. Review of finite fields7 2.2. Preliminaries: varieties over finite fields7 2.3. The Hasse-Weil zeta function 10 2.4. The statements of the Weil conjectures 12 2.5. Comments on the conjectures 13 2.6. Two examples: computing the Betti numbers for Grassmannians and full flag varieties 14 Chapter 3. The Weil conjectures for curves 17 3.1. Rationality of the zeta function 18 3.2. The functional equation 19 3.3. The analogue of the Riemann hypothesis 21 Chapter 4. Weil cohomology theories and the Weil conjectures 23 4.1. Weil cohomology theories 23 4.2. Rationality and the functional equation via Weil cohomology 29 4.3. A brief introduction to `-adic cohomology 32 Chapter 5. Fulton's trace formula for coherent sheaf cohomology 39 5.1. The statement of the main theorem 39 5.2. The proof of the Localization Theorem 42 5.3. Supersingular Calabi-Yau hypersurfaces 45 Chapter 6. The Lang-Weil estimate and the zeta function of an arithmetic scheme 49 6.1. The Chow variety 49 6.2. The Lang-Weil estimate 50 6.3. Estimating the number of points on arbitrary varieties 54 6.4. Review of Dirichlet series 55 6.5. The zeta function of an arithmetic scheme 62 v vi CONTENTS Chapter 7. The Grothendieck ring of varieties and Kapranov's motivic zeta function 69 7.1. The Grothendieck ring of algebraic varieties 69 7.2. Symmetric product and Kapranov's motivic zeta function 75 7.3. Rationality of the Kapranov zeta function for curves 80 7.4. Kapranov zeta function of complex surfaces 81 Chapter 8. Dwork's proof of rationality of zeta functions 87 8.1. A formula for the number of Fq-points on a hypersurface 87 8.2. The construction of Θ 89 8.3. Traces of certain linear maps on rings of formal power series 92 8.4. The rationality of the zeta function 98 Appendix A. Quotients by finite groups and ground field extensions 101 A.1. The general construction 101 A.2. Ground field extension for algebraic varieties 104 A.3. Radicial morphisms 106 A.4. Quotients of locally closed subschemes 108 Appendix B. Basics of p-adic fields 111 B.1. Finite extensions of Qp 111 B.2. Unramified extensions of Qp and Teichm¨ullerlifts 113 B.3. The field Cp 116 B.4. Convergent power series over complete non-Archimedean fields 118 B.5. Examples of analytic functions 121 Bibliography 125 Foreword These are lecture notes from a graduate course I taught in Spring 2011. They cover some elementary topics related to Hasse-Weil zeta functions. I hope to add in the near future a second part, largely independent of the first one, covering p-adic and motivic zeta functions. vii CHAPTER 1 Introduction: An overview of zeta functions Zeta functions encode the counting of certain objects of geometric, algebraic, or arithmetic behavior. What distinguishes them from other generating series are special analytic or algebraic properties. Zeta functions come up in a lot of area of mathematics. The ones we will deal with come in two flavors: local and global. Here local means relative to a prime p in Z, or in some ring of integers in a number field. In this case, one expects the zeta function to be a rational function, in a suitable variable. By a global zeta function we mean an object that takes into account all primes. In this case one expects to have a product formula in terms of local factors. The basic example is the well-known factorization of the Riemann zeta function: −1 Y 1 ζ(s) = 1 − : ps p prime A good understanding of the local factors of the zeta function can be used to show that the global zeta function is defined in some region fs 2 C j Re(s) > ηg, and then there are fundamental questions regarding analytic continuation and the ex- istence of a functional equation. Again, the model is provided by the Riemann zeta function. However, very little is known in a more general setting. The gen- eral philosophy is that the analytic properties of the zeta function encode a lot of information about the geometric/arithmetic/algebraic of the object that is studied. In what follows we give an overview of the types of zeta functions that we will discuss in the following chapters. In all this discussion, we restrict to the simplest possible setting. 