## Functiones et Approximatio Commentarii Mathematici

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

### Images of polynomial maps on ample fields

#### 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