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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Solid $ k$-varieties and Henselian fields

Author: Gustave Efroymson
Journal: Trans. Amer. Math. Soc. 170 (1972), 187-195
MSC: Primary 14G20; Secondary 13J15
MathSciNet review: 0318159
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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.

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Keywords: Henselian field, absolute value, $ k$-variety, real closed field, Noether normalization, dimension theorem
Article copyright: © Copyright 1972 American Mathematical Society