Functiones et Approximatio Commentarii Mathematici

Images of polynomial maps on ample fields

Michiel Kosters

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]

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

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