1.1. The Hasse-Weil zeta function This is one of the most famous zeta functions, and it played an important role in the development of algebraic geometry in the twentieth century. It is attached n to a variety over a finite field, say k = Fq. Suppose, for simplicity, that X ⊂ Ak is a closed subvariety defined by the equations f1; : : : ; fd. For every m ≤ 1, let n Nm := jfu 2 X(Fqm )j = jfu 2 Fqm j fi(u) = 0 for all igj: The Hasse-Weil zeta function of X is 0 1 X Nm Z(X; t) := exp tm 2 Q[[t]]: @ m A m≥1 A fundamental result is that Z(X; t) is a rational function. This was conjectured by Weil in [We2], who also proved it for curves and abelian varieties in [We1]. 1 2 1. INTRODUCTION: AN OVERVIEW OF ZETA FUNCTIONS The general case was proved by Dwork in [Dwo]. Another proof in the case of smooth projective varieties was later given by Grothendieck and its school using ´etalecohomology, see [Gro]. Both the methods of Grothendieck and of Dwork have been extremely influential for the development of arithmetic geometry. When X is a smooth projective variety, ZX (t) satisfies • The functional equation. • A connection with the Betti numbers defined over C. • An analogue of the Riemann hypothesis. These three properties, together with the rationality mentioned above, form the Weil conjectures [We2], now a theorem of Grothendieck [Gro] and Deligne [Del3]. See x2.4 for the precise statements. 1.2. The zeta function of an arithmetic variety n Suppose now that X ⊂ AZ is defined by the ideal (f1; : : : ; fd) ⊆ Z[x1; : : : ; xn]. For every prime p, we may consider f ;:::; f 2 F [x ; : : : ; x ] defining X ⊆ An , 1 d p 1 n p Fp and the corresponding Z(Xp; t). One then defines Y s LX (s) := Z(Xp; 1=p ): p6 ja n If X ⊂ AQ, then we may assume that the equations defining X have coefficients in some localization Z[1=a], where a is a positive integer. In this case we may still define Xp when p does not divide a, and we obtain LX as above, by taking the product over those p that do not divide a (this definition on the choice of a, but for us this is not important). Let us consider the case X = Spec Q, when we may take Xp = Spec Fp for every prime p. Note that 0 1 X te Z(X ; t) = exp = exp(−log(1 − t)) = (1 − t)−1: p @ e A e≥1 −1 Q 1 Therefore LX (s) = p 1 − ps = ζ(s) is the Riemann zeta function. In general, it is not hard to see that LX is defined in some half-plane fs 2 C j Re(s) > ηg (we will discuss this in Chapter6, giving a precise value for η, as a consequence of the Lang-Weil estimates, which in turn follow from the Weil conjectures for curves). It is conjectured that if X is a smooth projective variety over Q, then LX has an analytic continuation that is a meromorphic function. One also expects that after a suitable normalization (necessary for taking into account the infinite prime and the primes of bad reduction) LX satisfies a functional equation. Very little is know in this direction. Both properties are known for Pn and related varieties (such as toric varieties or flag varieties). The case of elliptic curves is known as a consequence of the Taniyama-Shimura conjecture (proved by Wiles [Wil], Taylor- Wiles [TW] and Breuil-Conrad-Diamond-Taylor [BCDT]), which implies that in this case LX can be described as the L-function attached to a modular form. 1.3. THE IGUSA ZETA FUNCTION 3 1.3. The Igusa zeta function Suppose now, for simplicity, that p is a prime in Z, and X,! An is defined Zp by f 2 Zp[x1; : : : ; xn]. The Igusa zeta function of f is defined by Z s Zf (s) := jf(x)jpdx: n Zp This is defined using the p-adic absolute value j·jp and the Haar measure on Zp. It is easy to see using the definition that Zf is analytic in the half-plane fs j Re(s) > 0g. Let us give some motivation for this definition. 1.3.1. The Archimedean analogue of Zf . The following analogue in the Archimedean setting (over R or C) appeared before Igusa's zeta function, in the setting of complex powers. Suppose, for example, that f 2 R[x1; : : : ; xn], and we want to define jf(x)js for s 2 C as a distribution. Given a test function Φ, consider the map Z s ! jf(x)jsΦ(x)dx: Rn It is not hard to see that this is well-defined and analytic in the half-space fs 2 C j Re(s) > 0g.
Recommended publications
  • Field Theory Pete L. Clark

