TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an algebraic function is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
that can be defined as the
root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large group ...
of a
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Quite often algebraic functions are
algebraic expressionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s using a finite number of terms, involving only the
algebraic operations In mathematics, a basic algebraic operation is any one of the common Operation (mathematics), operations of arithmetic, which include addition, subtraction, multiplication, Division (mathematics), division, raising to an integer exponentiation, powe ...
addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are: * $f\left(x\right) = 1/x$ * $f\left(x\right) = \sqrt$ * $f\left(x\right) = \frac$ Some algebraic functions, however, cannot be expressed by such finite expressions (this is the
Abel–Ruffini theorem In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general algebraic equation, polynomial equations of quintic equation, degree five or higher with arbitrary coef ...
). This is the case, for example, for the
Bring radical In algebra, the Bring radical or ultraradical of a real number ''a'' is the unique real root of a polynomial, root of the polynomial : x^5 + x + a. The Bring radical of a complex number ''a'' is either any of the five roots of the above pol ...
, which is the function implicitly defined by : $f\left(x\right)^5+f\left(x\right)+x = 0$. In more precise terms, an algebraic function of degree in one variable is a function $y = f\left(x\right),$ that is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
in its
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
and satisfies a
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
: $a_n\left(x\right)y^n+a_\left(x\right)y^+\cdots+a_0\left(x\right)=0$ where the coefficients are
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
s of , with integer coefficients. It can be shown that the same class of functions is obtained if
algebraic numbers An algebraic number is any complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...
are accepted for the coefficients of the 's. If
transcendental number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s occur in the coefficients the function is, in general, not algebraic, but it is ''algebraic over the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
'' generated by these coefficients. The value of an algebraic function at a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
, and more generally, at an
algebraic number An algebraic number is any complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...
is always an algebraic number. Sometimes, coefficients $a_i\left(x\right)$ that are polynomial over a ring are considered, and one then talks about "functions algebraic over ". A function which is not algebraic is called a
transcendental function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, as it is for example the case of $\exp x, \tan x, \ln x, \Gamma\left(x\right)$. A composition of transcendental functions can give an algebraic function: $f\left(x\right)=\cos \arcsin x = \sqrt$. As a polynomial equation of
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
''n'' has up to ''n'' roots (and exactly ''n'' roots over an
algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, such as the
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
), a polynomial equation does not implicitly define a single function, but up to ''n'' functions, sometimes also called
branches A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist or phytologist is a scientist who spe ...
. Consider for example the equation of the
unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

: $y^2+x^2=1.\,$ This determines ''y'', except only
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
an overall sign; accordingly, it has two branches: $y=\pm \sqrt.\,$ An algebraic function in ''m'' variables is similarly defined as a function $y=f\left(x_1,\dots ,x_m\right)$ which solves a polynomial equation in ''m'' + 1 variables: :$p\left(y,x_1,x_2,\dots,x_m\right) = 0.$ It is normally assumed that ''p'' should be an
irreducible polynomial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. The existence of an algebraic function is then guaranteed by the
implicit function theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Formally, an algebraic function in ''m'' variables over the field ''K'' is an element of the
algebraic closure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the field of
rational function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s ''K''(''x''1, ..., ''x''''m'').

# 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 In mathematics, a basic algebraic operation is any one of the common Operation (mathematics), operations of arithmetic, which include addition, subtraction, multiplication, Division (mathematics), division, raising to an integer exponentiation, powe ...
:
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

,
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
, and taking an ''n''th root. This is something of an oversimplification; because of the
fundamental theorem of Galois theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, algebraic functions need not be expressible by radicals. First, note that any
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
$y = p\left(x\right)$ is an algebraic function, since it is simply the solution ''y'' to the equation :$y-p\left(x\right) = 0.\,$ More generally, any
rational function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

