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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: if $f$ is meromorphic in $\mathbb{C}$ and $P(f,f^{(n)})=0$ for a nonzero polynomial $P$, then $f$ is rational, elliptic, a rational functi...
We prove a quantitative neighboring-denominator variant of the Erd\H{o}s-Mahler conjecture. Let $p_n/q_n$ be the convergents of an irrational real number $\xi$. If $p_nq_nq_{n+1}$ is $S$-smooth for infinitely many $n$, where $S$ is a fixed finite set of primes, then there exists an effectively computable constant $c=c(...
A classical theorem of Maillet states that every nonconstant rational function with rational coefficients maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether there exists a transcendental entire function with the same property. We resolve this question in the negative. More generally, we prove th...
A classical theorem of Maillet states that every nonconstant rational function with rational coefficients maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether there exists a transcendental entire function with the same property. We resolve this question in the negative. More generally, we prove th...
A classical theorem of Maillet asserts that every nonconstant rational function over $\mathbb{Q}$ maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether a transcendental entire function can have the same property. We prove a strong smooth counterpart: writing $\mathscr{L}$ for the set of Liouville n...
In 1939, Erd\H{o}s and Mahler conjectured that an irrational real number $\xi$ must be a Liouville number whenever $P(p_nq_n)$ is bounded for infinitely many convergents $p_n/q_n$, where $P(N)$ denotes the largest prime factor of a nonzero integer $N$. In this note, we prove a neighbouring-denominator case of their con...
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