Bulletin of the
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Bull. Korean Math. Soc. 2018; 55(4): 1023-1035

Online first article June 11, 2018      Printed July 1, 2018

https://doi.org/10.4134/BKMS.b170483

Copyright © The Korean Mathematical Society.

Morphisms of varieties over ample fields

Lior Bary-Soroker, Wulf-Dieter Geyer, Moshe Jarden

Tel Aviv University, Universitat Erlangen, Tel Aviv University

Abstract

We strengthen a result of Michiel Kosters by proving the following theorems: $(*)$ Let $\phi\colon W\to V$ be a finite surjective morphism of algebraic varieties over an ample field $K$. Suppose $V$ has a simple $K$-rational point ${\bf a}$ such that ${\bf a}\notin\phi(W(K_{ins}))$. Then, ${card}(V(K)\setminus \phi(W(K))={\rm card}(K)$. $(**)$ Let $K$ be an infinite field of positive characteristic and let $f\in K[X]$ be a non-constant monic polynomial. Suppose all zeros of $f$ in $\tilde K$ belong to $K_{\rm ins}\setminus K$. Then, ${\rm card}(K\setminus f(K))={\rm card}(K)$.

Keywords: ample fields, morphisms of varieties

MSC numbers: 12E30

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