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 (or even schemes), because the local ring at a smooth point P of an algebraic curve C (defined over an algebraically closed field) is always a discrete valuation ring. This valuation will endow us with a way to count the order (at the point P) of rational functions (which are natural generalizations for meromorphic functions in the non-complex realm) 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.
Read more about Local Parameter: Introduction, Definition, See Also
Famous quotes containing the word local:
“These native villages are as unchanging as the woman in one of their stories. When she was called before a local justice he asked her age. “I have 45 years.” “But,” said the justice, “you were forty-five when you appeared before me two years ago.” “Señor Judge,” she replied proudly, drawing herself to her full height, “I am not of those who are one thing today and another tomorrow!””
—State of New Mexico, U.S. public relief program (1935-1943)