Morphisms of varieties over ample fields
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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, ${\rm card}(K\hefresh f(K))={\rm card}(K)$.
Abstract
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, ${\rm card}(K\hefresh f(K))={\rm card}(K)$.
참고문헌 (15)
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