    Field Theory Pete L. Clark

    Field Theory Pete L. Clark Thanks to Asvin Gothandaraman and David Krumm for pointing out errors in these notes. Contents About These Notes 7 Some Conventions 9 Chapter 1. Introduction to Fields 11 Chapter 2. Some Examples of Fields 13 1. Examples From Undergraduate Mathematics 13 2. Fields of Fractions 14 3. Fields of Functions 17 4. Completion 18 Chapter 3. Field Extensions 23 1. Introduction 23 2. Some Impossible Constructions 26 3. Subfields of Algebraic Numbers 27 4. Distinguished Classes 29 Chapter 4. Normal Extensions 31 1. Algebraically closed fields 31 2. Existence of algebraic closures 32 3. The Magic Mapping Theorem 35 4. Conjugates 36 5. Splitting Fields 37 6. Normal Extensions 37 7. The Extension Theorem 40 8. Isaacs' Theorem 40 Chapter 5. Separable Algebraic Extensions 41 1. Separable Polynomials 41 2. Separable Algebraic Field Extensions 44 3. Purely Inseparable Extensions 46 4. Structural Results on Algebraic Extensions 47 Chapter 6. Norms, Traces and Discriminants 51 1. Dedekind's Lemma on Linear Independence of Characters 51 2. The Characteristic Polynomial, the Trace and the Norm 51 3. The Trace Form and the Discriminant 54 Chapter 7. The Primitive Element Theorem 57 1. The Alon-Tarsi Lemma 57 2. The Primitive Element Theorem and its Corollary 57 3 4 CONTENTS Chapter 8. Galois Extensions 61 1. Introduction 61 2. Finite Galois Extensions 63 3. An Abstract Galois Correspondence 65 4. The Finite Galois Correspondence 68 5. The Normal Basis Theorem 70 6. Hilbert's Theorem 90 72 7. Infinite Algebraic Galois Theory 74 8. A Characterization of Normal Extensions 75 Chapter 9.
  • Arxiv:1708.06494V1 [Math.AG] 22 Aug 2017 Proof

    Arxiv:1708.06494V1 [Math.AG] 22 Aug 2017 Proof

    CLOSED POINTS ON SCHEMES JUSTIN CHEN Abstract. This brief note gives a survey on results relating to existence of closed points on schemes, including an elementary topological characterization of the schemes with (at least one) closed point. X Let X be a topological space. For a subset S ⊆ X, let S = S denote the closure of S in X. Recall that a topological space is sober if every irreducible closed subset has a unique generic point. The following is well-known: Proposition 1. Let X be a Noetherian sober topological space, and x ∈ X. Then {x} contains a closed point of X. Proof. If {x} = {x} then x is a closed point. Otherwise there exists x1 ∈ {x}\{x}, so {x} ⊇ {x1}. If x1 is not a closed point, then continuing in this way gives a descending chain of closed subsets {x} ⊇ {x1} ⊇ {x2} ⊇ ... which stabilizes to a closed subset Y since X is Noetherian. Then Y is the closure of any of its points, i.e. every point of Y is generic, so Y is irreducible. Since X is sober, Y is a singleton consisting of a closed point. Since schemes are sober, this shows in particular that any scheme whose under- lying topological space is Noetherian (e.g. any Noetherian scheme) has a closed point. In general, it is of basic importance to know that a scheme has closed points (or not). For instance, recall that every affine scheme has a closed point (indeed, this is equivalent to the axiom of choice). In this direction, one can give a simple topological characterization of the schemes with closed points.
  • [Math.CA] 1 Jul 1992 Napiain.Freape Ecnlet Can We Example, for Applications

