Abstract
This article is the second installment in a series on the Berkovich ramification locus for nonconstant rational functions \(\varphi \in k(z)\). Here we show the ramification locus is contained in a strong tubular neighborhood of finite radius around the connected hull of the critical points of \(\varphi \) if and only if \(\varphi \) is tamely ramified. When the ground field \(k\) has characteristic zero, this bound may be chosen to depend only on the residue characteristic. We give two applications to classical non-Archimedean analysis, including a new version of Rolle’s theorem for rational functions.
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Notes
Simply calling such rational functions “tame” is appealing, but Trucco has already reserved this term for a function whose ramification locus is contained in a finitely branched subtree of \({\mathbf{P }^1}\) [7].
The visible ramification is related to Baldassarri’s notion of generic radius of convergence [3, Def. 2.0.9] via the formula \(\tau _\varphi (x) = -\log _{q_k} R_{D \subset {\mathbf{P }^1}} (\Sigma )\). Here \(\Sigma \) is the first order system of linear differential equations associated to the connection \((\varphi _*\mathcal{O }_{{\mathbf{P }^1}}, \nabla )\) alluded to in the discussion after Theorem D.
We could rephrase this condition by saying that \(\varphi \) has wildly ramified reduction at \(x\).
References
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Acknowledgments
This work was made possible by a National Science Foundation Postdoctoral Research Fellowship. Many thanks go to Bob Rumely for his enthusiasm during the discovery of these results, and for his insightful comments on an earlier draft of this manuscript. The anonymous referee also deserves acknowledgement for several suggested improvements to the exposition.
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Faber, X. Topology and geometry of the Berkovich ramification locus for rational functions, II. Math. Ann. 356, 819–844 (2013). https://doi.org/10.1007/s00208-012-0872-3
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DOI: https://doi.org/10.1007/s00208-012-0872-3