algebraic function

A functionMathworldPlanetmath of one variable is said to be algebraic if it satisfies a polynomial equation whose coefficientsMathworldPlanetmath are polynomialsMathworldPlanetmathPlanetmath in the same variable. Namely, the function f(x) is algebraic if y=f(x) is a solution of an equation of the form


where the p0(x),p1(x),,pn(x) are polynomials in x. A function that satisfies no such equation is said to be transcendental.

The graph of an algebraic function is an algebraic curve, which is, loosely speaking, the zero setMathworldPlanetmath of a polynomial in two variables.


Any rational functionMathworldPlanetmath f(x)=P(x)/Q(x) is algebraic, since y=f(x) is a solution to Q(x)y-P(x)=0.

The function f(x)=x is algebraic, since y=f(x) is a solution to y2-x=0. The same is true for any power functionDlmfDlmfPlanetmath xn/m, with n and m integers, it satisfies the equation ym-xn=0.

It is known that the functions ex and lnx are transcendental. Many special functions, such as Bessel functionsDlmfMathworldPlanetmathPlanetmath, elliptic integralsMathworldPlanetmath, and others are known to be transcendental.

Remark. There is also a version of an algebraic function defined on algebraic systems. Given an algebraic system A, an n-ary algebraic function on A is an n-ary operator f(x1,,xn) on A such that there is an (n+m)-ary polynomial ( p(x1,,xn,xn+1,,xn+m) on A for some non-negative integer m, and elements a1,,amA such that


Equivalently, given an algebraic system A, if we associate each element a of A a corresponding symbol, also written a, we may form an algebraic system A from A by adjoining every symbol a to the type of A considered as a unary operator symbol, and leaving everything else the same. Then an algebraic function on A is just a polynomial on A (and vice versa).

For example, in a ring R, a function f on R given by f(x)=anxn++a1x+a0 where aiR is a unary algebraic function on R, as f(x)=p(x,a0,,an), where p is an (n+2)-ary polynomial on R given by p(x,x0,,xn)=xnxn++x1x+x0.


  • 1 G. Grätzer: Universal AlgebraMathworldPlanetmath, 2nd Edition, Springer, New York (1978).
  • 2 S. Burris, H.P. Sankappanavar: A Course in Universal Algebra, Springer, New York (1981).
Title algebraic function
Canonical name AlgebraicFunction
Date of creation 2013-03-22 15:19:24
Last modified on 2013-03-22 15:19:24
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 15
Author CWoo (3771)
Entry type Definition
Classification msc 08A40
Classification msc 26A09
Related topic ElementaryFunction
Related topic PropertiesOfEntireFunctions
Related topic PolynomialsInAlgebraicSystems
Defines transcendental function
Defines transcendental
Defines algebraic curve