Local parameter

In the geometry of algebraic curves, a local parameter for a curve C at a (smooth) point P is a rational function that has a simple zero at P. This has an algebraic ressemblance with the concept of Discrete_valuation_ring#Uniformizing_parameter (or just uniformizer) found in the context of discrete valuation rings in commutative algebra; a uniformizing parameter for the DVR (R, m) is just a generator of the maximal ideal m. The link comes from the fact that the local ring at a smooth point of an algebraic curve is always a discrete valuation ring. If this is the case of $P\in C$, then the maximal ideal of ${\mathcal {O}}_{C,P}$ consists of all those regular functions defined around P which vanishes at P, and finally a local parameter at P will be a uniformizing parameter for the DVR (${\mathcal {O}}_{C,P}$, $m_{P}$).

Definition

Let (R, m) be a discrete valuation ring for the field K. An element $t\in K$ is

See also

• Discrete valuation ring