Editing Algebraic function

{{No footnotes|date=June 2023}}
{{Short description|Mathematical function}}
In [[mathematics]], an '''algebraic function''' is a [[Function (mathematics)|function]] that can be defined 
as the [[Zero of a function|root]] of an [[Irreducible polynomial|irreducible]] [[polynomial equation]]. Algebraic functions are often [[algebraic expression]]s using a finite number of terms, involving only the [[algebraic operations]] addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:
* <math>f(x) = 1/x</math>
* <math>f(x) = \sqrt{x}</math>
* <math>f(x) = \frac{\sqrt{1 + x^3}}{x^{3/7} - \sqrt{7} x^{1/3}}</math>

Some algebraic functions, however, cannot be expressed by such finite expressions (this is the [[Abel–Ruffini theorem]]). This is the case, for example, for the [[Bring radical]], which is the function [[implicit function|implicitly]] defined by

: <math>f(x)^5+f(x)+x = 0</math>.

In more precise terms, an algebraic function of degree {{math|''n''}} in one variable {{math|''x''}} is a function <math>y = f(x),</math> that is [[Continuous function|continuous]] in its [[domain of a function|domain]] and satisfies a [[polynomial equation]] of positive [[degree of a polynomial |degree]]

: <math>a_n(x)y^n+a_{n-1}(x)y^{n-1}+\cdots+a_0(x)=0</math>

where the coefficients {{math|''a''<sub>''i''</sub>(''x'')}} are [[polynomial function]]s of {{math|''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 {{math|''a''<sub>''i''</sub>(''x'')}}'s. If [[transcendental number]]s occur in the coefficients the function is, in general, not algebraic, but it is ''algebraic over the [[Field (mathematics)|field]]'' generated by these coefficients.

The value of an algebraic function at a [[rational number]], and more generally, at an [[algebraic number]] is always an algebraic number.
Sometimes, coefficients <math>a_i(x)</math> that are polynomial over a [[Ring (mathematics)|ring]] {{mvar|R}} are considered, and one then talks about "functions algebraic over {{mvar|R}}".

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

As a polynomial equation of [[Degree of a polynomial|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 [[branch cut|branches]]. Consider for example the equation of the [[unit circle]]:
<math>y^2+x^2=1.\,</math>
This determines ''y'', except only [[up to]] an overall sign; accordingly, it has two branches:
<math>y=\pm \sqrt{1-x^2}.\,</math>

An '''algebraic function in ''m'' variables''' is similarly defined as a function <math>y=f(x_1,\dots ,x_m)</math> which solves a polynomial equation in ''m''&thinsp;+&thinsp;1 variables:

:<math>p(y,x_1,x_2,\dots,x_m) = 0.</math>

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 ''m'' variables over the field ''K'' is an element of the [[algebraic closure]] of the field of [[rational function]]s ''K''(''x''<sub>1</sub>,&thinsp;...,&thinsp;''x''<sub>''m''</sub>).

== Algebraic functions in one variable ==
=== Introduction and overview ===
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 (mathematics)|division]], and taking an [[nth root|''n''th root]].  This is something of an oversimplification; because of the [[fundamental theorem of Galois theory]], algebraic functions need not be expressible by radicals.

First, note that any [[polynomial function]] <math>y = p(x)</math> is an algebraic function, since it is simply the solution ''y'' to the equation

:<math> y-p(x) = 0.\,</math>

More generally, any [[rational function]] <math>y=\frac{p(x)}{q(x)}</math> is algebraic, being the solution to

:<math>q(x)y-p(x)=0.</math>

Moreover, the ''n''th root of any polynomial <math display="inline">y=\sqrt[n]{p(x)}</math> is an algebraic function, solving the equation

:<math>y^n-p(x)=0.</math>

Surprisingly, the [[inverse function]] of an algebraic function is an algebraic function.  For supposing that ''y'' is a solution to

:<math>a_n(x)y^n+\cdots+a_0(x)=0,</math>

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,

:<math>b_m(y)x^m+b_{m-1}(y)x^{m-1}+\cdots+b_0(y)=0.</math>

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''&thinsp;=&thinsp;''x''<sup>2</sup> fails the [[horizontal line test]]: it fails to be [[one-to-one function|one-to-one]].  The inverse is the algebraic "function" <math>x = \pm\sqrt{y}</math>. 
Another way to understand this, is that the [[Set (mathematics)|set]] of branches of the polynomial equation defining our 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 an [[algebraically closed field]].  Hence any [[polynomial]] relation ''p''(''y'',&thinsp;''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 number|real]] values.  Thus, problems to do with the [[domain of a function|domain]] of an algebraic function can safely be minimized.

