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, 888-890 (1939)] 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 𝛾 is a limit ordinal greater than the lengths of all of the coefficients, then the roots all have length less than ωω𝛾.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Hahn, “Über die nichtarchimedischen Grössensysteme, Wien. Ber.,” 116, 601-655 (1907); reprinted in H. Hahn, Collected Works, vol. 1, Springer-Verlag, Wien (1995).
A. Mal’tsev, “On the embedding of group algebras in division algebras,” Dokl. Akad. Nauk SSSR, 60, 1499-1501 (1948).
B. H. Neumann, “On ordered division rings,” Trans. Am. Math. Soc., 66, 202-252 (1949).
S. Maclane, “The universality of formal power series fields,” Bull. Am. Math. Soc., 45, 888-890 (1939).
J. F. Knight, K. Lange, and R. Solomon, “Roots of polynomials in fields of generalized power series,” to appear in Proc. Aspects of Computation, World Scientific.
I. Newton, “Letter to Oldenburg dated Oct. 24, 1676,” in The Correspondence of Isaac Newton, Vol. II: 1676-1687, H. W. Turnbull (ed.), Cambridge Univ. Press, Cambridge (1960), pp. 126/127.
I. Newton, The Method of Fluxions and Infinite Series; with Its Application to the Geometry of Curve-Lines (translated by John Colson), printed by H. Woodfall; and sold by J. Nourse (1736); https://www.loc.gov/item/42048007/.
V. A. Puiseux, “Recherches sur les fonctions algébriques,” J. Math. Pures Appl., 15, 365-480 (1850).
V. A. Puiseux, “Nouvelles recherches sur les fonctions algébriques,” J. Math. Pures Appl., 16, 228-240 (1851).
R. J. Walker, Algebraic Curves, Princeton Math. Ser., 13, Oxford Univ. Press, London (1950).
S. Basu, R. Pollack, and M.-F. Roy, Algorithms in Real Algebraic Geometry, Algorithms Comput. Math., 10, 2nd ed., Springer-Verlag, Berlin (2006).
A. Fornasiero, “Embedding Henselian fields into power series,” J. Alg., 304, No. 1, 112-156 (2006).
F.-V. Kuhlmann, S. Kuhlmann, and J. W. Lee, “Valuation bases for generalized algebraic series fields,” J. Alg., 322, No. 5, 1430-1453 (2009).
J. F. Knight and K. Lange, “Lengths of developments in K((G)),” Sel. Math., New Ser., 25, No. 1 (2019), paper No. 14.
M. H. Mourgues and J. P. Ressayre, “Every real closed field has an integer part,” J. Symb. Log., 58, No. 2, 641-647 (1993).
J. F. Knight and K. Lange, “Complexity of structures associated with real closed fields,” Proc. London Math. Soc. (3),” 107, No. 1, 177-197 (2013).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Algebra i Logika, Vol. 60, No. 2, pp. 145-165, March-April, 2021. Russian https://doi.org/10.33048/alglog.2021.60.203.
J. F. Knight is supported by National Science Foundation, grant No. DMS-1800692.
K. Lange is supported by National Science Foundation (grant No. DMS-1100604), by Simons Foundation Collaboration (grant No. 523234), and by Wellesley College faculty awards.
Rights and permissions
About this article
Cite this article
Knight, J.F., Lange, K. Lengths of Roots of Polynomials in a Hahn Field. Algebra Logic 60, 95–107 (2021). https://doi.org/10.1007/s10469-021-09632-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-021-09632-0