Transcendence results on the generating functions of the characteristic functions of certain self-generating sets, II

Peter Bundschuh, and Keijo Väänänen. "Transcendence results on the generating functions of the characteristic functions of certain self-generating sets, II." Acta Arithmetica 167.3 (2015): 239-249. <http://eudml.org/doc/279222>.

@article{PeterBundschuh2015,
abstract = {This article continues a previous paper by the authors. Here and there, the two power series F(z) and G(z), first introduced by Dilcher and Stolarsky and related to the so-called Stern polynomials, are studied analytically and arithmetically. More precisely, it is shown that the function field ℂ(z)(F(z),F(z⁴),G(z),G(z⁴)) has transcendence degree 3 over ℂ(z). This main result contains the algebraic independence over ℂ(z) of G(z) and G(z⁴), as well as that of F(z) and F(z⁴). The first statement is due to Adamczewski, whereas the second is our previous main result. Moreover, an arithmetical consequence of the transcendence degree claim is that, for any algebraic α with 0 < |α| < 1, the field ℚ(F(α),F(α⁴),G(α),G(α⁴)) has transcendence degree 3 over ℚ.},
author = {Peter Bundschuh, Keijo Väänänen},
journal = {Acta Arithmetica},
keywords = {transcendence degree; algebraic independence},
language = {eng},
number = {3},
pages = {239-249},
title = {Transcendence results on the generating functions of the characteristic functions of certain self-generating sets, II},
url = {http://eudml.org/doc/279222},
volume = {167},
year = {2015},
}

TY - JOUR
AU - Peter Bundschuh
AU - Keijo Väänänen
TI - Transcendence results on the generating functions of the characteristic functions of certain self-generating sets, II
JO - Acta Arithmetica
PY - 2015
VL - 167
IS - 3
SP - 239
EP - 249
AB - This article continues a previous paper by the authors. Here and there, the two power series F(z) and G(z), first introduced by Dilcher and Stolarsky and related to the so-called Stern polynomials, are studied analytically and arithmetically. More precisely, it is shown that the function field ℂ(z)(F(z),F(z⁴),G(z),G(z⁴)) has transcendence degree 3 over ℂ(z). This main result contains the algebraic independence over ℂ(z) of G(z) and G(z⁴), as well as that of F(z) and F(z⁴). The first statement is due to Adamczewski, whereas the second is our previous main result. Moreover, an arithmetical consequence of the transcendence degree claim is that, for any algebraic α with 0 < |α| < 1, the field ℚ(F(α),F(α⁴),G(α),G(α⁴)) has transcendence degree 3 over ℚ.
LA - eng
KW - transcendence degree; algebraic independence
UR - http://eudml.org/doc/279222
ER -