Algebraic Independence of Certain Values of Exponential Function

The algebraic independence of e^θ1,…,e^θs is proved, where θ1,… ,θs are certain gap series or power series of algebraic numbers, or certain transcendental continued fractions with algebraic elements.     

Necessary Conditions for the Localization of the Spectrum of the Sturm-Liouville Problem on a Curve

We consider the Sturm-Liouville operator on a convex smooth curve lying in the complex plane and connecting the points 0 and 1. We prove that if the eigenvalues λ_k with large numbers are localized near a single ray, then this ray is the positive real semiaxis. Moreover, if the eigenvalues λ_k are numbered with algebraic multiplicities taken into account, then λ_k ∼ π · k as k → +∞.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 72–84.Original Russian Text Copyright © 2005 by Kh. K. Ishkin.     

Algebraic Independence of Certain Generalized Mahler Type Numbers

In this paper the generalized Mahler type number Mh（g;A,T） is defined, and in the case of multiplicatively dependent parameters gi, hi（1 ≤ i ≤ s） the algebraic independence of the numbers Mhi （gi; A, T）（1 ≤ i ≤ s） is proved, where A and T are certain infinite sequences of non-negative integers and of positive integers, respectively. Furthermore, the algebraic independence result on values of a certain function connected with the generalized Mahler type number and its derivatives at algebraic numbers is also given.     

Polynomials with small discriminants and regular systems of real algebraic numbers

The distribution of special algebraic numbers is studied, and an optimal regular system is constructed. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 10–16.     

On the periodicity of trigonometric functions generalized to quotient rings of R[x]

We apply a method of Euler to algebraic extensions of sets of numbers with compound additive inverse which can be seen as quotient rings of R[x]. This allows us to evaluate a generalization of Riemann’s zeta function in terms of the period of a function which generalizes the function sin z. It follows that the functions generalizing the trigonometric functions on these sets of numbers are not periodic.     

On a measure of algebraic independence of values of Jacobi elliptic functions

In this paper, an estimate for the measure of algebraic independence is proved for values of the Jacobi elliptic function sn(z) at different algebraic points. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 209–219, 2005.     

Exact calculation of inverse functions

We represent continuous functions on compact intervals by sequences of functions defined on finite sets of rational numbers. We call this an exact representation. This enables us to calculate the values of the function arbitrarily exactly, without roundoff errors. As an application we develop a procedure to transfer an exact representation of an increasing function into an exact representation of the corresponding inverse function. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Subgroups of ideal class groups of real quadratic algebraic function fields

Necessary and sufficient condition on real quadratic algebraic function fields K is given for their ideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic function fields K are obtained whose ideal class groups contain cyclic subgroups of order n. In particular, the ideal class numbers of these function fields are divisible by n.     

A generalization of the global relation method

We generalize the notions of global and nontrivial global relations to the case where the variables of the series in the definitions of these relations are series well approximable by algebraic numbers in any non-Archimedean local field. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 220–238.     

The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface

For a simple complete ideal ℘ of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincaré series P_℘, that gathers in a unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to ℘. This paper is devoted to prove that P_℘ is a rational function giving an explicit expression for it.     

某些无穷乘积的代数无关性(Ⅱ)

本文证明了某些具有代数系数的无穷乘积和幂级数定义的函数在某些代数数和超越数上值的代数无关性.     

Multiplicative relations with conjugate algebraic numbers

We study what algebraic numbers can be represented by a product of algebraic numbers conjugate over a fixed number field K in fixed integer powers. The problem is nontrivial if the sum of these integer powers is equal to zero. The norm of such a number over K must be a root of unity. We show that there are infinitely many algebraic numbers whose norm over K is a root of unity and which cannot be represented by such a product. Conversely, every algebraic number can be expressed by every sufficiently long product in algebraic numbers conjugate over K. We also construct nonsymmetric algebraic numbers, i.e., algebraic numbers such that no elements of the corresponding Galois group acting on the full set of their conjugates form a Latin square. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 890–900, July, 2007.     

Common prime factors of the numbersA_n = a^{2^n } + 1

A method to find certain common factors of numbers of the form , when theA _n are lacking algebraic factors, is discussed. Using known factors of the Fermat numbers , otherA _n , containing these factors, are deduced.     

Mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif

This work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group, defined over the field of algebraic numbers . Let G be such a group. Let W be a hyperplane of the tangent space at the origin of G, defined over , and u be a complex point of this tangent space, such that the image of u by the exponential map of the Lie group G(ℂ) is an algebraic point. Then we obtain a lower bound for the distance between u and W⊗ℂ, which improves the results known before and which is, in particular, the best possible for the height of the hyperplane W. The proof rests on Baker’s method and Hirata’s reduction as well as a new arithmetic argument (Chudnovsky’s process of variable change) which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof.

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Mathematics Subject Classification (2000) 11J86, 11J20, 14L10     

The Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at mnT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m + n − 2 containing all but one point of them also contains the last point.     

A sequentially computable function that is not effectively continuous at any point

P. Hertling [Lecture Notes in Computer Science, vol. 2380, Springer, Berlin, 2002, pp. 962–972; Ann. Pure Appl. Logic 132 (2005) 227–246] showed that there exists a sequentially computable function mapping all computable real numbers to computable real numbers that is not effectively continuous. Here, that result is strengthened: a sequentially computable function on the computable real numbers is constructed that is not effectively continuous at any point.     

Divisors,measures and critical functions

In [4] we have introduced a new distance between Galois orbits over ℚ. Using generalized divisors, we have extended the notion of trace of an algebraic number to other transcendental quantities. In this article we continue the work started in [4]. We define the critical function for a class of transcendental numbers, that generalizes the notion of minimal polynomial of an algebraic number. Our results extend the results obtained by Popescu et al [5].     

Euclidean distance degree and limit points in a Morsification

Motivated by finding an effective way to compute the algebraic complexity of the nearest point problem for algebraic models, we introduce an efficient method for detecting the limit points of the stratified Morse trajectories in a small perturbation of any polynomial function on a complex affine variety. We compute the multiplicities of these limit points in terms of vanishing cycles. In the case of functions with only isolated stratified singularities, we express the local multiplicities in terms of polar intersection numbers.     

Monotonicity Properties of Determinants of Special Functions

We prove the absolute monotonicity or complete monotonicity of some determinant functions whose entries involve