Algebraic function

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In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions 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:

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

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

${\displaystyle f(x)^{5}+f(x)+x=0}$.

In more precise terms, an algebraic function of degree n in one variable x is a function ${\displaystyle y=f(x),}$ that is continuous in its domain and satisfies a polynomial equation of positive degree

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

where the coefficients a_i(x) are polynomial functions of x, 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 a_i(x)'s. If transcendental numbers occur in the coefficients the function is, in general, not algebraic, but it is algebraic over the 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 ${\displaystyle a_{i}(x)}$ that are polynomial over a ring R are considered, and one then talks about "functions algebraic over R".

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

As a polynomial equation of 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 branches. Consider for example the equation of the unit circle: ${\displaystyle y^{2}+x^{2}=1.\,}$ This determines y, except only up to an overall sign; accordingly, it has two branches: ${\displaystyle y=\pm {\sqrt {1-x^{2}}}.\,}$

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

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

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 functions K(x₁, ..., x_m).