Functiones et Approximatio Commentarii Mathematici

Images of polynomial maps on ample fields

Michiel Kosters

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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

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

First available in Project Euclid: 19 September 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

ample field large field polynomial map surjective valuation theory curve


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.

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  • 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:/\!/ 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:/\!/ 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.