# Regular function

﻿
Regular function

:"In complex analysis, see holomorphic function."In mathematics, a regular function in the sense of algebraic geometry is an everywhere-defined, polynomial function on an algebraic variety "V" with values in the field "K" over which "V" is defined.

For example, if "V" is the affine line over "K", the regular functions on "V" make up a commutative ring, under pointwise multiplication of functions, isomorphic with the polynomial ring in one variable over "K". In other words, the regular functions are just polynomials in some natural parameter on the affine line.

More generally, for any affine variety "V", the regular functions make up the coordinate ring of "V", often written "K" ["V"] . This can be expressed in other ways. A regular function is the same as a morphism to the affine line, or in the language of scheme theory a global section of the structure sheaf.

The reason for looking at regular functions becomes more apparent when one allows "V" to be a projective variety. Then regular functions on "V" become rare. For example morphisms from a projective space to the affine line must be constant: regular functions on a projective space are constant functions. The same is true for any connected projective variety (this can be viewed as an algebraic analogue of Liouville's theorem in complex analysis).

In fact taking the function field "K"("V") of an irreducible algebraic curve "V", the functions "F" in the function field may all be realised as morphisms from "V" to the projective line over "K". The image will either be a single point, or the whole projective line (this is a consequence of the completeness of projective varieties). That is, unless "F" is actually constant, we have to attribute to "F" the value &infin; at some points of "V". Now in some sense "F" is no worse behaved at those points than anywhere else: &infin; is just the chosen point at infinity on the projective line, and by using a Möbius transformation we can move it anywhere we wish. But it is in some way inadequate to the needs of geometry to use only the affine line as target for functions, since we shall end up only with constants.

For those reasons, the larger class of rational functions are constantly used in algebraic geometry. For the needs of birational geometry, more generally, morphisms are replaced with morphisms defined on open dense subsets. This brings fresh phenomena in dimension &ge; 1.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• regular function — noun a meromorphic function …   Wiktionary

• Function field (scheme theory) — In algebraic geometry, the function field KX of a scheme X is a generalization of the notion of a sheaf of rational functions on a variety. In the case of varieties, such a sheaf associates to each open set U the ring of all rational functions on …   Wikipedia

• Regular conditional probability — is a concept that has developed to overcome certain difficulties in formally defining conditional probabilities for continuous probability distributions. It is defined as an alternative probability measure conditioned on a particular value of a… …   Wikipedia

• Function — Func tion (f[u^][ng]k sh[u^]n), Functionate Func tion*ate, v. i. To execute or perform a function; to transact one s regular or appointed business. [1913 Webster] …   The Collaborative International Dictionary of English

• Regular singular point — In mathematics, in the theory of ordinary differential equations in the complex plane , the points of are classified into ordinary points, at which the equation s coefficients are analytic functions, and singular points, at which some coefficient …   Wikipedia

• Regular number — The numbers that evenly divide the powers of 60 arise in several areas of mathematics and its applications, and have different names coming from these different areas of study. As an example, 602 = 3600 = 48 times; 75, so both 48 and 75 are… …   Wikipedia

• Regular solution — A regular solution is a solution that diverges from the behavior of an ideal solution only moderately [Simon McQuarrie Physical Chemistry: A molecular approach] .More precisely it can be described by Raoult s law modified with a Margules function …   Wikipedia

• Regular economy — A regular economy is an economy characterized by an excess demand function which has the property that its slope at any equilibrium price vector is non zero. In other words, if we graph the excess demand function against prices, then the excess… …   Wikipedia

• regular — I. adjective Etymology: Middle English reguler, from Anglo French, from Late Latin regularis regular, from Latin, of a bar, from regula rule more at rule Date: 14th century 1. belonging to a religious order 2. a. formed, built, arranged, or… …   New Collegiate Dictionary

• Regular — The term regular can mean normal or obeying rules. Regular may refer to:In organizations: * Regular Army for usage in the U.S. Army * Regular clergy, members of a religious order subject to a rule of life * Regular Force for usage in the Canadian …   Wikipedia