# analytic algebraic function

Let $k$ be a field, and let $k\{{x}_{1},\mathrm{\dots},{x}_{n}\}$ 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 $\u2102$ 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 $f\in k\{{x}_{1},\mathrm{\dots},{x}_{n}\}$ is said to be *$k$-analytic
algebraic* 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 a
neighbourhood of the origin in ${k}^{n}$.
If $k=\u2102$ 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 $f:U\subset {k}^{n}\to {k}^{m}$ 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 |
---|---|

Canonical name | AnalyticAlgebraicFunction |

Date of creation | 2013-03-22 15:36:05 |

Last modified on | 2013-03-22 15:36:05 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 7 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 14-00 |

Classification | msc 14P20 |

Synonym | $k$-analytic algebraic function |

Synonym | analytic algebraic |

Defines | holomorphic algebraic function |

Defines | real-analytic algebraic function |

Defines | Nash function |

Defines | analytic algebraic mapping |