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 ai(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,...,xn). 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.
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Famous quotes containing the words algebraic and/or function:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
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