A note on a generalization of the Hadamard quotient theorem

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 $f:={\sum }_{n\ge 0}a(n)x^n$ and $g:={\sum }_{n\ge 0}b(n)x^n$ represent, respectively, an algebraic and a rational function over a global field $K$ such that $b(n)\ne 0$ for all $n$ and the coefficients of the power series $h:={\sum }_{n\ge 0}a(n)/b(n)x^n$ 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$andatwhichall$b(n)$areunits,andwith$d := [K(t,f):K(f)]$,thelocalradiiofconvergence$R_v$of$h$attheplaces$v$of$K$satisfy$\sum_v \log^+{R_v^{-1}} \leq δ/12d^4$.

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