analytic algebraic function

Let $k$ be a field, and let $k\{x_1,\mathrm{\dots },x_n\}$ be the ring of convergentpower series in $n$ variables. An element in this ring can be thought of asa function defined in a neighbourhood of the origin in $k^n$ to $k$. The most common cases for $k$ are $ℂ$ or $ℝ$, where the convergence is with respect to the standard euclidean metric. These definitions can also be generalized to other fields.

Definition.

A function $f\in k\{x_1,\mathrm{\dots },x_n\}$ is said to be $k$-analyticalgebraic if there exists a nontrivial polynomial $p\in k[x_1,\mathrm{\dots },x_n,y]$ such that $p(x,f(x))\equiv 0$ for all $x$ in aneighbourhood of the origin in $k^n$.If $k=ℂ$ then $f$ is said to be holomorphic algebraic and if$k=ℝ$ then $f$ is said to be real-analytic algebraic or aNash function.

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

Definition.

A mapping $f:U\subset k^n\to k^m$ where $U$ is a neighbourhood of the origin is said to be $k$-analytic algebraic if each component function is analytic algebraic.