8 papers · 1 filter
Eremenko's conjecture on higher-order Briot--Bouquet equations
Diego Marques
The Briot--Bouquet theorem describes the meromorphic solutions of autonomous first-order algebraic differential equations. We prove Eremenko's conjecture extending it to all orders…
On Mahler -numbers under rational maps
Diego Marques
In 1906, Maillet proved that every nonconstant rational function over maps Liouville numbers to Liouville numbers. Let be a nonconstant rational function with real…
Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers
Diego Marques
Let denote the set of Liouville numbers. We prove a local arithmetic rigidity theorem for real-analytic functions. If is an open interval and $f:U\…
Smooth Diffeomorphisms and Mahler's Problem on Liouville Numbers
Diego Marques
A classical theorem of Maillet asserts that every nonconstant rational function over maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether a trans…
A Neighboring-Denominator Variant of the Erdős--Mahler Conjecture
Diego Marques
We prove a quantitative neighboring-denominator variant of the Erdős-Mahler conjecture. Let be the convergents of an irrational real number . If is …
Height Rigidity for Entire Functions
Diego Marques
We prove that algebraic values of bounded degree and polynomially bounded height are sparse on rational translates of the graph of a transcendental entire function. More precisely,…
Algebraic values of transcendental power series with geometric coefficient moduli
Diego Marques
Let be a real algebraic number. We construct continuum many power series of radius of convergence exactly one such that every nonzero coefficient $…
Arithmetical structure of sumset intersections
Diego Marques, Melvyn B. Nathanson
The -fold sumset of a set of integers is the set of all sums of not necessarily distinct elements of . Let be a strictly decreasing sequence of…