Showing papers on "Field (mathematics) published in 1987"

Geometry of Numbers

Abstract: The geometry. of numbers can be traced back at least to Lagrange [1773], who proved important results about quadratic forms in two variables. The proofs as well as the formulations of results were purely arithmetic. Reviewing a book of Seeber [1831B] on ternary quadratic forms, Gaus [1831] introduced for the first time geometric methods. Geometric methods were predominant in the work of Dirichlet [1850]. On the other hand Hermite [1850] and Korkine and Zolotareff [1872], [1873], [1877] gave arithmetic proofs for their results on quadratic forms in more than three variables. Finally Minkowski [1891] noticed that a simple geometric argument which he used to give a new proof of a theorem of Hermite could be adapted to much more general situations. Then Minkowski [1896B], [7B], [11B] started a systematic study of geometric methods in number theory and called this new branch of number theory geometry of numbers. Many results and most concepts of modern geometry of numbers have their origin in the work of Minkowski. After Minkowski many eminent mathematicians made contributions to this field. In order to avoid controversies I will not mention any of them. Geometry of numbers is closely related to other branches of number theory such as algebraic number theory and Diophantine approximation. A flourishing offspring is discrete geometry, developed mainly by Fejes Toth and his school.

Mechanical Geometry Theorem Proving

Abstract: I: Methods in Mechanical Geometry Theorem Proving.- 1. An Introduction to Wu's Method.- 1. The Defects in Traditional Proofs.- 1.1. The Traditional Euclidean Proof.- 1.2. The Traditional Analytic Proof.- 2. Four Examples.- 3. A Summary of Wu's Method.- 4. Pseudo Division and Successive Pseudo Division.- 5. A Simple Triangulation Procedure.- 6. Geometry Statements of Constructive Type.- 7. Further Discussion of Geometry Statements of Constructive Type.- 2. Ritt's Characteristic Set Method.- 1. The Prerequisite in Algebra.- 2. Ascending Chains and Characteristic Sets.- 3. Irreducible Ascending Chains.- 4. A Complete Triangulation Procedure: Ritt's Principle.- 5. Ritt's Decomposition Algorithm.- 3. Algebra and Geometry.- 1. Axiomatic Geometries and Number Systems.- 1.1. Affine Geometry.- 1.2. Metric Geometry.- 1.3. Hilbert Geometry.- 1.4. Tarski Geometry.- 2. On the Algebraic formulation of Geometry Statements.- 2.1. Formulation Fl.- 2.2. Formulation F2.- 3. Formulation F3.- 3.1. The Generic Validity of a Geometry Statement.- 3.2. Identifying Nondegenerate Conditions.- 3.3. The Generic Validity of a Geometry Statement in an Arbitrary Field.- 4. The Complete Method of Wu.- 1. Ritt's Principle Revised.- 2. Ritt's Decomposition Algorithm Revised.- 3. Complete Method of Wu - Irreducible Cases.- 4. Complete Method of Wu - General Cases.- 5. Examples.- 5. Geometry Theorem Proving Using The Grobner Basis Method.- 1. A Review of the Grobner Basis Method.- 2. Proof Methods for Formulation F3.- 3. A Proof Method for Formulation Fl.- 4. Connections Between Characteristic Sets and Grobner Bases.- 5. A Comparison of the Grobner Basis Method with Wu's Method.- 5.1. The Scope.- 5.2. The Efficiency.- References.- II: 512 Theorems Mechanically Proved.- Explanations.- 1. General Remarks.- 2. Algebraic Representations of Geometric Conditions.- Theorems Proved Mechanically by Wu's Method.- 2. Algebraic Representations of Geometric Conditions.- Appendix. The Timing For the Grobner Basis Method.- Index of Examples.

A fast parallel algorithm to compute the rank of a matrix over an arbitrary field

Abstract: It is shown that the rank of a matrix over an arbitrary field can be computed inO(log2 n) time using a polynomial number of processors.

Fast parallel computation of hermite and smith forms of polynomial matrices

Abstract: Boolean circuits of polynomial size and polylogarithmic depth are given for computing the Hermite and Smith normal forms of polynomial matrices over finite fields and the field of rational numbers. The circuits for the Smith normal form computation are probabilistic ones and also determine very efficient sequential algorithms. Furthermore, we give a polynomial-time deterministic sequential algorithm for the Smith normal form over the rationals. The Smith normal form algorithms are applied to the rational canonical form of matrices over finite fields and the field of rational numbers.

New generalizations of poincar~'s geometric theorem

Abstract: In this work we show that the number of closed trajectories of the field of kernels of a closed, nondegenerate, and center of gravity preserving 2-form on the total space of an oriented fiber bundle with fiber the circle over an orientable compact two-dimensional base is not less than the minimal number of critical points of a smooth function on the basis, under the assumption that the field of kernels is C1-close to a vertical field. Counting multiplicities, this number is not smaller than the minimal number of critical points of a Morse function on the base. We also give a lower estimate for the number of closed trajectories in the case of a higher-dimensional base. A form preserves.the center of gravity if its cohomology class is the lift of a class from the base. As an application we prove that the number of closed trajectories of a particle on a surface under the action of a strong and little varying magnetic field perpendicular to the surface is not smaller than the minimal number of critical points of a function on the given surface. The author is grateful to V. I. Arnol'd for formulating the problem on the closed trajectories of fields of kernels of differential forms preserving the center of gravity and for helping in work. I. Condition of Preservation of the Center of Gravity. Let S I + M ~ B be an orientable bundle over a closed (compact and boundaryless) orientable manifold B.

