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Algebraic function

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Mathematical function

Inmathematics, analgebraic function is afunction that can be defined as theroot of anirreducible polynomial equation. Algebraic functions are oftenalgebraic expressions using a finite number of terms, involving only thealgebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:

$f(x)=1/x$
$f(x)={\sqrt {x}}$
$f(x)={\frac {\sqrt {1+x^{3}}}{x^{3/7}-{\sqrt {7}}x^{1/3}}}$

Some algebraic functions, however, cannot be expressed by such finite expressions (this is theAbel–Ruffini theorem). This is the case, for example, for theBring radical, which is the functionimplicitly defined by

f(x)^{5}+f(x)+x=0

In more precise terms, an algebraic function of degreen in one variablex is a function $y=f(x),$ that iscontinuous in itsdomain and satisfies apolynomial equation of positivedegree

a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0

where the coefficientsa_i(x) arepolynomial functions ofx, with integer coefficients. It can be shown that the same class of functions is obtained ifalgebraic numbers are accepted for the coefficients of thea_i(x)'s. Iftranscendental numbers occur in the coefficients the function is, in general, not algebraic, but it isalgebraic over thefield generated by these coefficients.

The value of an algebraic function at arational number, and more generally, at analgebraic number is always an algebraic number.Sometimes, coefficients $a_{i}(x)$ that are polynomial over aringR are considered, and one then talks about "functions algebraic overR".

A function which is not algebraic is called atranscendental function, as it is for example the case of $\exp x,\tan x,\ln x,\Gamma (x)$ . A composition of transcendental functions can give an algebraic function: $f(x)=\cos \arcsin x={\sqrt {1-x^{2}}}$ .

As a polynomial equation ofdegreen has up ton roots (and exactlyn roots over analgebraically closed field, such as thecomplex numbers), a polynomial equation does not implicitly define a single function, but up tonfunctions, sometimes also calledbranches. Consider for example the equation of theunit circle: $y^{2}+x^{2}=1.\,$ This determinesy, except onlyup to an overall sign; accordingly, it has two branches: $y=\pm {\sqrt {1-x^{2}}}.\,$

Analgebraic function inm variables is similarly defined as a function $y=f(x_{1},\dots ,x_{m})$ which solves a polynomial equation inm + 1 variables:

p(y,x_{1},x_{2},\dots ,x_{m})=0.

It is normally assumed thatp should be anirreducible polynomial. The existence of an algebraic function is then guaranteed by theimplicit function theorem.

Formally, an algebraic function inm variables over the fieldK is an element of thealgebraic closure of the field ofrational functionsK(x₁, ..., x_m).

Algebraic functions in one variable

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Introduction and overview

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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 usualalgebraic operations:addition,multiplication,division, and taking annth root. This is something of an oversimplification; because of thefundamental theorem of Galois theory, algebraic functions need not be expressible by radicals.

First, note that anypolynomial function $y=p(x)$ is an algebraic function, since it is simply the solutiony to the equation

y-p(x)=0.\,

More generally, anyrational function $y={\frac {p(x)}{q(x)}}$ is algebraic, being the solution to

q(x)y-p(x)=0.

Moreover, thenth root of any polynomial ${\textstyle y={\sqrt[{n}]{p(x)}}}$ is an algebraic function, solving the equation

y^{n}-p(x)=0.

Surprisingly, theinverse function of an algebraic function is an algebraic function. For supposing thaty is a solution to

a_{n}(x)y^{n}+\cdots +a_{0}(x)=0,

for each value ofx, thenx is also a solution of this equation for each value ofy. Indeed, interchanging the roles ofx andy and gathering terms,

b_{m}(y)x^{m}+b_{m-1}(y)x^{m-1}+\cdots +b_{0}(y)=0.

Writingx as a function ofy gives the inverse function, also an algebraic function.

However, not every function has an inverse. For example,y = x² fails thehorizontal line test: it fails to beone-to-one. The inverse is the algebraic "function" $x=\pm {\sqrt {y}}$ . Another way to understand this, is that theset of branches of the polynomial equation defining our algebraic function is the graph of analgebraic curve.

The role of complex numbers

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From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. First of all, by thefundamental theorem of algebra, the complex numbers are analgebraically closed field. Hence anypolynomial relationp(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions not exceeding the degree ofp iny) fory at each pointx, provided we allowy to assume complex as well asreal values. Thus, problems to do with thedomain of an algebraic function can safely be minimized.

