Mahler型函数值的代数无关度量

In this paper,we give the algebraic independence measures for the values of Mahler type functions in complex number field and p-adic number field,respectively.     

Properties of Algebraic Equations on the Set of E-Functions over the Field of Rational Functions

The paper presents several theorems on the linear and algebraic independence of the values at algebraic points of the set of E-functions related by algebraic equations over the field of rational functions, as well as some estimates of the absolute values of polynomials with integer coefficients in the values of such functions. The results are obtained by using the properties of ideals in the ring of polynomials of several variables formed by equations relating the above functions over the field of rational functions.     

p-adic Measures of Algebraic Independence for the Values of Ramanujan Functions and Klein Modular Functions

In this paper, we give the p-adic measures of algebraic independence for the values of Ramanujan functions and Klein modular functions at algebraic points.     

Arithmetical properties of the values ofE-functions

For a collection ofE-functions which is algebraically dependent over the field of rational functions, theorems on the algebraic independence of values of subcollections at algebraic points are proved. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 452–458, September, 1999.     

Regular functions of several quaternionic variables and the Cauchy-Fueter complex

We employ a classical idea of Ehrenpreis, together with a new algebraic result, to give a new proof that regular functions of several quaternionic variables cannot have compact singularities. As a byproduct we characterize those inhomogeneous Cauchy-Fueter systems which admit solutions on convex sets. Our method readily extends to the case of monogenic functions on Clifford Algebras. We finally study a free resolution of the Cauchy-Fueter complex of differential operators and we obtain some new duality theorems which hint at a hyperfunction theory of several quaternionic variables.     

Algebraic independence results for the values of certain Mahler functions and their application to infinite products

In this paper we establish algebraic independence criteria for the values at an algebraic point of Mahler functions each of which satisfies either a multiplicative type of functional equation or an additive one. As application we construct, using a linear recurrence sequence, an entire function defined by an infinite product such that its values as well as its all successive derivatives at algebraic points other than its zeroes are algebraically independent. Zeroes of such an entire function form a subsequence of the linear recurrence sequence. We prove the algebraic independency by reducing those values at algebraic points to those of Mahler functions.     

Global variants of Hartogs’ theorem

Hartogs’ theorem asserts that a separately holomorphic function, defined on an open subset of $$\mathbb {C}^n$$ , is holomorphic in all the variables. We prove a global variant of this theorem for functions defined on an open subset of the product of complex algebraic manifolds. We also obtain global Hartogs-type theorems for complex Nash functions and complex regular functions.     

Interpolation Formulas and Auxiliary Functions

We prove an interpolation formula for “semi-cartesian products” and use it to study several constructions of auxiliary functions. We get in this way a criterion for the values of the exponential map of an elliptic curve E defined over Q. It reduces the analogue of Schanuel's conjecture for the elliptic logarithms of E to a statement of the form of a criterion of algebraic independence. We also consider a construction of auxiliary function related to the four exponentials conjecture and show that it is essentially optimal. For analytic functions vanishing on a semi-cartesian product, we get a version of the Schwarz lemma in which the exponent involves a condition of distribution reminiscent of the so-called technical hypotheses in algebraic independence. We show by two examples that such a condition is unavoidable.     

Another proof of Joseph and Letzter's separation of variables theorem for quantum groups

Let g be a simple finite-dimensional complex Lie algebra and letG be the corresponding simply-connected algebraic group. A theorem of Kostant states that the universal enveloping algebra of g is a free module over its center. A theorem of Richardson states that the algebra of regular functions ofG is a free module over the subalgebra of regular class functions. Joseph and Letzter extended Kostant's theorem to the case of the quantized enveloping algebra of g. Using the theory of crystal bases as the main tool, we prove a quantum analogue of Richardson's theorem. From it, we recover Joseph and Letzter's result by a kind of quantum duality principle.     

Algebraic Independence of Solutions of First-Order Rational Difference Equations

We prove a theorem on algebraic independence of solutions of first order rational difference equations. By the theorem, we are able to prove algebraic independence of x, the exponential function e ^x and the Weierstrass function ${\wp(x)}$ over ${\mathbb{C}}$ only by seeing degrees of polynomials associated with their double angle formulas. As a corollary, we obtain a result on unsolvability of a first-order rational difference equation by solutions of other first-order rational difference equations, which implies its irreducibility. Additionally, we introduce some applications to algebraic independence of functions f(x), f(x ²), . . . , f(x ⁿ ).     

用逼近方法建立代数无关性

应用逼近方法建立了一个关于复数代数无关性的一般性判别法则，并用来研究某些缺项级数在代数点和超越点上值的代数无关性．     

The joint value distribution of the Riemann zeta function and Hurwitz zeta functions II

In the previous paper [9] the author proved the joint limit theorem for the Riemann zeta function and the Hurwitz zeta function attached with a transcendental real number. As a corollary, the author obtained the joint functional independence for these two zeta functions. In this paper, we study the joint value distribution for the Riemann zeta function and the Hurwitz zeta function attached with an algebraic irrational number. Especially we establish the weak joint functional independence for these two zeta functions. Received: 17 Apri1 2007     

Algebraic independence by approximation method

By the use of approximation method a general criterion of algebraic independence of complex numbers is established. As its application the algebraic independence of values of certain gap series at algebraic and transcendental points is given. Subject supported by the National Natural Science Foundation of China     

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.     

Triangular domain extension of algebraic trigonometric Bézier-like basis

In computer aided geometric design (CAGD), Bézier-like bases receive more and more considerations as new modeling tools in recent years. But those existing Bézier-like bases are all defined over the rectangular domain. In this paper, we extend the algebraic trigonometric Bézier-like basis of order 4 to the triangular domain. The new basis functions defined over the triangular domain are proved to fulfill non-negativity, partition of unity, symmetry, boundary representation, linear independence and so on. We also prove some properties of the corresponding Bézier-like surfaces. Finally, some applications of the proposed basis are shown.     

Algebraic values of analytic functions

Given an analytic function of one complex variable f, we investigate the arithmetic nature of the values of f at algebraic points. A typical question is whether f() is a transcendental number for each algebraic number . Since there exist transcendental entire functions f such that for any t0 and any algebraic number , one needs to restrict the situation by adding hypotheses, either on the functions, or on the points, or else on the set of values.

Among the topics we discuss are recent results due to Andrea Surroca on the number of algebraic points where a transcendental analytic function takes algebraic values, new transcendence criteria by Daniel Delbos concerning entire functions of one or several complex variables, and Diophantine properties of special values of polylogarithms.     

Graded Associative Conformal Algebras of Finite Type

In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group Γ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group G such that the identity component G ⁰ is the affine line and G/G ⁰???Γ. A classification of simple and semisimple graded associative conformal algebras of finite type is obtained.     

某些无穷乘积的代数无关性

本文研究了某些用无穷乘积定义的函数在代数点和超越点上的值的代数无关 性.     

A Cauchy kernel for slice regular functions

In this article, we show how to construct a regular, non-commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results.     

Independence measures of arithmetic functions

The notion of algebraic dependence in the ring of arithmetic functions with addition and Dirichlet product is considered. Measures for algebraic independence are derived.