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.
@article{AL_2021_60_2_a2,
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/}
}
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/