- Algebraic function
mathematics, an algebraic function is informally a function which satisfies a polynomialequation whose coefficients are themselves polynomials. 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". A function which is not algebraic is called a
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
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.
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 closureof 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
More generally, any rational function is algebraic, being the solution of
Moreover, the "n"th root of any polynomial is an algebraic function, solving the equation
inverse functionof 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" . 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
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
cubic formula, one solution is (the red curve in the accompanying image)
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 analysisto discuss algebraic functions. In particular, the argument principlecan 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"0 ∈ C 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
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
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
discriminantvanishes. 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 functionassociated 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
since the "f""i" are by definition the distinct zeros of "p". The
monodromy groupacts by permuting the factors, and thus forms the monodromy representation of the Galois groupof "p". (The monodromy actionon the universal covering spaceis related but different notion in the theory of Riemann surfaces.)
Algebraic functions in several variables
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."
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