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On the algebraicity of generalized power series

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Abstract

Let K be an algebraically closed field of characteristic p. We exhibit a counterexample against a theorem asserted in one of our earlier papers, which claims to characterize the integral closure of K((t)) within the field of Hahn–Mal’cev–Neumann generalized power series. We then give a corrected characterization, generalizing our earlier description in terms of finite automata in the case where K is the algebraic closure of a finite field. We also characterize the integral closure of K(t), thus generalizing a well-known theorem of Christol and suggesting a possible framework for computing in this integral closure. We recover various corollaries on the structure of algebraic generalized power series; one of these is an extension of Derksen’s theorem on the zero sets of linear recurrent sequences in characteristic p.

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Authors and Affiliations

  1. Department of Mathematics, University of California, 9500 Gilman Drive #0112, La Jolla, San Diego, CA, 92093, USA

    Kiran S. Kedlaya

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  1. Kiran S. Kedlaya
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Correspondence to Kiran S. Kedlaya.

Additional information

Thanks to Rodrigo Salomão and Reillon Santos for initiating the discussion about Kedlaya (2001a) that led to the identification of Example 2.6. Thanks also to Jason Bell, Harm Derksen, Anna Medvedovsky, and David Speyer for additional discussions. The author was supported by NSF (Grants DMS-1101343, DMS-1501214) and UCSD (Stefan E. Warschawski chair).

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Kedlaya, K.S. On the algebraicity of generalized power series. Beitr Algebra Geom 58, 499–527 (2017). https://doi.org/10.1007/s13366-016-0325-3

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