Class groups of number fields: numerical heuristics

Abstract: Extending previous work of H. W. Lenstra, Jr. and the first author, we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four (given the signature and the Galois group of the Galois closure). The theoretical justifications for these conjectures will appear elsewhere, but the agreement with the existing tables is quite good. 1. Introduction and Notations. In (3), H. W. Lenstra, Jr., and the first author developed a method for conjecturing quantitative results on class groups of quadratic fields and cyclic extensions of prime degree. In a forthcoming paper (4) we shall show that this technique can be extended to a much wider class of number fields, and also to relative extensions. The aim of the present paper is to rapidly make available the numerical conjec- tures obtained, for people not really interested in our heuristic reasoning or not wanting to wait for (4) to appear. Hence, apart from a total lack of justifications for the conjectures that we present, this paper is essentially self-contained. The plan is as follows. In the rest of this section we present the notations used in the sequel. Some of them being nonstandard (and in general differing from the notations of (3)), we urge the reader to read the notations carefully before applying the conjectures. In the next section we present templates for the subsequent conjectures, and then the conjectures themselves, illustrated by numerical examples, first for their own sake, and second as a double check for the reader to understand the templates. These conjectures are given for all types of fields of degree less than or equal to four. In the final section we comment on the consistency of the conjectures with existing tables (which is quite good). Combinatorial Notations: * If X is a set, {XI denotes its cardinality.

First Order Topological Structures and Theories

Abstract: In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof. Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on o-minimal structures [PS] in a general topological context. Note, however, that the p-adic numbers, and structures definable therein, will also fit into our analysis. In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on (closed) definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.

A Course in Galois Theory

Abstract: Preface Part I. Algebraic Preliminaries: 1. Groups, fields and vector spaces 2. The axiom of choice, and Zorn's lemma 3. Rings Part II. The Theory of Fields, and Galois Theory: 4. Field extensions 5. Tests for irreducibility 6. Ruler-and-compass constructions 7. Splitting fields 8. The algebraic closure of a field 9. Normal extensions 10. Separability 11. Automorphisms and fixed fields 12. Finite fields 13. The theorem of the primative element 14. Cubics and quartics 15. Roots of unity 16. Cyclic extensions 17. Solution by radicals 18. Transcendental elements and algebraic independence 19. Some further topics 20. The calculation of Galois groups Index.

Stochastic-evolution equations driven by nuclear-space-valued martingales. Technical report, 1 October 1986-30 September 1987

Abstract: This paper presents a theory of stochastic-evolution equations for nuclear-space-valued processes and provides a unified treatment of several examples from the field of applications (C/sub O,1/) reversed evolution systems on countably Hilbertian nuclear spaces are also investigated

Codes on the Klein quartic, ideals, and decoding (Corresp.)

Abstract: A sequence of codes with particular symmetries and with large rates compared to their minimal distances is constructed over the field GF (2^{3}) . In the sequence there is, for instance, a code of length 21 and dimension 10 with minimal distance 9 , and a code of length 21 and dimension 16 with minimal distance 3 . The codes are constructed from algebraic geometry using the dictionary between coding theory and algebraic curves over finite fields established by Goppa. The curve used in the present work is the Klein quartic. This curve has the maximal number of rational points over GF (2^{3}) allowed by Serre's improvement of the Hasse-Weil bound, which, together with the low genus, accounts for the good code parameters. The Klein quartic has the Frobenius group G of order 21 acting as a group of automorphisms which accounts for the particular symmetries of the codes. In fact, the codes are given alternative descriptions as left ideals in the group-algebra GF (2^{3})[G] . This description allows for easy decoding. For instance, in the case of the single error correcting code of length 21 and dimension 16 with minimal distance 3 . decoding is obtained by multiplication with an idempotent in the group algebra.

Algebraic complexities and algebraic curves over finite fields

Abstract: We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, 169-180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields.

Calculation of the class numbers of imaginary cyclic quartic fields

Abstract: Any imaginary cyclic quartic field can be expressed uniquely in the form K = Q(\JA(D + b/d) ), where A is squarefree, odd and negative, D = B2 + C2 is squarefree, B > 0, C > 0, and (A,D)= 1. Explicit formulae for the discriminant and conductor of K are given in terms of A, B, C, D. The calculation of tables of the class numbers h(K) of such fields K is described. Let Q denote the field of rational numbers and let K be a cyclic extension of Q of degree 4. The unique quadratic subfield of K is denoted by k. The class number of K (resp. k) is denoted by h(K) (resp. h(k)). The conductor of K is denoted by / = /( K ). In the case of real cyclic quartic fields K, Gras (3) has given a table of the values of h(K) for all such fields with / < 4000. Recently, the authors have carried out the calculation of the class numbers of imaginary cyclic quartic fields (4). In this note we give a brief description of the computation of the tables given in (4). The following explicit representation of a cyclic quartic field is proved in (4,

Densities for ranks of certain parts of $p$-class groups

Abstract: Let K be a Galois extension of the field of rational numbers of prime degree p, and let CA be the p-class group of K. In this paper densities for the ranks of certain parts of such CK are calculated, and these densities suggest a way to extend conjectures of Cohen and Lenstra.

