- Algebraic function
In

mathematics , an**algebraic function**is informally a function which satisfies apolynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable "x" is a solution "y" for an equation: $a\_n(x)y^n+a\_\{n-1\}(x)y^\{n-1\}+cdots+a\_0(x)=0$

where the coefficients "a"

_{"i"}("x") are polynomial functions of "x". A function which is not algebraic is called atranscendental 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 ::$y^2+x^2=1.,$

This determines "y", except only up to an overall sign:

:$y=pm\; sqrt\{1-x^2\}.,$

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::$p(y,x\_1,x\_2,dots,x\_n)=0.,$

It is normally assumed that "p" should be an

irreducible polynomial . The existence of an algebraic function is then guaranteed by theimplicit function theorem .Formally, an algebraic function in "n" variables over the field "K" is an element of the

algebraic closure of the field ofrational function s "K"("x"_{1},...,"x"_{"n"}). In order to understand algebraic functions as functions, it becomes necessary to introduce ideas relating toRiemann surface s or more generallyalgebraic 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 ofcasus irreducibilis (and more generally thefundamental 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

:$y-p(x)\; =\; 0.,$

More generally, any rational function is algebraic, being the solution of

:$q(x)y-p(x)=0\; implies\; y=frac\{p(x)\}\{q(x)\}.$

Moreover, the "n"th root of any polynomial is an algebraic function, solving the equation

:$y^n-p(x)=0\; implies\; y=sqrt\; [n]\; \{p(x)\}.$

Surprisingly, the

inverse function of an algebraic function is an algebraic function. For supposing that "y" is a solution of:$a\_n(x)y^n+cdots+a\_0(x),$

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,

:$b\_m(y)x^m+b\_\{m-1\}(y)x^\{m-1\}+cdots+b\_0(y)=0.$

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 thehorizontal line test : it fails to beone-to-one . The inverse is the algebraic "function" $x=pmsqrt\{y\}$. In this sense, algebraic functions are often not true functions at all, but instead aremultiple valued function s.Another way to understand this, which will become important later in the article, is that an algebraic function is the graph of an

algebraic curve .**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 analgebraically 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:$y^3-xy+1=0.,$

Using the

cubic formula , one solution is (the red curve in the accompanying image):$y=-frac\{(1+isqrt\{3\})x\}\{2^\{2/3\}sqrt\; [3]\; \{729-108x^3-frac\{(1-isqrt\{3\})sqrt\; [3]\; \{-27+sqrt\{729-108x^3\}\{6sqrt\; [3]\; \{2.$

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 analysis to discuss algebraic functions. In particular, theargument principle can be used to show that any algebraic function is in fact ananalytic 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:$frac\{1\}\{2pi\; i\}oint\_\{partialDelta\_i\}\; frac\{p\_y(x\_0,y)\}\{p(x\_0,y)\},dy\; =\; 1.$

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 theresidue theorem ::$f\_i(x)\; =\; frac\{1\}\{2pi\; i\}oint\_\{partialDelta\_i\}\; yfrac\{p\_y(x,y)\}\{p(x,y)\},dy$

which is an analytic function.

**Monodromy**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 thediscriminant vanishes. 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 theentire function associated 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

:$p(x,y)\; =\; a\_n(x)(y-f\_1(x))(y-f\_2(x))cdots(y-f\_n(x))$

since the "f"

_{"i"}are by definition the distinct zeros of "p". Themonodromy group acts by permuting the factors, and thus forms the**monodromy representation**of theGalois group of "p". (Themonodromy action on theuniversal covering space is related but different notion in the theory of Riemann surfaces.)**Algebraic functions in several variables****Formal definitions****Properties****Connection with algebraic geometry****History**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 inEdward 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."**References***

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