We give a deepened version of a lemma of Gabrielov and then use it to prove the following fact: if h ∈ 𝕂[[X]] (𝕂 = ℝ or ℂ) is a root of a non-zero polynomial with convergent power series coefficients, then h is convergent.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap80-0-18,
author = {Rafa\l\ Pierzcha\l a},
title = {On roots of polynomials with power series coefficients},
journal = {Annales Polonici Mathematici},
volume = {81},
year = {2003},
pages = {211-217},
zbl = {1031.13013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap80-0-18}
}
Rafał Pierzchała. On roots of polynomials with power series coefficients. Annales Polonici Mathematici, Tome 81 (2003) pp. 211-217. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap80-0-18/