Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T18:15:58.085Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Northcott property and other properties related to polynomial mappings

Published online by Cambridge University Press:  27 February 2013

SARA CHECCOLI  and
MARTIN WIDMER
SARA CHECCOLI
Affiliation:
Institute of Mathematics, University of Basel, Rheinsprung 21, CH-4051 Basel, Switzerland. e-mail: [email protected]
MARTIN WIDMER
Affiliation:
Department for Analysis and Computational Number Theory, Graz University of Technology, Steyrergasse 30/II, 8010 Graz, Austria. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that if K/ℚ is a Galois extension of finite exponent and K(d) is the compositum of all extensions of K of degree at most d, then K(d) has the Bogomolov property and the maximal abelian subextension of K(d)/ℚ has the Northcott property.

Moreover, we prove that given any sequence of finite solvable groups {Gm}m there exists a sequence of Galois extensions {Km}m with Gal(Km/ℚ)=Gm such that the compositum of the fields Km has the Northcott property. In particular we provide examples of fields with the Northcott property with uniformly bounded local degrees but not contained in ℚ(d).

We also discuss some problems related to properties introduced by Liardet and Narkiewicz to study polynomial mappings. Using results on the Northcott property and a result by Dvornicich and Zannier we easily deduce answers to some open problems proposed by Narkiewicz.

Keywords

Primary 11G5011R04Secondary 37P05
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 1 - 12
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bombieri, E. and Gubler, W.Heights in Diophantine Geometry (Cambridge University Press, 2006).Google Scholar
[2]Bombieri, E. and Zannier, U.A note on heights in certain infinite extensions of ℚ. Rend. Mat. Acc. Lincei 12 (2001), 5–14.Google Scholar
[3]Bourbaki, N.Algèbre (Chapitre 4–5) (Hermann, Paris, 1959).Google Scholar
[4]Checcoli, S. and Zannier, U.On fields of algebraic numbers with bounded local degrees. C. R. Acad. Sci. Paris 349 (2011), no. 1–2, 11–14.CrossRefGoogle Scholar
[5]Checcoli, S.Fields of algebraic numbers with bounded local degrees and their properties. Trans. Amer. Math. Soc. 365 (2013), no. 4, 2223–2240.CrossRefGoogle Scholar
[6]Doerk, K. and Hawkes, T.Finite Soluble Groups (De Gruyter, Berlin, 1992).CrossRefGoogle Scholar
[7]Dvornicich, R. and Zannier, U.Cyclotomic Diophantine problems. Duke Math. J. 139 (2007), no. 3, 527–554.Google Scholar
[8]Dvornicich, R. and Zannier, U.On the properties of Northcott and Narkiewicz for fields of algebraic numbers. Funct. Approx. 39 (2008), no. 1, 163–173.Google Scholar
[9]Fried, M. D. and Jarden, M.Field Arithmetic, Third Edition (Springer-Verlag, Berlin, Heidelberg, 2008).Google Scholar
[10]Halter–Koch, F. and Narkiewicz, W.Finiteness properties of polynomial mappings. Math. Nachr. 159 (1992), 7–18.CrossRefGoogle Scholar
[11]Kubota, K. K.Note on a conjecture of W. Narkiewicz. J. Number Theory 4 (1972), 181–190.CrossRefGoogle Scholar
[12]Lang, S.Fundamentals of Diophantine Geometry (Springer, 1983).CrossRefGoogle Scholar
[13]Liardet, P.Sur les transformationes polynomiales et rationelles. Sém. Th. Nombres Bordeaux 72 (1971), exp. 29.Google Scholar
[14]Narkiewicz, W.On polynomial transformations. Acta. Arith. 7 (1961/1962), 241–249.CrossRefGoogle Scholar
[15]Narkiewicz, W.Problem 415. Colloq. Math. 10 (1963), no. 1, 186.Google Scholar
[16]Narkiewicz, W.Remark on rational transformations. Colloq. Math. 10 (1963), 139–142.CrossRefGoogle Scholar
[17]Narkiewicz, W.Some unsolved problems. Colloque de Théorie des Nombres (Univ. Bordeaux, Bordeaux, 1969), pp. 159–164. Bull. Soc. Math. France, Mem. No. 25, Soc. Math. France (Paris, 1971).Google Scholar
[18]Narkiewicz, W.Polynomial Mappings. Lecture Notes in Mathematics 1600, Springer, 1995.CrossRefGoogle Scholar
[19]Northcott, D. G.An inequality in the theory of arithmetic on algebraic varieties. Proc. Cambridge Phil. Soc. 45 (1949), 502–509 and 510–518.CrossRefGoogle Scholar
[20]Widmer, M.On certain infinite extensions of the rationals with Northcott property. Monatsh. Math. 162 (2011), no. 3, 341–353.CrossRefGoogle Scholar
[21]Zannier, U.Lecture Notes on Diophantine Analysis (with an Appendix by F. Amoroso). Appunti. Scuola Normale Superiore di Pisa (Nuova serie), vol.8 (Edizioni Della Normale, Pisa, 2009).Google Scholar