Solid $k$-varieties and Henselian fields

Abstract

Let $k$ be a field with a nontrivial absolute value. Define property $( \ast )$ for $k$: Given any polynomial $f(x)$ in $k[x]$ with a simple root $\alpha$ in $k$; then if $g(x)$ is a polynomial near enough to $f(x),g(x)$ has a simple root $\beta$ near $\alpha$. A characterization of fields with property $( \ast )$ is given. If $Y$ is an affine $k$-variety, $Y \subset {\bar k^{(n)}}$, define ${Y_k} = Y \cap {k^{(n)}}$. Define $Y$ to be solid if $I(Y) = I({Y_k})$ in $k[{x_1}, \cdots ,{x_n}]$. If $\pi :Y \to {\bar k^d}$ is a projection induced by Noether normalization, and if $k$ has property $( \ast )$, then $Y$ is a solid $k$-variety if and only if $\pi ({Y_k})$ contains a sphere in ${k^d}$. Using this characterization of solid $k$-varieties and Bertini’s theorem, a dimension theorem is proven.