$y=\frac$ is algebraic, being the solution to :$q\left(x\right)y-p\left(x\right)=0.$ Moreover, the ''n''th root of any polynomial
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
of an algebraic function is an algebraic function. For supposing that ''y'' is a solution to :$a_n\left(x\right)y^n+\cdots+a_0\left(x\right)=0,$ 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, :$b_m\left(y\right)x^m+b_\left(y\right)x^+\cdots+b_0\left(y\right)=0.$ 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 the
horizontal line test Horizontal may refer to: * Horizontal plane, in astronomy, geography, geometry and other sciences and contexts *Horizontal coordinate system The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon ...
: it fails to be
one-to-one One-to-one or one to one may refer to: Mathematics and communication *One-to-one function, also called an injective function *One-to-one correspondence, also called a bijective function *One-to-one (communication), the act of an individual commun ...
. The inverse is the algebraic "function" $x = \pm\sqrt$. Another way to understand this, is that the set of branches of the polynomial equation defining our algebraic function is the graph of an
algebraic curve In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.

## 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 fundamental theorem of algebra states that every non- constant single-variable polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
, the complex numbers are an
algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
values. Thus, problems to do with the
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
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 In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
). For example, consider the algebraic function determined by the equation :$y^3-xy+1=0.\,$ Using the cubic formula, we get :$y=-\frac+\frac.$ For $x\le \frac,$ the square root is real and the cubic root is thus well defined, providing the unique real root. On the other hand, for $x>\frac,$ 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 Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
to discuss algebraic functions. In particular, the
argument principle frame, The simple contour ''C'' (black), the zeros of ''f'' (blue) and the poles of ''f'' (red). Here we have \frac\oint_ \, dz=4-5.\, In complex analysis, the argument principle (or Cauchy's argument principle) relates the difference between the n ...
can be used to show that any algebraic function is in fact an
analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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 A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
of ''x''0. Choose a system of ''n'' non-overlapping discs Δ''i'' containing each of these zeros. Then by the argument principle :$\frac\oint_ \frac\,dy = 1.$ 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 the
residue theorem In complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigat ...
: :$f_i\left(x\right) = \frac\oint_ y\frac\,dy$ 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 the
discriminant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 150px, The real line with the point at infinity; it is called the real projective line. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...

). Thus the holomorphic extension of 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 :$p\left(x,y\right) = a_n\left(x\right)\left(y-f_1\left(x\right)\right)\left(y-f_2\left(x\right)\right)\cdots\left(y-f_n\left(x\right)\right)$ since the ''f''''i'' are by definition the distinct zeros of ''p''. The
monodromy group In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a Mathematical singularity, singularity. As the name implies, the fundam ...
acts by permuting the factors, and thus forms the monodromy representation of the
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of ''p''. (The on the
universal covering space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
is related but different notion in the theory of Riemann surfaces.)

# History

The ideas surrounding algebraic functions go back at least as far as
René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

. The first discussion of algebraic functions appears to have been in
Edward Waring Edward Waring (15 August 1798) was a UK, British mathematician. He entered Magdalene College, Cambridge as a sizar and became Senior wrangler in 1757. He was elected a Fellow of Magdalene and in 1760 Lucasian Professor of Mathematics, holding t ...
'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.

*
Algebraic expressionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
*
Analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
*
Complex function of the function . Hue represents the Argument (complex analysis), argument, brightness the Absolute_value#Complex_numbers, magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of m ...
*
Elementary function In mathematics, an elementary function is a function (mathematics), function of a single variable (mathematics), variable (typically Function of a real variable, real or Complex analysis#Complex functions, complex) that is defined as taking addit ...
*
Function (mathematics) In mathematics, a functionThe words map, mapping, transformation, correspondence, and operator are often used synonymously. . from a set (mathematics), set to a set assigns to each element of exactly one element of . The set is called the Dom ...
*
Generalized function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
* List of special functions and eponyms *
List of types of functions Function (mathematics), Functions can be identified according to the properties they have. These properties describe the functions' behaviour under certain conditions. A parabola is a specific type of function. Relative to set theory These properti ...
*
Polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

*
Rational function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

*
Special functions Special functions are particular function (mathematics), mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. ...
*
Transcendental function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

* *