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MORPHISMS OF VARIETIES OVER AMPLE FIELDS
Bary-Soroker, Lior,Geyer, Wulf-Dieter,Jarden, Moshe Korean Mathematical Society 2018 대한수학회보 Vol.55 No.4
We strengthen a result of Michiel Kosters by proving the following theorems: (*) Let ${\phi}:W{\rightarrow}V$ be a finite surjective morphism of algebraic varieties over an ample field K. Suppose V has a simple K-rational point a such that $a{\not\in}{\phi}(W(K_{ins}))$. Then, card($V(K){\backslash}{\phi}(W(K))$ = 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 $K_{ins}{\backslash}K$. Then, card(K \ f(K)) = card(K).
Morphisms of varieties over ample fields
Lior Bary-Soroker,Wulf-Dieter Geyer,Moshe Jarden 대한수학회 2018 대한수학회보 Vol.55 No.4
We strengthen a result of Michiel Kosters by proving the following theorems: \noindent $(*)$ 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 $\bfa\notin\phi(W(K_\ins))$. Then, ${\rm card}(V(K)\hefresh \phi(W(K))={\rm card}(K)$. \medskip\noindent $(**)$ 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 $K_{\rm ins}\hefresh K$. Then, ${\rm card}(K\hefresh f(K))={\rm card}(K)$.