Arithmetische Eigenschaften Analytischer Functionen

Stäckel, Paul

doi:10.1007/bf02419032

Cited by 13 publications

(10 citation statements)

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“…Definition 2.11. We say that a map φ : A → A is correct with respect to the ideal P ⊂ A if for every ideal I, such that all its associated primes are contained in P, the inclusion φ(I) ⊂ eq(I) (31) implies φ(eq(I)) ⊂ eq(I) (32) (recall that eq(I) is introduced in Definition 2.10).…”

Section: Definitions and Properties Related To Commutative Algebramentioning

confidence: 99%

Multiplicity estimates for algebraically dependent analytic functions

Zorin

2013

Proceedings of the London Mathematical Society

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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.

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Section: Definitions and Properties Related To Commutative Algebramentioning

confidence: 99%

Multiplicity estimates for algebraically dependent analytic functions

Zorin

2013

Proceedings of the London Mathematical Society

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“…This assertion was proved in 1895 by Stäckel [2] who established a much more general result: for each countable subset Σ ⊆ C and each dense subset T ⊆ C, there exists a transcendental entire function f such that f (Σ) ⊆ T . In another construction, Stäckel [3] produced a transcendental function f (z), analytic in a neighbourhood of the origin, and with the property that both f (z) and its inverse function assume, in this neighbourhood, algebraic values at all algebraic points. Based on this result, in 1976, Mahler [1, p. 53] suggested the following question a n z n , with rational coefficients a n and such that the image and the preimage of Q under f are subsets of Q?…”

Section: Introductionmentioning

confidence: 99%

A positive answer for a question proposed by K. Mahler

Marques

Moreira

2016

Math. Ann.

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In 1902, P. Stäckel proved the existence of a transcendental function f (z), analytic in a neighbourhood of the origin, and with the property that both f (z) and its inverse function assume, in this neighbourhood, algebraic values at all algebraic points. Based on this result, in 1976, K. Mahler raised the question of the existence of such functions which are analytic in C. In this work, we provide a positive answer for this question by showing the existence of uncountable many of these functions.

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“…Later, Stäckel [6] proved that for each countable subset Σ ⊆ C and each dense subset T ⊆ C, there is a transcendental entire function f such that f (Σ) ⊆ T (F. Gramain showed that Stäckel's theorem is valid if Σ and T are subsets of R). Another construction due to Stäckel [7] produces a transcendental entire function f whose derivatives f (t) , for t = 0, 1, 2, . .…”

Section: Introductionmentioning

confidence: 99%