Regular function
Encyclopedia
In mathematics
, a regular function is a function that is analytic
and single-valued (unique) in a given region. In complex analysis, any complex regular function is known as a holomorphic function
. In algebraic geometry
the term takes up a more specific definition, referring to 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 ∞ at some points of V. Now in some sense F is no worse behaved at those points than anywhere else: ∞ 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 function
s 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 ≥ 1.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a regular function is a function that is analytic
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
and single-valued (unique) in a given region. In complex analysis, any complex regular function is known as a holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
. In algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
the term takes up a more specific definition, referring to an everywhere-defined, polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
function on an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V with values in the field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
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
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
, under pointwise multiplication of functions, isomorphic with the polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
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
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
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
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
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
Liouville's theorem (complex analysis)
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f| ≤ M for all z in C is constant.The theorem is considerably improved by...
in complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
).
In fact taking the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
K(V) of an irreducible algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
V, the functions F in the function field may all be realised as morphisms from V to the projective line
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
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 ∞ at some points of V. Now in some sense F is no worse behaved at those points than anywhere else: ∞ 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 function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s are constantly used in algebraic geometry. For the needs of birational geometry
Birational geometry
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
, more generally, morphisms are replaced with morphisms defined on open dense subsets. This brings fresh phenomena in dimension ≥ 1.