Content deleted Content added
No edit summary Tags: Reverted extraneous markup |
m Reverted edit by 5.195.71.25 (talk) to last version by D.Lazard |
||
Line 4:
{{More footnotes needed|date=September 2015}}
In [[mathematics]], a '''rational function''' is any [[function (mathematics)|function]] that can be defined by a '''rational fraction''', which is an [[algebraic fraction]] such that both the [[numerator]] and the [[denominator]] are [[polynomial]]s. The [[coefficient]]s of the polynomials need not be [[rational number]]s; they may be taken in any [[field (mathematics)|field]] {{mvar|K}}. In this case, one speaks of a rational function and a rational fraction ''over {{mvar|K}}''. The values of the [[variable (mathematics)|variable]]s may be taken in any field {{mvar|L}} containing {{mvar|K}}. Then the [[___domain (function)|___domain]] of the function is the set of the values of the variables for which the denominator is not zero, and the [[codomain]] is {{mvar|L}}.
The set of rational functions over a field {{mvar|K}} is a field, the [[field of fractions]] of the [[ring (mathematics)|ring]] of the [[polynomial function]]s over {{mvar|K}}.
==Definitions==
A function <math>f</math> is called a rational function if it can be written in the form
Line 145 ⟶ 143:
Like [[Polynomial ring#The polynomial ring in several variables|polynomials]], rational expressions can also be generalized to ''n'' indeterminates ''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>, by taking the field of fractions of ''F''[''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>], which is denoted by ''F''(''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>).
An extended version of the abstract idea of rational function is used in algebraic geometry. There the [[function field of an algebraic variety]] ''V'' is formed as the field of fractions of the [[coordinate ring]] of ''V'' (more accurately said, of a [[Zariski topology|Zariski]]-[[dense subset|dense]] affine open set in ''V''). Its elements ''f'' are considered as regular functions in the sense of algebraic geometry on non-empty open sets ''U'', and also may be seen as morphisms to the [[projective line]].
| |||