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In the geometry of complex [[algebraic curve]]s, 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 <math>\mathbb{C}</math> (or even [[scheme (mathematics)|scheme]]s),
Local parameters, as its name indicates, are used mainly to properly ''count multiplicities'' in a local way.
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Now, the valuation function on <math>\mathcal{O}_{C,P}</math> is given by
:<math>\operatorname{ord}_P(f)=\max\{d=0,1,2,\ldots: f\in m^d_P\};</math>
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 <math>\mathcal{O}_{C,P}</math>. Hence the idea of ''having a simple zero at a point P'' is now complete: it will be a rational function <math>f\in K(C)</math> such that its germ falls into <math>m_P^d</math>, with ''d'' at most 1.
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==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 <math>P\in C</math> is defined as
<math>\operatorname{ord}_P(f/g)=\operatorname{ord}_P(f)-\operatorname{ord}_P(g)</math>, where <math>\operatorname{ord}_P</math> is the usual valuation on the local ring (<math>\mathcal{O}_{C,P}</math>, <math>m_P</math>). A '''local parameter''' for ''C'' at ''P'' is a function <math>t\in K(C)</math> such that <math>\operatorname{ord}_P(t)=1</math>.
==See also==
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