Local parameter: Difference between revisions

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In the geometry of [[algebraic curves]], a '''local parameter''' for a curve ''C'' at a (smooth) point ''P'' is a rational function that has a simple zero at ''P''.

In [[commutative algebra]], a '''local parameter''' for the [[discrete valuation ring]] (''R,m'') is an element which generates the maximal ideal ''m''. They can be characterized either by means of divisibility (as an [[irreducible]] element of ''R'') or by using the [[discrete valuation]] defined in the [[field of fractions]] of ''R''.

This has an algebraic ressemblance with the concept of [[Discrete_valuation_ring#Uniformizing_parameter]] (or just '''uniformizer''') found in the context of [[discrete valuation ring]]s in [[commutative algebra]]; a uniformizing parameter for the DVR (''R, m'') is just a generator of the maximal ideal ''m''.

Terms such as '''uniformizing parameter''' or just '''uniformizer''' are also found in the litterature <ref>J. H. Silverman (1986). ''The arithmetic of elliptic curves''. Springer. pp. 22.</ref>.

The link comes from the fact that the local ring at a smooth point of an algebraic curve is always a discrete valuation ring<ref>J. H. Silverman (1986). ''The arithmetic of elliptic curves''. Springer. p. 21</ref>. If this is the case of <math>P\in C</math>, then the maximal ideal of <math>\mathcal{O}_{C,P}</math> consists of all those regular functions defined around ''P'' which vanishes at ''P'', and finally a local parameter at ''P'' will be a uniformizing parameter for the DVR (<math>\mathcal{O}_{C,P}</math>, <math>m_P</math>).

==Definition==

Let (''R, m'') be a discrete valuation ring for the field ''K''. An element <math>t\in K</math> is

==See also==

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[[Category:Commutative algebra]]

[[Category:Algebraic geometry]]