Note on a differential algebra bound

Léo Jimenez

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arXiv · 2412.06034

Note on a differential algebra bound

Léo Jimenez

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Abstract

In a recent article, Freitag, Moosa and the author showed that in differentially closed fields of characteristic zero, if two types are nonorthogonal, then their n+3 and m+3 Morley powers are not weakly orthogonal, where n and m are their respective Lascar ranks. In this short note, we prove that the bound is tight: there are such types with weakly orthogonal n+2 and m+2 Morley powers. The types in question were constructed by Freitag and Moosa as examples of types with degree of nonminimality 2. As interesting as our result are our methods: we rely mostly on Galois theory and some descent argument for types, combined with the failure of the inverse Galois problem over constant parameters.

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Léo Jimenez. 2024-12-08. Note on a differential algebra bound. https://arxiv.org/abs/2412.06034

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