A note on a generalization of the Hadamard quotient theorem

TL;DR

This paper generalizes the Hadamard quotient theorem, proposing a conjecture about algebraic and rational functions over global fields, and proves it under specific conditions related to poles and valuations.

Contribution

It introduces a new conjecture extending the Hadamard quotient theorem and proves it in cases involving pole behavior and valuation conditions.

Findings

Proved the conjecture when g has a simple pole of maximal absolute value.

Established the conjecture under conditions of poles being simple and certain valuation density.

Demonstrated algebraicity of h under specified pole and valuation constraints.

Abstract

We consider a generalization of the "Hadamard quotient theorem" of Pourchet and van der Poorten. A particular case of our conjecture states that if $f := \sum_{n \geq 0} a(n)x^n$ and $g := \sum_{n \geq 0} b(n)x^n$ represent, respectively, an algebraic and a rational function over a global field $K$ such that $b(n) \neq 0$ for all $n$ and the coefficients of the power series $h := \sum_{n \geq 0} a(n)/b(n)x^n$ are contained in a finitely generated ring, then $h$ is algebraic. We prove this conjecture if either (i) $g$ has a simple pole of a strictly maximal absolute value at some place; or (ii) or poles of $g$ are simple, there is a positive density $\delta > 0$ of places which split completely in the field generated by the poles of g$ and at which all $b(n)$ are units, and with $d := [K(t,f):K(f)]$, the local radii of convergence $R_v$ of $h$ at the places $v$ of $K$ satisfy $\sum_v \log^+{R_v^{-1}} \leq \delta/12d^4$.