# analytic algebraic function

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.

###### Definition.

A function $f\in k\{x_{1},\ldots,x_{n}\}$ is said to be $k$-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 holomorphic algebraic and if $k=\mathbb{R}$ then $f$ is said to be real-analytic algebraic or a Nash function.

The same definition applies near any other point other then the origin by just translation.

###### Definition.

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.

## References

• 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
Title analytic algebraic function AnalyticAlgebraicFunction 2013-03-22 15:36:05 2013-03-22 15:36:05 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 14-00 msc 14P20 $k$-analytic algebraic function analytic algebraic holomorphic algebraic function real-analytic algebraic function Nash function analytic algebraic mapping