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.
3
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
This article continues two papers which recently appeared in this same journal. First, Dilcher and Stolarsky [140 (2009)] introduced two new power series, F(z) and G(z), related to the so-called Stern polynomials and having coefficients 0 and 1 only. Shortly later, Adamczewski [142 (2010)] proved, inter alia, that G(α),G(α⁴) are algebraically independent for any algebraic α with 0 < |α| < 1. Our first key result is that F and G have large blocks of consecutive zero coefficients. Then, a Roth-type argument shows that F(a/b) and G(a/b), for any (a,b) ∈ ℤ × ℕ with 0 < |a| < √b, are transcendental but not U-numbers. Moreover, reasonably good upper bounds for the irrationality exponent of these numbers are obtained. Another main result for which an elementary (or poor men's) proof is presented concerns the algebraic independence of F(z),F(z⁴) over ℂ(z) leading to the F-analogue of Adamczewski's above-mentioned theorem.
4
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW