Factorization in generalized power series

Berarducci, Alessandro

doi:10.1090/S0002-9947-99-02172-8

Factorization in generalized power series
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by Alessandro Berarducci

Trans. Amer. Math. Soc. 352 (2000), 553-577

DOI: https://doi.org/10.1090/S0002-9947-99-02172-8

Published electronically: May 20, 1999

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Abstract:

The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group $\mathbf {G}$ is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring $\mathbf {R}(( \mathbf {G}^{\leq 0}))$ consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): $\sum _n t^{-1/n}+1$. Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If $\mathbf {G}= ( \mathbf {R}, +, 0, \leq )$ we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either $\omega$ or of the form $\omega ^{\omega ^\alpha }$ and the series is not divisible by any monomial, then it is irreducible. To handle the general case we use a suggestion of M.-H. Mourgues, based on an idea of Gonshor, which allows us to reduce to the special case $\mathbf {G}=\mathbf {R}$. In the final part of the paper we study the irreducibility of series with finite support.

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