In the geometry of complex algebraic curves, a local parameter for a curve C at a smooth point P is just a meromorphic function on C that has a simple zero at P. This concept can be generalized to curves defined over fields other than
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Local parameters, as its name indicates, are used mainly to properly count multiplicities in a local way.
Introduction
When C is a complex algebraic curve, we know how to count multiplicities of zeroes and poles of meromorphic functions defined on it. However, when discussing curves defined over fields other than
Now, the valuation function on
this valuation can naturally be extended to K(C) (which is the field of rational functions of C) because it is the field of fractions of
This has an algebraic resemblance with the concept of a 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 a local parameter at P will be a uniformizing parameter for the DVR (
Definition
Let C be an algebraic curve defined over an algebraically closed field K, and let K(C) be the field of rational functions of C. The valuation on K(C) corresponding to a smooth point