Sen’s theorem on iteration of power series
| Autor: | Jonathan Lubin |
|---|---|
| Rok vydání: | 1995 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 123:63-66 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-1995-1215030-8 |
| Popis: | In the group of continuous automorphisms of the field of Laurent series in one variable over a field of characteristic p > 0 , Sen's Theorem de- scribes the rapidity of convergence to the identity of the sequence formed by taking successive pth powers of a given element. This paper gives a short proof of Sen's Theorem, utilizing the methods of p-adic analysis in characteristic zero. The theorem in question appears in Sen's thesis (Sen), and is concerned with the group .%_ i (k) of formal power series in one variable with no constant term, and first degree coefficient equal to 1, over a field k of characteristic p > 0, where the group law is composition of series. If we call the variable t, this group is a closed subset of the discrete valuation ring K((t)), namely, the set of all u(t) for which u = t (mod?2). For the (Z)-adic filtration of group %,x, the successive quotients are isomorphic to the additive group k . Thus if we call u°" the «-fold iteration of u with itself, any time that u = t (mod/"), we necessarily have u°p = t (modz"+1). Sen's Theorem says much more and is best stated in terms of the additive valuation v of K((t)) normalized so that v(t) = 1. According to the theorem, if u°p" is not the identity, then v(u°p"(t)-t) = v(uop"\t)-t) (rnodp"). Let us abbreviate notation by setting iu(n) = i(n) := v(u°p"(t) - t). Sen's Theorem now says that if u°p" is not the identity, then i(n) = i(n - 1) (mod/?"). As examples of this phenomenon, we have, in characteristic 2, if u(t) = t + t4, then iu(n) = 22"+l ; if u(t) = t + t4 + /5, then iu(n) = 2n+2 ; and if u(t) = t + t3, then iu(n) = 1 + 2"+1 . It is easy to see why the first two of these facts hold, since each of t + t4 and t + Z4 +15 is an endomorphism of a formal group, and since in a formal-group endomorphism ring, the multiplication comes from substitution of power series. The first-mentioned series is an endomorphism of the additive formal group sf(x,y) = x + y, whose endomorphism ring has characteristic 2, and in that ring t + tA is g = 1 + tj> |
| Databáze: | OpenAIRE |
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