Showing papers on "Algebraic number published in 1983"

Lower bounds for algebraic computation trees

Abstract: A topological method is given for obtaining lower bounds for the height of algebraic computation trees, and algebraic decision trees. Using this method we are able to generalize, and present in a uniform and easy way, almost all the known nonlinear lower bounds for algebraic computations. Applying the method to decision trees we extend all the apparently known lower bounds for linear decision trees to bounded degree algebraic decision trees, thus answering the open questions raised by Steele and Yao [20]. We also show how this new method can be used to establish lower bounds on the complexity of constructions with ruler and compass in plane Euclidean geometry.

Symmetric Designs: An Algebraic Approach

Abstract: Symmetric designs are an important class of combinatorial structures which arose first in the statistics and are now especially important in the study of finite geometries. This book presents some of the algebraic techniques that have been brought to bear on the question of existence, construction and symmetry of symmetric designs – including methods inspired by the algebraic theory of coding and by the representation theory of finite groups – and includes many results. Rich in examples and containing over 100 problems, the text also provides an introduction to many of the modern algebraic approaches used, through six lengthy appendices and supplementary problems. The book will be of interest to both combinatorialists and algebraists, and could be used as a course text for a graduate course.

Galois Module Structure of Algebraic Integers

Abstract: Notation and Conventions- I Survey of Results- 1 The Background- 2 The Classgroup- 3 Ramification and Module Structure- 4 Resolvents- 5 L-Functions and Galois Gauss Sums- 6 Symplectic Root Numbers and the Class UN/K- 7 Some Problems and Examples- Notes to Chapter I- II Classgroups and Determinants- 1 Hom-Description- 2 Localization- 3 Change in Basefield and Change in Group- 4 Reduction mod l and Some Computations- 5 The Logarithm for Group Rings- 6 Galois Properties of the Determinant- Notes to Chapter II- III Resolvents, Galois Gauss Sums, Root Numbers, Conductors- 1 Preliminaries- 2 Localization of Galois Gauss Sums and of Resolvents- 3 Galois Action- 4 Signatures- 5 The Local Main Theorems- 6 Non-Ramified Base Field Extension- 7 Abelian Characters, Completion of Proofs- 8 Module Conductors and Module Resolvents- Notes to Chapter III- IV Congruences and Logarithmic Values- 1 The Non-Ramified Characteristic- 2 Proof of Theorem 31- 3 Reduction Steps for Theorem 30- 4 Strategy for Theorem 32- 5 Gauss Sum Logarithm- 6 The Congruence Theorems- 7 The Arithmetic Theory of Tame Local Galois Gauss Sums- Notes to Chapter IV- V Root Number Values- 1 The Arithmetic of Quaternion Characters- 2 Root Number Formulae- 3 Density Results- 4 The Distribution Theorem- VI Relative Structure- 1 The Background- 2 Galois Module Structure and the Embedding Problem- 3 An Example- 4 Generalized Kummer Theory- 5 The Generalized Class Number Formula and the Generalized Stickelberger Relation- Literature List- List of Theorems- Some Further Notation

Solving algebraic problems with high accuracy

Abstract: Publisher Summary This chapter presents the new methods of solving algebraic problems with high accuracy. Examples of such problems are the solving of linear systems, eigenvalue/eigenvector determination, computing zeros of polynomials, sparse matrix problems, computation of the value of an arbitrary arithmetic expression, in particular, the value of a polynomial at a point, nonlinear systems, linear, quadratic, and convex programming over the field of real or complex numbers as well as over the corresponding interval spaces. All the algorithms based on new methods have some key properties in common: (1) every result is automatically verified to be correct by the algorithm; (2) the results are of high accuracy, that is, the error of every component of the result is of the magnitude of the relative rounding error unit; (3) the solution of the given problem is automatically shown to exist and to be unique within the given error bounds; and (4) the computing time is of the same order as comparable floating-point algorithm. The key property of the algorithms is that error control is performed automatically by the computer without any requirement on the part of the user, such as estimating spectral radii. The error bounds for all components of the inverse of the Hilbert 15 × 15 matrix are as small as possible, that is, left and right bounds differ only by one in the 12 place of the mantissa of each component. It is called least significant bit accuracy.

On Conic Bundle Structures

Abstract: This paper determines under which conditions an algebraic variety can be uniquely (up to equivalence) represented in the form of a conic bundle The results are used to show that many conic bundles over rational varieties are nonrational, and to construct examples of nonrational algebraic threefolds whose three-dimensional integral cohomology group is trivial Bibliography: 16 titles

Neighborhoods of algebraic sets

Abstract: In differential topology, a smooth submanifold in a manifold has a tubular neighborhood, and in piecewise-linear topology, a subcomplex of a simplicial complex has a regular neighborhood. The purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets. The neighbor- hoods will be defined as level sets of polynomial or semialgebraic functions. Introduction. Let M be an algebraic set in real n-space R", and let X be a compact algebraic subset of M containing its singular locus, if any. An algebraic neighborhood of A' in M is defined to be a_1(0, 8), where o > 0 is sufficiently small and a: M -> R is a proper polynomial function for which a > 0 and a_1(0) = X. Such an a will be called a rug function. Occasionally we will need rational or analytic a, but this is not a significant generalization. Algebraic neighborhoods always exist. The curve selec- tion lemma is used to prove uniqueness ; anyone familiar with (Milnor 2) will recognize the technique. Since uniqueness is a crucial result, and since there are a few trouble- some small points, this proof is given in detail. The uniqueness theorem shows that the "link" of a singularity of an algebraic set M is independent of the embedding of M in its ambient space; I have been unable to find a proof of this result in the litera- ture. In addition, an algebraic neighborhood of a nonsingular X in M is shown to be a tubular neighborhood in the sense of differential topology. This material is in §1. In §2 the theory is rapidly developed for real and complex projective space by embedding these spaces in real affine space. As an application, it is shown that when M is an affine algebraic set with projective completion M, then the complement in M of a large ball centered at the origin of affine space is an alge- braic neighborhood of the intersection of M with the hyperplane at infinity. In §3, the theory is generalized to the case where M is a semialgebraic subset of R" and X is a compact semialgebraic subset of M. (For example, X could be a non- isolated singular point of M.) A semialgebraic neighborhood of X in M is defined to be a-1(0, o), where o > 0 is sufficiently small and a: M -> R is a proper semialgebraic function for which a > 0 and a_1(0) = X. Again these neighborhoods exist and are unique ; this is shown by mimicking the proof in the algebraic case, using semialgebraic

Real Zeros of Polynomials

Abstract: Let A be a polynomial over Z, Q or Q(α) where α is a real algebraic number. The problem is to compute a sequence of disjoint intervals with rational endpoints, each containing exactly one real zero of A and together containing all real zeros of A. We describe an algorithm due to Kronecker based on the minimum root Separation, Sturm’s algorithm, an algorithm based on Rolle’s theorem due to Collins and Loos and the modified Uspensky algorithm due to Collins and Aritas. For the last algorithm a recursive version with correctness proof is given which appears in print for the first time.

Necessary Condition for the Existence of Algebraic First Integrals. II: Condition for Algebraic Integrability

Abstract: Necessary condition for the existence of a sufficient number of algebraic first integrals is given for a class of dynamical systems. It is proved that in order that a given system is algebraically integrable, all possible Kowalevski's exponents, which characterize a singularity of the solution, must be rational number. For example, the classical 3-body problem and the Henon-Heiles system are shown to be not algebraically integrable.

Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods

Abstract: Lie algebraic methods are developed to describe the behavior of trajectories near a given trajectory for general Hamiltonian systems. A procedure is presented for the computation of nonlinear effects of arbitrarily high degree, and explicit formulas are given through effects of degree 5. Expected applications include accelerator design, charged particle beam and light optics, other problems in the general area of nonlinear dynamics, and, perhaps, with suitable modification, the area of S‐matrix expansions in quantum field theory.

A Method for Computing All Solutions to Systems of Polynomials Equations

Abstract: Small polynomial systems of equations arise in many apphcations areas: computer-aided design, mechanical design, chemmal kinetics modeling, and nonlinear circuit analysis. Use of local iterative methods, such as Newton's method, can be a hit-or-miss process. Purely algebraic schemes, such as the method of resultants, can lead to severe numerical difficulties. Imbedding methods provide the most reliable techniques for computing all solutmns to small polynomial systems. One such method is described and computational experience with it is reported on.