    [Math.CA] 1 Jul 1992 Napiain.Freape Ecnlet Can We Example, for Applications

    Convolution Polynomials Donald E. Knuth Computer Science Department Stanford, California 94305–2140 Abstract. The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically. A family of polynomials F (x), F (x), F (x),... forms a convolution family if F (x) has degree n 0 1 2 n ≤ and if the convolution condition F (x + y)= F (x)F (y)+ F − (x)F (y)+ + F (x)F − (y)+ F (x)F (y) n n 0 n 1 1 · · · 1 n 1 0 n holds for all x and y and for all n 0. Many such families are known, and they appear frequently ≥ n in applications. For example, we can let Fn(x)= x /n!; the condition (x + y)n n xk yn−k = n! k! (n k)! kX=0 − is equivalent to the binomial theorem for integer exponents. Or we can let Fn(x) be the binomial x coefficient n ; the corresponding identity n x + y x y = n k n k Xk=0 − is commonly called Vandermonde’s convolution. How special is the convolution condition? Mathematica will readily find all sequences of polynomials that work for, say, 0 n 4: ≤ ≤ F[n_,x_]:=Sum[f[n,j]x^j,{j,0,n}]/n! arXiv:math/9207221v1 [math.CA] 1 Jul 1992 conv[n_]:=LogicalExpand[Series[F[n,x+y],{x,0,n},{y,0,n}] ==Series[Sum[F[k,x]F[n-k,y],{k,0,n}],{x,0,n},{y,0,n}]] Solve[Table[conv[n],{n,0,4}], [Flatten[Table[f[i,j],{i,0,4},{j,0,4}]]]] Mathematica replies that the F ’s are either identically zero or the coefficients of Fn(x) = fn0 + f x + f x2 + + f xn /n! satisfy n1 n2 · · · nn f00 = 1 , f10 = f20 = f30 = f40 = 0 , 2 3 f22 = f11 , f32 = 3f11f21 , f33 = f11 , 2 2 4 f42 = 4f11f31 + 3f21 , f43 = 6f11f21 , f44 = f11 .
  • Formal Power Series - Wikipedia, the Free Encyclopedia

    Formal Power Series - Wikipedia, the Free Encyclopedia

    Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series
  • Geometry on Grassmannians and Applications to Splitting Bundles and Smoothing Cycles

    Geometry on Grassmannians and Applications to Splitting Bundles and Smoothing Cycles

    PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. STEVEN L. KLEIMAN Geometry on grassmannians and applications to splitting bundles and smoothing cycles Publications mathématiques de l’I.H.É.S., tome 36 (1969), p. 281-297. <http://www.numdam.org/item?id=PMIHES_1969__36__281_0> © Publications mathématiques de l’I.H.É.S., 1969, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ GEOMETRY ON GRASSMANNIANS AND APPLICATIONS TO SPLITTING BUNDLES AND SMOOTHING CYCLES by STEVEN L. KLEIMAN (1) INTRODUCTION Let k be an algebraically closed field, V a smooth ^-dimensional subscheme of projective space over A:, and consider the following problems: Problem 1 (splitting bundles). — Given a (vector) bundle G on V, find a monoidal transformation f \ V'->V with smooth center CT such thatyG contains a line bundle P. Problem 2 (smoothing cycles). — Given a cycle Z on V, deform Z by rational equivalence into the difference Z^ — Zg of two effective cycles whose prime components are all smooth. Strengthened form. — Given any finite number of irreducible subschemes V, of V, choose a (resp. Z^, Zg) such that for all i, the intersection V^n a (resp.
  • 3 Lecture 3: Spectral Spaces and Constructible Sets

    3 Lecture 3: Spectral Spaces and Constructible Sets

    3 Lecture 3: Spectral spaces and constructible sets 3.1 Introduction We want to analyze quasi-compactness properties of the valuation spectrum of a commutative ring, and to do so a digression on constructible sets is needed, especially to define the notion of constructibility in the absence of noetherian hypotheses. (This is crucial, since perfectoid spaces will not satisfy any kind of noetherian condition in general.) The reason that this generality is introduced in [EGA] is for the purpose of proving openness and closedness results on the locus of fibers satisfying reasonable properties, without imposing noetherian assumptions on the base. One first proves constructibility results on the base, often by deducing it from constructibility on the source and applying Chevalley’s theorem on images of constructible sets (which is valid for finitely presented morphisms), and then uses specialization criteria for constructible sets to be open. For our purposes, the role of constructibility will be quite different, resting on the interesting “constructible topology” that is introduced in [EGA, IV1, 1.9.11, 1.9.12] but not actually used later in [EGA]. This lecture is organized as follows. We first deal with the constructible topology on topological spaces. We discuss useful characterizations of constructibility in the case of spectral spaces, aiming for a criterion of Hochster (see Theorem 3.3.9) which will be our tool to show that Spv(A) is spectral, our ultimate goal. Notational convention. From now on, we shall write everywhere (except in some definitions) “qc” for “quasi-compact”, “qs” for “quasi-separated”, and “qcqs” for “quasi-compact and quasi-separated”.
  • What Is a Generic Point?

    What Is a Generic Point?