The Arcata Conference on Representations of Finite Groups

Abstract: Researchers in the fields of finite groups, representation theory, and algebra will appreciate this volume for it presents an excellent view of the many significant developments in the representation theory of finite groups. The papers in these proceedings were presented at the Summer Research Institute on Representations of Finite Groups, held at Humboldt State University in Arcata, California in July 1986. There has been intense research in the representation theory of finite groups in recent years.Particularly striking is the influence of algebraic geometry and cohomology theory in the modular representation theory and the character theory of reductive groups over finite fields, and in the general modular representation theory of finite groups. Noteworthy too are the continuing developments in block theory and the general character theory of finite groups. These proceedings contain nearly 100 papers of an expository and research nature and will give readers a sense of the new and important influences in this vital field.

Characters of $p$-groups

Abstract: The following theorem is proved for any prime p: Every irreducible rational represention of a finite p-group is the difference of two transitive permutation representations Also given are two useful results about representations of p-groups, which are known to experts in the field 1 Statement of results Presented here are three results about characters of finite p-groups The first two are probably known to people familiar with the subject The first follows directly from a theorem of Solomon [7, p 156, Theorem 4] and the second from a generalization of this in Feit [2, p 73, 143] Since they do not seem to have been stated and proved explicitly our purpose in doing so here is to make them available to a wider audience The third result generalizes a theorem of Graeme Segal [6] that may not be well known among group theorists As Segal states in his paper, Feit has observed that this theorem may be proved using the result above We present such a proof here and obtain the following result THEOREM An irreducible rational representation of a finite p-group, p a prime, is the difference of two transitive permutation representations Let Q denote the rational field and QG the rational group algebra of a finite group G Let p be a prime number Gal denotes a Galois group [X2 : Xi] denotes the index of X1 in X2 for either fields or groups Absolute value bars denote order All groups are finite here THEOREM 1 Let G be a p-group and X an irreducible complex character of G Then one of the following holds: (i) There exists a linear character A on a subgroup H of G which induces X and generates the same field as X; that is AG = X and Q(A) = Q(X) (ii) p = 2 and there exist subgroups H < K in G with [K : H] = 2 and a linear character A of H such that with AK, [Q(A) Q(f)] = 2, 5G = X, and Q(M) = Q(X) THEOREM 2 Let G be a p-group and X an irreducible complex character of G for which the Schur index mQ(X) = 2 Then G contains a generalized quaternion section More specifically there is an irreducible character f on a subgroup K of G Received by the editors August 20, 1986 Presented at the San Antonio Meeting of the AMS on January 22, 1987 1980 Mathematics Subject Classification (1985 Revision) Primary 20C15

Algebraic dispersion fields on ternary diagrams

Abstract: Previously published dispersion fields on ternary diagrams have been constructed variously, and their derivations have not been well-specified. Here an explanation of their bases is provided through an algebraic method for calculating two related forms, designated thesilhouette dispersion field and thegirth dispersion field. Such dispersion estimates can be made more precise by specifying the percentage of samples that fall within the field. Because such fields represent a mechanistic rather than a probabilistic approach, their use in comparison of sample sets must be viewed with caution.

On isomorphisms of chain geometries

Abstract: Isomorphisms (automorphisms) of chain geometries over a ring L were studied mainly for a skewfield [2], [3], [8] alternating field [9], a commutative algebra (see the survey in [4]) a local ring [7] or an algebra of characteristic , [1]-in most cases by means of harmonic quadruples, cross rations- or (in the last example) generalized harmonic quadruples. Here instead of this are studied isomorphisms of chain geometries over any K-algebra A of finite dimension with (local algebra of finite dimension with ).

Distribution of effective field in the ising spin glass of the +/- J model at T = O. Solutions as superpositions of delta-functions.

Abstract: The integral equation for the distribution function of effective field of the ±J random Ising model in the pair (Bethe) approximation is investigated. Its exact solutions atH (magnetic field)=0,T (temperature)=0 and forz (coordination number)=3 expressed as superpositions of 2N+1 (more than 3), delta functions are considered. Then the integral equation is reduced to a system of algebraic equations ofz −1th degree withN+1 unknowns. The system of the equations is solved by the Grobner basis method withN=1, 2, 3, 4. The number of physically acceptable solutions for a givenN is ω(N)+1 where ω(N) is the number of divisors ofN. The ground-state energy and entropy for these are calculated. They are very close in value (entropies are positive), and it is suggested that a number of physically acceptable solutions correspond to local stationary spin-glass states, as discussed in the literatures.