Measure Of Algebraic Independence Research Articles

Topical problems of the theory of Transcendental numbers: Development of approaches to their solution in the works of Yu. V. Nesterenko

The present paper is a survey of a part of the theory devoted to certain problems concerning the algebraic independence of the values of analytic functions, to quantitative results on estimates for the measure of transcendence or the measure of algebraic independence of numbers, to functional analogs of these results on the algebraic independence of solutions of algebraic differential equations, and estimates for the multiplicities of zeros for polynomials in these solutions, which play an important role in the proof of numerical results. This choice is related to the fact that, in December 2016, the head of the Department of Number Theory of Moscow State University, Corresponding Member of the RAS Yu.V. Nesterenko, who did a lot to develop these directions of the theory Transcendental numbers and whose works are marked by many awards, became seventy. He is a laureate of the Markov RAS Prize, 2006, of the Ostrovsky international prize, 1997, of the Hardy–Ramanujan Society Prize, 1997, and the Alexander von Humboldt Prize, 2003. Since the article is dedicated to the 70th anniversary of the birth of Yurii Valentinovich, we preface the scientific part with a brief biography.

Multiplicity estimates for algebraically dependent analytic functions

We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of functions, for instance, within the context of Mahler's method. For this reason, our result provides an important tool for the proofs of algebraic independence of complex numbers. At the same time, these estimates can be considered as a measure of algebraic independence of functions themselves. Hence, our result provides, under some conditions, the measure of algebraic independence of elements in Fq[[T]], where Fq denotes a finite field.

On a measure of algebraic independence of modulus and values of Jacobi elliptic function

It is proved an estimation of a measure of algebraic independence for the modulus and values at various algebraic points of the Jacobi elliptic function.

ZERO ORDER ESTIMATES FOR ANALYTIC FUNCTIONS

In this article we develop an important tool in transcendental number theory. More precisely, we study multiplicity estimates (or multiplicity lemmas) for analytic functions. Our main theorem reduces multiplicity estimates at zero to the study of ideals in polynomial ring stable under an appropriate map. In particular, in the case of algebraic morphisms this result gives a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. Specialized to the case of differential operators this theorem improves Nesterenko's conditional result on solutions of systems of differential equations. We also deduce an analog of Nesterenko's theorem for Mahler's functions and for solutions of q-difference equations. Further, analyzing stable ideals we prove the unconditional optimal result in the case of linear functional systems of generalized Mahler's type. The latter result generalizes a famous theorem of Nishioka (1986) previously conjectured by Mahler (1969). This new multiplicity estimate allows to prove new results on algebraic independence and on measures of algebraic independence, as done in Zorin (2010 and 2011).

New results on algebraic independence with Mahlerʼs method

We give some new results on algebraic independence in the frame of Mahlerʼs method, including algebraic independence of values at transcendental points. We also give some new measures of algebraic independence. In particular, our results furnish for n ⩾ 1 arbitrarily large new examples of families of members ( θ 1 , … , θ n ) ∈ R n normal in the sense of the definition formulated by G. Chudnovsky (1980) [2].

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.

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.

On the measure of algebraic independence of values of derivatives of ap-adic modular function

The measure of the relative transcendence for the values of the derivatives of ap-adic Weyl-Lutz modular functionJ(q) at algebraic points is estimated.

On a measure of algebraic independence of values of an elliptic function

An estimate is obtained for a measure of algebraic independence of values of a Weierstrass elliptic function with algebraic invariants at various algebraic points. The estimate is sharp as a function of the degree and height of polynomials.

A Note on Some Elementary Measures of Algebraic Independence

A Note on Some Elementary Measures of Algebraic Independence

A note on some elementary measures of algebraic independence

We investigate the algebraic independence of some numbers associated with elliptic functions when one of the numbers is a "Liouville-type" number. Suppose ℘ ( z ) \wp (z) is a Weierstrass elliptic function with algebraic invariants and β \beta is an algebraic number, not belonging to the field of multiplications for ℘ ( z ) \wp (z) . We establish the algebraic independence of ℘ ( u ) \wp (u) and ℘ ( β u ) \wp (\beta u) (respectively, of u u and ℘ ( β u ) \wp (\beta u) ) when ℘ ( u ) \wp (u) (respectively, u u ) is a "Liouville-type" number. We also give quantitative versions of these results.

Grands degrés de transcendance pour des familles d'exponentielles

We improve lower bounds obtained by P. Philippon for the transcendence degree over Q of families of the type {exp( u i v j ); i, j}, { v j , exp( u i v j ); i, j}. As a corollary we replace the lower bound [ d 2 ] by [ (d + 1) 2 ] for the family {exp( β j log α); 1 ≤ j ≤ d − 1} where α and β are algebraic and β of degree d ≥ 2; when d = 3 we give also a measure of algebraic independence of exp(β log α) and exp( β 2 log α).

On the measure of algebraic independence of certain values of elliptic functions

We provide a measure for the algebraic independence of some special values of the Weierstrass elliptic function with complex multiplication and algebraic invariants. Specifically, suppose ℘(z) is such an elliptic function, u is a nontorsion algebraic point for ℘(z), and β is an algebraic number which is cubic over the field of multiplications of ℘(z). We then give the following result: For every ε > 0 there exists a positive real number t( ε) > 0 such that for any nonzero integral polynomial P( X, Y), with t( P) = deg P + log height P > t( ε), log|P(℘(βu), ℘(β 2u))| > − exp(t(P) 4 + ε) .

Effectivity in independence measures for values ofE-functions

AbstractWe establish a measure of algebraic independence for values ofE-functions which is more nearly effectively computable than the previous one. When the system of equations meets either of two criteria, then the measure becomes entirely effectively computable.

Measure of algebraic independence for almost all pairs of p-adic numbers

Measure of algebraic independence for almost all pairs of p-adic numbers

Algebraic grounds for the proof of algebraic independence. How to obtain measure of algebraic independence using elementary methods part I. Elementary algebra

Algebraic grounds for the proof of algebraic independence. How to obtain measure of algebraic independence using elementary methods part I. Elementary algebra

ESTIMATES FOR THE ORDERS OF ZEROS OF FUNCTIONS OF A CERTAIN CLASS AND APPLICATIONS IN THE THEORY OF TRANSCENDENTAL NUMBERS

By the use of estimates for the multiplicities of the zeros of polynomials in certain fixed functions, an estimate is proved for the measure of algebraic independence of the values of the functions which is effective in the degree and height.Bibliography: 10 titles.

Simultaneous approximations of exponents by transcendental numbers of certain classes

We obtain a theorem on simultaneous approximations of values of the exponential function by elements of fields of finite transcendence degree for whose generators we have a “sufficiently good” estimate of the measure of algebraic independence.

A criterion for algebraic dependence of transcendental numbers

We obtain a theorem concerning the algebraic dependence of the p+1 numbersθ,θ 1 ...θ p subject to the condition that the numbersθ 1 , ...,θ p are algebraically independent and possess a “sufficiently good” estimate of measure of algebraic independence.