On the algebraicity of generalized power series

TL;DR

This paper corrects a previous theorem on the algebraic structure of generalized power series over algebraically closed fields of characteristic p, providing new characterizations and extending known results.

Contribution

It offers a corrected characterization of the integral closure of K((t)) and K(t), generalizes automata-based descriptions, and extends results on algebraic power series in characteristic p.

Findings

Counterexample to previous theorem on integral closure

New characterization of integral closure of K((t)) and K(t)

Extension of Derksen's theorem on zero sets of linear recurrent sequences

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 $p$.