    Generic Point. Eric Brussel, Emory University We define and prove the existence of generic points of schemes, and prove that the irreducible components of any scheme correspond bijectively to the scheme's generic points, and every open subset of an irreducible scheme contains that scheme's unique generic point. All of this material is standard, and [Liu] is a great reference. Let X be a scheme. Recall X is irreducible if its underlying topological space is irre- ducible. A (nonempty) topological space is irreducible if it is not the union of two proper distinct closed subsets. Equivalently, if the intersection of any two nonempty open subsets is nonempty. Equivalently, if every nonempty open subset is dense. Since X is a scheme, there can exist points that are not closed. If x 2 X, we write fxg for the closure of x in X. This scheme is irreducible, since an open subset of fxg that doesn't contain x also doesn't contain any point of the closure of x, since the compliment of an open set is closed. Therefore every open subset of fxg contains x, and is (therefore) dense in fxg. Definition. ([Liu, 2.4.10]) A point x of X specializes to a point y of X if y 2 fxg. A point ξ 2 X is a generic point of X if ξ is the only point of X that specializes to ξ. Ring theoretic interpretation. If X = Spec A is an affine scheme for a ring A, so that every point x corresponds to a unique prime ideal px ⊂ A, then x specializes to y if and only if px ⊂ py, and a point ξ is generic if and only if pξ is minimal among prime ideals of A.
  • Etale and Crystalline Companions, I

    Etale and Crystalline Companions, I

    ETALE´ AND CRYSTALLINE COMPANIONS, I KIRAN S. KEDLAYA Abstract. Let X be a smooth scheme over a finite field of characteristic p. Consider the coefficient objects of locally constant rank on X in ℓ-adic Weil cohomology: these are lisse Weil sheaves in ´etale cohomology when ℓ 6= p, and overconvergent F -isocrystals in rigid cohomology when ℓ = p. Using the Langlands correspondence for global function fields in both the ´etale and crystalline settings (work of Lafforgue and Abe, respectively), one sees that on a curve, any coefficient object in one category has “companions” in the other categories with matching characteristic polynomials of Frobenius at closed points. A similar statement is expected for general X; building on work of Deligne, Drinfeld showed that any ´etale coefficient object has ´etale companions. We adapt Drinfeld’s method to show that any crystalline coefficient object has ´etale companions; this has been shown independently by Abe–Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of ´etale coefficient objects; this subject will be pursued in a subsequent paper. 0. Introduction 0.1. Coefficients, companions, and conjectures. Throughout this introduction (but not beyond; see §1.1), let k be a finite field of characteristic p and let X be a smooth scheme over k. When studying the cohomology of motives over X, one typically fixes a prime ℓ =6 p and considers ´etale cohomology with ℓ-adic coefficients; in this setting, the natural coefficient objects of locally constant rank are the lisse Weil Qℓ-sheaves. However, ´etale cohomology with p-adic coefficients behaves poorly in characteristic p; for ℓ = p, the correct choice for a Weil cohomology with ℓ-adic coefficients is Berthelot’s rigid cohomology, wherein the natural coefficient objects of locally constant rank are the overconvergent F -isocrystals.
  • Skew and Infinitary Formal Power Series

    Skew and Infinitary Formal Power Series

    Appeared in: ICALP’03, c Springer Lecture Notes in Computer Science, vol. 2719, pages 426-438. Skew and infinitary formal power series Manfred Droste and Dietrich Kuske⋆ Institut f¨ur Algebra, Technische Universit¨at Dresden, D-01062 Dresden, Germany {droste,kuske}@math.tu-dresden.de Abstract. We investigate finite-state systems with costs. Departing from classi- cal theory, in this paper the cost of an action does not only depend on the state of the system, but also on the time when it is executed. We first characterize the terminating behaviors of such systems in terms of rational formal power series. This generalizes a classical result of Sch¨utzenberger. Using the previous results, we also deal with nonterminating behaviors and their costs. This includes an extension of the B¨uchi-acceptance condition from finite automata to weighted automata and provides a characterization of these nonter- minating behaviors in terms of ω-rational formal power series. This generalizes a classical theorem of B¨uchi. 1 Introduction In automata theory, Kleene’s fundamental theorem [17] on the coincidence of regular and rational languages has been extended in several directions. Sch¨utzenberger [26] showed that the formal power series (cost functions) associated with weighted finite automata over words and an arbitrary semiring for the weights, are precisely the rational formal power series. Weighted automata have recently received much interest due to their applications in image compression (Culik II and Kari [6], Hafner [14], Katritzke [16], Jiang, Litow and de Vel [15]) and in speech-to-text processing (Mohri [20], [21], Buchsbaum, Giancarlo and Westbrook [4]).
  • Euler and His Work on Infinite Series

