Algebraic function

Algebraic function

In mathematics, an algebraic function is informally a function which satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable "x" is a solution "y" for an equation

: a_n(x)y^n+a_{n-1}(x)y^{n-1}+cdots+a_0(x)=0

where the coefficients "a""i"("x") are polynomial functions of "x". A function which is not algebraic is called a transcendental function.

In more precise terms, an algebraic function may not be a function at all, at least not in the conventional sense. Consider for example the equation of a circle:


This determines "y", except only up to an overall sign:

:y=pm sqrt{1-x^2}.,

However, both branches are thought of as belonging to the "function" determined by the polynomial equation.

An algebraic function in "n" variables is similarly defined as a function "y" which solves a polynomial equation in "n" + 1 variables:


It is normally assumed that "p" should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem.

Formally, an algebraic function in "n" variables over the field "K" is an element of the algebraic closure of the field of rational functions "K"("x"1,...,"x""n"). In order to understand algebraic functions as functions, it becomes necessary to introduce ideas relating to Riemann surfaces or more generally algebraic varieties, and sheaf theory.

Algebraic functions in one variable

Introduction and overview

The informal definition of an algebraic function provides a number of clues about the properties of algebraic functions. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual algebraic operations: addition, multiplication, division, and taking an "n"th root. Of course, this is somewhat of an oversimplification; because of casus irreducibilis (and more generally the fundamental theorem of Galois theory), algebraic functions need not be expressible by radicals.

First, note that any polynomial is an algebraic function, since polynomials are simply the solutions for "y" of the equation

: y-p(x) = 0.,

More generally, any rational function is algebraic, being the solution of

:q(x)y-p(x)=0 implies y=frac{p(x)}{q(x)}.

Moreover, the "n"th root of any polynomial is an algebraic function, solving the equation

:y^n-p(x)=0 implies y=sqrt [n] {p(x)}.

Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that "y" is a solution of


for each value of "x", then "x" is also a solution of this equation for each value of "y". Indeed, interchanging the roles of "x" and "y" and gathering terms,


Writing "x" as a function of "y" gives the inverse function, also an algebraic function.

However, not every function has an inverse. For example, "y" = "x"2 fails the horizontal line test: it fails to be one-to-one. The inverse is the algebraic "function" x=pmsqrt{y}. In this sense, algebraic functions are often not true functions at all, but instead are multiple valued functions.

Another way to understand this, which will become important later in the article, is that an algebraic function is the graph of an algebraic curve.

The role of complex numbers

From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. First of all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field. Hence any polynomial relation

: "p"("y", "x") = 0

is guaranteed to have at least one solution (and in general a number of solutions not exceeding the degree of "p" in "x") for "y" at each point "x", provided we allow "y" to assume complex as well as real values. Thus, problems to do with the domain of an algebraic function can safely be minimized.

Furthermore, even if one is ultimately interested in real algebraic functions, there may be no adequate means to express the function in a simple manner without resorting to complex numbers (see casus irreducibilis). For example, consider the algebraic function determined by the equation


Using the cubic formula, one solution is (the red curve in the accompanying image)

:y=-frac{(1+isqrt{3})x}{2^{2/3}sqrt [3] {729-108x^3-frac{(1-isqrt{3})sqrt [3] {-27+sqrt{729-108x^3}{6sqrt [3] {2.

There is no way to express this function in terms of real numbers only, even though the resulting function is real-valued on the domain of the graph shown.

On a more significant theoretical level, using complex numbers allow one to use the powerful techniques of complex analysis to discuss algebraic functions. In particular, the argument principle can be used to show that any algebraic function is in fact an analytic function, at least in the multiple-valued sense.

Formally, let "p"("x", "y") be a complex polynomial in the complex variables "x" and "y". Suppose that "x"0C is such that the polynomial "p"("x"0,"y") of "y" has "n" distinct zeros. We shall show that the algebraic function is analytic in a neighborhood of "x"0. Choose a system of "n" non-overlapping discs Δ"i" containing each of these zeros. Then by the argument principle

:frac{1}{2pi i}oint_{partialDelta_i} frac{p_y(x_0,y)}{p(x_0,y)},dy = 1.