A graph of three branches of the algebraic functiony, wherey³ − xy + 1 = 0, over the domain 3/2^2/3 <x < 50.

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 takingnth roots without resorting to complex numbers (seecasus irreducibilis). For example, consider the algebraic function determined by the equation

y^{3}-xy+1=0.\,

Using thecubic formula, we get

y=-{\frac {2x}{\sqrt[{3}]{-108+12{\sqrt {81-12x^{3}}}}}}+{\frac {\sqrt[{3}]{-108+12{\sqrt {81-12x^{3}}}}}{6}}.

For $x\leq {\frac {3}{\sqrt[{3}]{4}}},$ the square root is real and the cubic root is thus well defined, providing the unique real root. On the other hand, for $x>{\frac {3}{\sqrt[{3}]{4}}},$ the square root is not real, and one has to choose, for the square root, either non-real square root. Thus the cubic root has to be chosen among three non-real numbers. If the same choices are done in the two terms of the formula, the three choices for the cubic root provide the three branches shown, in the accompanying image.

It may be proven that there is no way to express this function in terms ofnth roots using 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 allows one to use the powerful techniques ofcomplex 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, letp(x, y) be a complex polynomial in the complex variablesx andy. Suppose thatx₀ ∈ C is such that the polynomialp(x₀, y) ofy hasn distinct zeros. We shall show that the algebraic function is analytic in aneighborhood ofx₀. Choose a system ofn non-overlapping discs Δ_i containing each of these zeros. Then by the argument principle

{\frac {1}{2\pi i}}\oint _{\partial \Delta _{i}}{\frac {p_{y}(x_{0},y)}{p(x_{0},y)}}\,dy=1.

By continuity, this also holds for allx in a neighborhood ofx₀. In particular,p(x, y) has only one root in Δ_i, given by theresidue theorem:

f_{i}(x)={\frac {1}{2\pi i}}\oint _{\partial \Delta _{i}}y{\frac {p_{y}(x,y)}{p(x,y)}}\,dy

which is an analytic function.

Monodromy

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Note that the foregoing proof of analyticity derived an expression for a system ofn differentfunction elementsf_i(x), provided thatx is not acritical point ofp(x, y). Acritical point is a point where the number of distinct zeros is smaller than the degree ofp, and this occurs only where the highest degree term ofp or thediscriminant vanish. Hence there are only finitely many such pointsc₁, ..., c_m.

A close analysis of the properties of the function elementsf_i near the critical points can be used to show that themonodromy cover isramified over the critical points (and possibly thepoint at infinity). Thus theholomorphic extension of thef_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 thef_i are by definition the distinct zeros ofp. Themonodromy group acts by permuting the factors, and thus forms themonodromy representation of theGalois group ofp. (Themonodromy action on theuniversal covering space is related but different notion in the theory ofRiemann surfaces.)

History

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The ideas surrounding algebraic functions go back at least as far asRené Descartes. The first discussion of algebraic functions appears to have been inEdward Waring's 1794An Essay on the Principles of Human Knowledge in which he writes:

let a quantity denoting the ordinate, be an algebraic function of the abscissax, by the common methods of division and extraction of roots, reduce it into an infinite series ascending or descending according to the dimensions ofx, and then find the integral of each of the resulting terms.

References

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Ahlfors, Lars (1979).Complex Analysis. McGraw Hill.
van der Waerden, B.L. (1931).Modern Algebra, Volume II. Springer.

External links

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Wikimedia Commons has media related toAlgebraic functions.

Definition of "Algebraic function" in theEncyclopedia of Math
Weisstein, Eric W."Algebraic Function".MathWorld.
Algebraic Function atPlanetMath.
Definition of "Algebraic function"Archived 2020-10-26 at theWayback Machine inDavid J. Darling's Internet Encyclopedia of Science

v t e Function
History List of specific functions
Types by domain and codomain	X → 𝔹 𝔹 → X 𝔹ⁿ → X X → ℤ ℤ → X X → ℝ ℝ → X ℝⁿ → X X → ℂ ℂ → X ℂⁿ → X
Classes/properties	Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions	Restriction Composition λ Inverse
Generalizations	Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
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