Algebraic Function

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

where the coefficients a_i(x) are polynomial functions of x with rational coefficients. A function that 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:

However, both branches are thought of as belonging to the "function" determined by the polynomial equation. Thus an algebraic function is most naturally considered as a multiple valued function.

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(x₁,...,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.