By continuity, this also holds for all "x" in a neighborhood of "x"0. In particular, "p"("x","y") has only one root in Δ"i", given by the residue theorem:

:f_i(x) = frac{1}{2pi i}oint_{partialDelta_i} yfrac{p_y(x,y)}{p(x,y)},dy

which is an analytic function.


Note that the foregoing proof of analyticity derived an expression for a system of "n" different function elements "f""i"("x"), provided that "x" is not a critical point of "p"("x", "y"). A "critical point" is a point where the number of distinct zeros is smaller than the degree of "p", and this occurs only where the highest degree term of "p" vanishes, and where the discriminant vanishes. Hence there are only finitely many such points "c"1, ..., "c""m".

A close analysis of the properties of the function elements "f""i" near the critical points can be used to show that the monodromy cover is ramified over the critical points (and possibly the point at infinity). Thus the entire function associated to the "f""i" has at worst algebraic poles and ordinary algebraic branchings over the critical points.

Note that, away from the critical points, we have

:p(x,y) = a_n(x)(y-f_1(x))(y-f_2(x))cdots(y-f_n(x))

since the "f""i" are by definition the distinct zeros of "p". The monodromy group acts by permuting the factors, and thus forms the monodromy representation of the Galois group of "p". (The monodromy action on the universal covering space is related but different notion in the theory of Riemann surfaces.)

Algebraic functions in several variables

Formal definitions


Connection with algebraic geometry


Algebraic functions have a long history. The ideas surrounding algebraic functions go back as far at least as René Descartes. The first discussion of algebraic functions appears to have been in Edward Waring's 1794 "An Essay on the Principles of Human Knowledge" in which he writes::"let a quantity denoting the ordinate, be an algebraic function of the abscissa x, by the common methods of division and extraction of roots, reduce it into an infinite series ascending or descending according to the dimensions of x, and then find the integral of each of the resulting terms."



Wikimedia Foundation. 2010.

Поможем написать реферат

Look at other dictionaries:

  • Algebraic function — Function Func tion, n. [L. functio, fr. fungi to perform, execute, akin to Skr. bhuj to enjoy, have the use of: cf. F. fonction. Cf. {Defunct}.] 1. The act of executing or performing any duty, office, or calling; performance. In the function of… …   The Collaborative International Dictionary of English

  • algebraic function — noun : a function whose dependence on the independent variable or variables is determined by an algebraic equation * * * Math. a function that can be expressed as a root of an equation in which a polynomial, in the independent and dependent… …   Useful english dictionary

  • algebraic function — Math. a function that can be expressed as a root of an equation in which a polynomial, in the independent and dependent variables, is set equal to zero. * * * …   Universalium

  • algebraic function — noun Any function that only uses the operations of addition, subtraction, multiplication, division and raising to a rational power …   Wiktionary

  • Function — Func tion, n. [L. functio, fr. fungi to perform, execute, akin to Skr. bhuj to enjoy, have the use of: cf. F. fonction. Cf. {Defunct}.] 1. The act of executing or performing any duty, office, or calling; performance. In the function of his public …   The Collaborative International Dictionary of English

  • 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… …   Wikipedia

  • Algebraic differential equation — Note: Differential algebraic equation is something different. In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the… …   Wikipedia

  • Algebraic space — In mathematics, an algebraic space is a generalization of the schemes of algebraic geometry introduced by Michael Artin for use in deformation theory.DefinitionAn algebraic space X comprises a schemeOne can always assume that U is an affine… …   Wikipedia

  • Algebraic equation — In mathematics, an algebraic equation over a given field is an equation of the form:P = Qwhere P and Q are (possibly multivariate) polynomials over that field. For example:y^4+frac{xy}{2}=frac{x^3}{3} xy^2+y^2 frac{1}{7} is an algebraic equation… …   Wikipedia

  • function — /fungk sheuhn/, n. 1. the kind of action or activity proper to a person, thing, or institution; the purpose for which something is designed or exists; role. 2. any ceremonious public or social gathering or occasion. 3. a factor related to or… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”