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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 C {\displaystyle \mathbb {C} } , because the local ring at a smooth point P of an algebraic curve C is always a discrete valuation ring. This valuation will endow us with a way to count the order of rational functions having a zero or a pole at P.
Local parameters, as its name indicates, are used mainly to properly count multiplicities in a local way.
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