    Euler and His Work on Infinite Series

    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 4, October 2007, Pages 515–539 S 0273-0979(07)01175-5 Article electronically published on June 26, 2007 EULER AND HIS WORK ON INFINITE SERIES V. S. VARADARAJAN For the 300th anniversary of Leonhard Euler’s birth Table of contents 1. Introduction 2. Zeta values 3. Divergent series 4. Summation formula 5. Concluding remarks 1. Introduction Leonhard Euler is one of the greatest and most astounding icons in the history of science. His work, dating back to the early eighteenth century, is still with us, very much alive and generating intense interest. Like Shakespeare and Mozart, he has remained fresh and captivating because of his personality as well as his ideas and achievements in mathematics. The reasons for this phenomenon lie in his universality, his uniqueness, and the immense output he left behind in papers, correspondence, diaries, and other memorabilia. Opera Omnia [E], his collected works and correspondence, is still in the process of completion, close to eighty volumes and 31,000+ pages and counting. A volume of brief summaries of his letters runs to several hundred pages. It is hard to comprehend the prodigious energy and creativity of this man who fueled such a monumental output. Even more remarkable, and in stark contrast to men like Newton and Gauss, is the sunny and equable temperament that informed all of his work, his correspondence, and his interactions with other people, both common and scientific. It was often said of him that he did mathematics as other people breathed, effortlessly and continuously.
  • 3 Formal Power Series

    3 Formal Power Series

    MT5821 Advanced Combinatorics 3 Formal power series Generating functions are the most powerful tool available to combinatorial enu- merators. This week we are going to look at some of the things they can do. 3.1 Commutative rings with identity In studying formal power series, we need to specify what kind of coefficients we should allow. We will see that we need to be able to add, subtract and multiply coefficients; we need to have zero and one among our coefficients. Usually the integers, or the rational numbers, will work fine. But there are advantages to a more general approach. A favourite object of some group theorists, the so-called Nottingham group, is defined by power series over a finite field. A commutative ring with identity is an algebraic structure in which addition, subtraction, and multiplication are possible, and there are elements called 0 and 1, with the following familiar properties: • addition and multiplication are commutative and associative; • the distributive law holds, so we can expand brackets; • adding 0, or multiplying by 1, don’t change anything; • subtraction is the inverse of addition; • 0 6= 1. Examples incude the integers Z (this is in many ways the prototype); any field (for example, the rationals Q, real numbers R, complex numbers C, or integers modulo a prime p, Fp. Let R be a commutative ring with identity. An element u 2 R is a unit if there exists v 2 R such that uv = 1. The units form an abelian group under the operation of multiplication. Note that 0 is not a unit (why?).
  • Formal Power Series License: CC BY-NC-SA

    Formal Power Series License: CC BY-NC-SA

    Formal Power Series License: CC BY-NC-SA Emma Franz April 28, 2015 1 Introduction The set S[[x]] of formal power series in x over a set S is the set of functions from the nonnegative integers to S. However, the way that we represent elements of S[[x]] will be as an infinite series, and operations in S[[x]] will be closely linked to the addition and multiplication of finite-degree polynomials. This paper will introduce a bit of the structure of sets of formal power series and then transfer over to a discussion of generating functions in combinatorics. The most familiar conceptualization of formal power series will come from taking coefficients of a power series from some sequence. Let fang = a0; a1; a2;::: be a sequence of numbers. Then 2 the formal power series associated with fang is the series A(s) = a0 + a1s + a2s + :::, where s is a formal variable. That is, we are not treating A as a function that can be evaluated for some s0. In general, A(s0) is not defined, but we will define A(0) to be a0. 2 Algebraic Structure Let R be a ring. We define R[[s]] to be the set of formal power series in s over R. Then R[[s]] is itself a ring, with the definitions of multiplication and addition following closely from how we define these operations for polynomials. 2 2 Let A(s) = a0 + a1s + a2s + ::: and B(s) = b0 + b1s + b1s + ::: be elements of R[[s]]. Then 2 the sum A(s) + B(s) is defined to be C(s) = c0 + c1s + c2s + :::, where ci = ai + bi for all i ≥ 0.