In the geometry of complex
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
s, a local parameter for a curve ''C'' at a smooth point ''P'' is a
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
on ''C'' that has a
simple zero
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root.
The notion of multip ...
at ''P.'' This concept can be generalized to curves defined over fields other than
(or
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
s), because the
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
at a smooth point ''P'' of an algebraic curve ''C'' (defined over an
algebraically closed field) is always a
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R'' i ...
. This valuation will show 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.
Introduction
If ''C'' is a complex algebraic curve, count multiplicities of zeroes and poles of meromorphic functions defined on it.
[R. Miranda (1995). ''Algebraic curves and Riemann surfaces''. American Mathematical Society. p. 26] However, when discussing curves defined over fields other than
, if there is no access to the power of the complex analysis, a replacement must be found in order to define multiplicities of zeroes and poles of rational functions defined on such curves. In this last case, say that the germ of the regular function
vanishes at
if
. This is in complete analogy with the complex case, in which the maximal ideal of the local ring at a point ''P'' is actually conformed by the germs of holomorphic functions vanishing at ''P''.
The valuation function on
is given by
:
This valuation can naturally be extended to ''K''(''C'') (which is the field of
rational functions
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
of C) because it is the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of
. Hence, the idea of ''having a simple zero at a point P'' is now complete: it will be a rational function
such that its germ falls into
, with ''d'' at most 1.
This has an algebraic resemblance with the concept of a
uniformizing parameter (or just uniformizer) found in the context of
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R'' i ...
s in
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
; 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 (
,
), whence the name.
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
is defined as
, where
is the usual valuation on the local ring (
,
). A local parameter for ''C'' at ''P'' is a function
such that
.
References
Commutative algebra
Algebraic geometry