AnalyticAlgebraicFunction 2005-12-05 20:51:43 2005-12-09 12:52:22 Definition holomorphic algebraic function real-analytic algebraic function Nash function analytic algebraic mapping % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{theorem} \newtheorem*{thm}{Theorem} \newtheorem*{lemma}{Lemma} \newtheorem*{conj}{Conjecture} \newtheorem*{cor}{Corollary} \newtheorem*{example}{Example} \newtheorem*{prop}{Proposition} \theoremstyle{definition} \newtheorem*{defn}{Definition} \theoremstyle{remark} \newtheorem*{rmk}{Remark} Let $k$ be a field, and let $k\{x_1,\ldots,x_n\}$ be the ring of convergent power series in $n$ variables. An element in this ring can be thought of as a function defined in a neighbourhood of the origin in $k^n$ to $k$. The most common cases for $k$ are $\mathbb{C}$ or $\mathbb{R}$, where the convergence is with respect to the standard euclidean metric. These definitions can also be generalized to other fields. \begin{defn} A function $f \in k\{x_1,\ldots,x_n\}$ is said to be \emph{$k$-analytic algebraic} if there exists a nontrivial polynomial $p \in k[x_1,\ldots,x_n,y]$ such that $p(x,f(x)) \equiv 0$ for all $x$ in a neighbourhood of the origin in $k^n$. If $k=\mathbb{C}$ then $f$ is said to be \emph{holomorphic algebraic} and if $k=\mathbb{R}$ then $f$ is said to be \emph{real-analytic algebraic} or a \emph{Nash function}. \end{defn} The same definition applies near any other point other then the origin by just translation. \begin{defn} A mapping $f \colon U \subset k^n \to k^m$ where $U$ is a neighbourhood of the origin is said to be $k$-analytic algebraic if each component function is analytic algebraic. \end{defn} \begin{thebibliography}{9} \bibitem{ber:submanifold} M.\@ Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. {\em \PMlinkescapetext{Real Submanifolds in Complex Space and Their Mappings}}, Princeton University Press, Princeton, New Jersey, 1999. \end{thebibliography}