Algebraic function
that is continuous in its domain and satisfies a polynomial equation of positive degree where the coefficients ai(x) are polynomial functions of x, with integer coefficients.
It can be shown that the same class of functions is obtained if algebraic numbers are accepted for the coefficients of the ai(x)'s.
that are polynomial over a ring R are considered, and one then talks about "functions algebraic over R".
As a polynomial equation of degree n has up to n roots (and exactly n roots over an algebraically closed field, such as the complex numbers), a polynomial equation does not implicitly define a single function, but up to n functions, sometimes also called branches.
Consider for example the equation of the unit circle:
Formally, an algebraic function in m variables over the field K is an element of the algebraic closure of the field of rational functions K(x1, ..., xm).
The informal definition of an algebraic function provides a number of clues about their properties.
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 nth root.
This is something of an oversimplification; because of the fundamental theorem of Galois theory, algebraic functions need not be expressible by radicals.
is algebraic, being the solution to Moreover, the nth root of any polynomial
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 y) 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 means to express the function in terms of addition, multiplication, division and taking nth roots without resorting to complex numbers (see casus irreducibilis).
For example, consider the algebraic function determined by the equation Using the cubic formula, we get For
Thus the cubic root has to be chosen among three non-real numbers.
It may be proven that there is no way to express this function in terms of nth roots using real numbers only, even though the resulting function is real-valued on the domain of the graph shown.
We shall show that the algebraic function is analytic in a neighborhood of x0.
Choose a system of n non-overlapping discs Δi containing each of these zeros.
In particular, p(x, y) has only one root in Δi, given by the residue theorem: which is an analytic function.
Note that the foregoing proof of analyticity derived an expression for a system of n different function elements fi (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 or the discriminant vanish.
Note that, away from the critical points, we have since the fi 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.)
The ideas surrounding algebraic functions go back at least as far 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:
