TY - JOUR

T1 - Semi-galois categories II

T2 - An arithmetic analogue of Christol's theorem

AU - Uramoto, Takeo

N1 - Funding Information:
We are grateful to Isamu Iwanari, who told us the papers[13,18], and also to Go Yamashita, who continuously encouraged us. We are also indebted to the anonymous reviewer, who provided us several corrections and suggestions, which improved the original version of this paper. This work was supported by JSPS KAKENHI Grant number JP16K21115.
Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2018/8/15

Y1 - 2018/8/15

N2 - In connection with our previous work on semi-galois categories [1,2], this paper proves an arithmetic analogue of Christol's theorem concerning an automata-theoretic characterization of when a formal power series ξ=∑ξntn∈Fq[[t]] over finite field Fq is algebraic over the polynomial ring Fq[t]. There are by now several variants of Christol's theorem, all of which are concerned with rings of positive characteristic. This paper provides an arithmetic (or F1-) variant of Christol's theorem in the sense that it replaces the polynomial ring Fq[t] with the ring OK of integers of a number field K and the ring Fq[[t]] of formal power series with the ring of Witt vectors. We also study some related problems.

AB - In connection with our previous work on semi-galois categories [1,2], this paper proves an arithmetic analogue of Christol's theorem concerning an automata-theoretic characterization of when a formal power series ξ=∑ξntn∈Fq[[t]] over finite field Fq is algebraic over the polynomial ring Fq[t]. There are by now several variants of Christol's theorem, all of which are concerned with rings of positive characteristic. This paper provides an arithmetic (or F1-) variant of Christol's theorem in the sense that it replaces the polynomial ring Fq[t] with the ring OK of integers of a number field K and the ring Fq[[t]] of formal power series with the ring of Witt vectors. We also study some related problems.

UR - http://www.scopus.com/inward/record.url?scp=85047069376&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047069376&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2018.04.033

DO - 10.1016/j.jalgebra.2018.04.033

M3 - Article

AN - SCOPUS:85047069376

VL - 508

SP - 539

EP - 568

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -