In his paper 'A simple proof of the Skolem Mahler Lech theorem' Hansel gives a proof of the theorem in the case that the coefficients of the rational series belong to $\mathbb{Q}$. He claims that the case for purely transcendental extensions can be found in Section III and the general case in Section IV - this paper covers only section I and II. Does anyone know where Sections III and IV can be found or how this extension of the proof to these cases can be done?
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What you found is essentially a research announcement; publishing these seems to be common practice in computer science. The full paper was published the next year in Theoretical Computer Science, https://doi.org/10.1016/0304-3975(86)90168-4 (in French).
