Geodesic
Lengths of roots of polynomials in a Hahn field
Algebra i logika, Tome 60 (2021) no. 2, pp. 145-165

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $K$ be an algebraically closed field of characteristic $0$, and let $G$ be a divisible ordered Abelian group. Maclane [Bull. Am. Math. Soc., 45 (1939), 888—890] showed that the Hahn field $K((G))$ is algebraically closed. Our goal is to bound the lengths of roots of a polynomial $p(x)$ over $K((G))$ in terms of the lengths of its coefficients. The main result of the paper says that if $\gamma$ is a limit ordinal greater than the lengths of all of the coefficients, then the roots all have length less than $\omega^{\omega^\gamma}$.
Keywords: Hahn field, generalized power series, truncation-closed field, length. 
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     author = {J. F. Knight and K. Lange},
     title = {Lengths of roots of polynomials in a {Hahn} field},
     journal = {Algebra i logika},
     pages = {145--165},
     publisher = {mathdoc},
     volume = {60},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2021_60_2_a2/}
}
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J. F. Knight; K. Lange. Lengths of roots of polynomials in a Hahn field. Algebra i logika, Tome 60 (2021) no. 2, pp. 145-165. http://geodesic.mathdoc.fr/item/AL_2021_60_2_a2/