Operateurs conjugués et propriétés de propagation

Abstract: We use an algebraic criteria (based on local positivity of a commutator) which asserts the existence of a direction of propagation for the flowe−iHt associated to a self-adjoint operatorH.

The prolongation structures of quasi-polynomial flows

Abstract: We look closely at the process of finding a Wahlquist-Estabrook prolongation structure for a given (system of) nonlinear evolution equation(s). There are two main steps in this calculation: the first, to reduce the problem to the investigation of a finitely generated, free Lie algebra with constraints; the second, to find a finite-dimensional linear representation of these generators. We discuss some of the difficulties that arise in this calculation. For quasi-polynomial flows (defined later) we give an algorithm for the first step. We do not totally solve the problems of the second step, but do give an algebraic framework and a number of techniques that are quite generally applicable. We illustrate these methods with many examples, several of which are new.

Algebraic techniques for nonlinear codes

Abstract: We introduce some techniques for nonlinear systematic error-correcting codes over the two-element alphabet. As an application, we construct new perfect 1-error-correcting (15,11)-codes.

Subdirect decomposition of concept lattices

Abstract: In [1], G.Birkhoff exhibited the subdirect product of algebraic structures as a universal tool, which since has been extensively used in the study of algebraic theories. Although a subdirect product is not uniquely determined by its factors, there are useful construction methods based on subdirect products (cf. Wille [8], [9], [10]). The aim of this paper is to make these methods available for handling the “Determination Problem” of concept lattices as it is exposed in Wille [11]. In particular, a useful method for determining concept lattices via its scaffoldings will be developed under some finiteness condition.

Factorization of Polynomials

Abstract: Algorithms for factoring polynomials in one or more variables over various coefficient domains are discussed. Special emphasis is given to finite fields, the integers, or algebraic extensions of the rationals, and to multivariate polynomials with integral coefficients. In particular, various squarefree decomposition algorithms and Hensel lifting techniques are analyzed. An attempt is made to establish a complete historic trace for today’s methods. The exponential worst case complexity nature of these algorithms receives attention.

Algebra in a Localic Topos with Applications to Ring Theory

Abstract: Categories of algebraic sheaves.- The formal initial segments.- Localizations and algebraic sheaves.- Integral theories and characterization theorem.- Spectrum of a theory.- Applications to module theory.- Pure representation of rings.- Gelfand rings.

An accurate closed-form approximation for ground return impedance calculations

Abstract: The calculation of ground return impedances generally requires the evaluation of an infinite complex integral. Algebraic series expressions that usually converge to the value of this integral have been known for about 50 years. Several approximate "closed-form" expressions for the integral have been proposed in recent years. Under most conditions, these approximate models give values within 15 percent of the correct value. This letter presents an improved "closed-form" approximation which gives values within 2.5 percent of the correct ones in all cases tested. It does so without any convergence difficulties. Furthermore, the results obtained are better than the ones obtained from series expression approximations if premature truncation takes place.

The hyperelliptic equation over function fields

Abstract: Siegel, in a letter to Mordell of 1925(9), proved that the hyper-elliptic equation y2 = g(x) has only finitely many solutions in integers x and y, where g denotes a square-free polynomial of degree at least three with integer coefficients. Siegel's method reduces the hyperelliptic equation to a finite set of Thue equations f(x, y) = 1, where f denotes a binary form with algebraic coefficients and at least three distinct linear factors; x and y are integral in a fixed algebraic number field. Siegel had already proved that the Thue equations so obtained have only finitely many solutions. However, as is well known, the work of Siegel is ineffective in that it fails to provide bounds on the integer solutions of y2 = g(x). In 1969 Baker (1), using the theory of linear forms in logarithms, employed Siegel's technique to establish explicit bounds on x and y; Baker's result thus reduced the problem of determining all integer solutions of the hyperelliptic equation to a finite amount of computation.

An algebraic formula for the output of a system with large-signal, multifrequency excitation

Abstract: A formula is derived for the output components of a non-linearity which can be described by a power series, with complex coefficients and frequency-dependent time delays, when the input is a sum of sinusoids.