[[Image:y^3-xy+1=0.png|thumb|A graph of three branches of the algebraic function ''y'', where ''y''<sup>3</sup>&nbsp;&minus;&nbsp;''xy''&nbsp;+&nbsp;1&nbsp;=&nbsp;0, over the domain 3/2<sup>2/3</sup> < ''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 taking ''nth'' roots without resorting to complex numbers (see [[casus irreducibilis]]).  For example, consider the algebraic function determined by the equation

:<math>y^3-xy+1=0.\,</math>

Using the [[cubic formula]], we get

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

For <math>x\le \frac{3}{\sqrt[3]{4}},</math> the square root is real and the cubic root is thus well defined, providing the unique real root. On the other hand, for <math>x>\frac{3}{\sqrt[3]{4}},</math> 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 of ''nth'' 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 of [[complex analysis]] to discuss algebraic functions.  In particular, the [[argument principle]] can 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'',&thinsp;''y'') be a complex polynomial in the complex variables ''x'' and ''y''.  Suppose that
''x''<sub>0</sub>&thinsp;∈&thinsp;'''C''' is such that the polynomial ''p''(''x''<sub>0</sub>,&thinsp;''y'') of ''y'' has ''n'' distinct zeros.  We shall show that the algebraic function is analytic in a [[Neighborhood (mathematics)|neighborhood]] of ''x''<sub>0</sub>.  Choose a system of ''n'' non-overlapping discs Δ<sub>''i''</sub> containing each of these zeros.  Then by the argument principle

:<math>\frac{1}{2\pi i}\oint_{\partial\Delta_i} \frac{p_y(x_0,y)}{p(x_0,y)}\,dy = 1.</math>

By continuity, this also holds for all ''x'' in a neighborhood of ''x''<sub>0</sub>.  In particular, ''p''(''x'',&thinsp;''y'') has only one root in Δ<sub>''i''</sub>, given by the [[residue theorem]]:

:<math>f_i(x) = \frac{1}{2\pi i}\oint_{\partial\Delta_i} y\frac{p_y(x,y)}{p(x,y)}\,dy</math>

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''<sub>''i''{{space|hair}}</sub>(''x''), provided that ''x'' is not a '''critical point''' of ''p''(''x'',&thinsp;''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.  Hence there are only finitely many such points ''c''<sub>1</sub>,&thinsp;...,&thinsp;''c''<sub>''m''</sub>.

A close analysis of the properties of the function elements ''f''<sub>''i''</sub> near the critical points can be used to show that the  [[monodromy theorem|monodromy cover]] is [[Ramification (mathematics)|ramified]] over the critical points (and possibly the [[Riemann sphere|point at infinity]]).  Thus the [[Holomorphic function|holomorphic]] extension of the ''f''<sub>''i''</sub> has at worst algebraic poles and ordinary algebraic branchings over the critical points.

Note that, away from the critical points, we have

:<math>p(x,y) = a_n(x)(y-f_1(x))(y-f_2(x))\cdots(y-f_n(x))</math>

since the ''f''<sub>''i''</sub> 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 surface]]s.)

== History ==

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:
: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.

==See also==
* [[Algebraic expression]]
* [[Analytic function]]
* [[Complex function]]
* [[Elementary function]]
* [[Function (mathematics)]]
* [[Generalized function]]
* [[List of special functions and eponyms]]
* [[List of types of functions]]
* [[Polynomial]]
* [[Rational function]]
* [[Special functions]]
* [[Transcendental function]]

==References==
* {{cite book|authorlink=Lars Ahlfors|first = Lars|last = Ahlfors|title = Complex Analysis|publisher = McGraw Hill|year = 1979}}
* {{cite book|author = van der Waerden, B.L.|authorlink=Bartel Leendert van der Waerden| title = Modern Algebra, Volume II|publisher = Springer|year=1931}}

==External links==
{{Commons category|Algebraic functions}}
*[https://www.encyclopediaofmath.org/index.php/Algebraic_function Definition of "Algebraic function"] in the ''Encyclopedia of Math''
*{{MathWorld |title=Algebraic Function |id=AlgebraicFunction}}
*{{PlanetMath|urlname=AlgebraicFunction|title=Algebraic Function}}
*[http://www.daviddarling.info/encyclopedia/A/algebraic_function.html Definition of "Algebraic function"] {{Webarchive|url=https://web.archive.org/web/20201026100400/http://www.daviddarling.info/encyclopedia/A/algebraic_function.html |date=2020-10-26 }} in [[David J. Darling]]'s Internet Encyclopedia of Science

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