Functiones et Approximatio Commentarii Mathematici

Images of polynomial maps on ample fields

Michiel Kosters

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this article we study the following problem. Let $k$ be an infinite field and let $f \in k[x]$. Consider the evaluation map $f_k: k \to k$. Assume that $f_k$ is not surjective. Is $k \smallsetminus f_k(k)$ infinite? We give a positive answer to this question when $k$ is a perfect ample field. In fact, we prove that $|k \smallsetminus f_k(k)|=|k|$. This conclusion follows from a similar statement about finite morphisms between normal projective curves over perfect ample fields.

Article information

Source
Funct. Approx. Comment. Math., Volume 55, Number 1 (2016), 23-30.

Dates
First available in Project Euclid: 19 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.facm/1474301227

Digital Object Identifier
doi:10.7169/facm/2016.55.1.2

Mathematical Reviews number (MathSciNet)
MR3549010

Zentralblatt MATH identifier
06862550

Subjects
Primary: 12J10: Valued fields
Secondary: 11R58: Arithmetic theory of algebraic function fields [See also 14-XX]

Keywords
ample field large field polynomial map surjective valuation theory curve

Citation

Kosters, Michiel. Images of polynomial maps on ample fields. Funct. Approx. Comment. Math. 55 (2016), no. 1, 23--30. doi:10.7169/facm/2016.55.1.2. https://projecteuclid.org/euclid.facm/1474301227


Export citation

References

  • L. Bary-Soroker, A. Fehm, Open problems in the theory of ample fields, in Geometric and differential Galois theories, vol. 27 of Sémin. Congr. Soc. Math. France, Paris, 2013, 1–11.
  • M. Jarden, Algebraic Patching, Springer Monographs in Mathematics. Springer, Heidelberg, 2011.
  • M. Kosters, The algebraic theory of valued fields, http:/\!/arxiv.org/abs/ 1404.3916, 2014, preprint.
  • Q. Liu, Algebraic geometry and arithmetic curves, vol. 6 of Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2002; translated from the French by Reinie Erné, Oxford Science Publications.
  • F. Pop, Little survey on large fields, http:/\!/www.math.upenn.edu/~ pop/Research/files-Res/LF_6Oct2013.pdf, 2013.
  • H. Stichtenoth, Algebraic function fields and codes, second ed., vol. 254 of Graduate Texts in Mathematics, Springer-Verlag, Berlin, 2009.
  • D.Q. Wan, A $p$-adic lifting lemma and its applications to permutation polynomials, in Finite fields, coding theory, and advances in communications and computing (Las Vegas, NV, 1991), vol. 141 of Lecture Notes in Pure and Appl. Math. Dekker, New York, 1993, pp. 209–216.