arXiv subjects

Diego Marques

Publications and source records attributed to Diego Marques.

60 records · Page 4Linked to original sources

arXiv2011

Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental

We give a brief history of transcendental number theory, including Schanuel's conjecture (S). Assuming (S), we prove that if z and w are complex numbers, not 0 or 1, with z^w and w^z algebraic, then z and w are either both rational or both transcendental. A corollary is that if (S) is true, then we can find four distinct transcendental positive real numbers x, y, s, t such that the three numbers x^y=/=y^x and s^t=t^s are all integers. Another application (possibly known) is that (S) implies the transcendence of the numbers sqrt(2)^sqrt(2)^sqrt(2), i^i^i, and i^e^pi. We also prove that if (S) holds and a^a^z=z, where a=/=0 is algebraic and z is irrational, then z is transcendental.

math.NT↗

arXiv2010

A Geometric Proof to Cantor's Theorem and an Irrationality Measure for Some Cantor's Series

Generalizing a geometric idea due to J. Sondow, we give a geometric proof for the Cantor's Theorem. Moreover, it is given an irrationality measure for some Cantor series.

math.HO↗

arXiv2010

Some field theoretical properties and an application concerning transcendental numbers

For a proper subfield $K$ of $\QQ$ we show the existence of an algebraic number $α$ such that no power $α^n$, $n\geq 1$, lies in $K$. As an application it is shown that these numbers, multiplied by convenient Gaussian numbers, can be written in the form $P(T)^{Q(T)}$ for some transcendental numbers $T$ where $P$ and $Q$ are arbitrarily prescribed non-constant rational functions over $\QQ$.

math.NT↗

arXiv2009

Transcendence Measures for some $U_m$-numbers related to Liouville's constant

In this note, we shall prove that the sum and the product of an algebraic number $α$ by the \textit{Liouville constant} $L=\sum_{j=1}^{\infty}10^{-j!}$ is a $U$-number with type equals to the degree of $α$ (with respect to $\mathbb{Q}$). Moreover, we shall have that $\max\{w^{\ast}_n(αL),w^{\ast}_n(α+ L)\}\leq 2m^2n+m-1$, for $n=1,...,m-1$.

math.NT↗

arXiv2008

Algebraic Values of Transcendental Functions at Algebraic Points

In this paper, the authors will prove that any subset of $\overline{\QQ}$ can be the exceptional set of some transcendental entire function. Furthermore, we could generalize this theorem to a much more general version and present a unified proof.

math.NT↗

arXiv2008

Some consequences of Schanuel's Conjecture

During the Arizona Winter School 2008 (held in Tucson, AZ) we worked on the following problems: a) (Expanding a remark by S. Lang). Define $E_0 = \overline{\mathbb{Q}}$ Inductively, for $n \geq 1$, define $E_n$ as the algebraic closure of the field generated over $E_{n-1}$ by the numbers $\exp(x)=e^x$, where $x$ ranges over $E_{n-1}$. Let $E$ be the union of $E_n$, $n \geq 0$. Show that Schanuel's Conjecture implies that the numbers $π, \log π, \log \log π, \log \log \log π, \ldots $ are algebraically independent over $E$. b) Try to get a (conjectural) generalization involving the field $L$ defined as follows. Define $L_0 = \overline{\mathbb{Q}}$. Inductively, for $n \geq 1$, define $L_n$ as the algebraic closure of the field generated over $L_{n-1}$ by the numbers $y$, where $y$ ranges over the set of complex numbers such that $e^y\in L_{n-1}$. Let $L$ be the union of $L_n$, $n \geq 0$. We were able to prove that Schanuel's Conjecture implies $E$ and $L$ are linearly disjoint over $\overline{\mathbb{Q}}$.

math.NT↗