algebraic function

A function of one variable is said to be algebraic if it satisfies a polynomial equation whose coefficients are polynomials in the same variable.Namely, the function $f(x)$ is algebraic if $y=f(x)$ is a solution of an equation of the form

$$p_n(x)y^n+\mathrm{\cdots }+p_1(x)y+p_0(x)=0,$$

where the $p_0(x),p_1(x),\mathrm{\dots },p_n(x)$ are polynomials in $x$. A function that satisfies no such equation is said to be transcendental.

The graph of an algebraic function is an algebraic curve, which is, loosely speaking, the zero set of a polynomial in two variables.

Examples

Any rational function $f(x)=P(x)/Q(x)$ is algebraic, since $y=f(x)$ is a solution to $Q(x)y-P(x)=0$.

The function $f(x)=\sqrt{x}$ is algebraic, since $y=f(x)$ is asolution to $y^2-x=0$. The same is true for any power function$x^{n/m}$, with $n$ and $m$ integers, it satisfies the equation $y^m-x^n=0$.

It is known that the functions $e^x$ and $\ln x$ are transcendental. Many special functions, such as Bessel functions, elliptic integrals, and others are known to be transcendental.

Remark. There is also a version of an algebraic function defined on algebraic systems. Given an algebraic system $A$, an $n$-ary algebraic function on $A$ is an $n$-ary operator $f(x_1,\mathrm{\dots },x_n)$ on $A$ such that there is an $(n+m)$-ary polynomial (http://planetmath.org/PolynomialsInAlgebraicSystems) $p(x_1,\mathrm{\dots },x_n,x_{n+1},\mathrm{\dots },x_{n+m})$ on $A$ for some non-negative integer $m$, and elements $a_1,\mathrm{\dots },a_m\in A$ such that

$$f(x_1,\mathrm{\dots },x_n)=p(x_1,\mathrm{\dots },x_n,a_1,\mathrm{\dots },a_m).$$

Equivalently, given an algebraic system $A$, if we associate each element $a$ of $A$ a corresponding symbol, also written $a$, we may form an algebraic system $A^{\prime }$ from $A$ by adjoining every symbol $a$ to the type of $A$ considered as a unary operator symbol, and leaving everything else the same. Then an algebraic function on $A$ is just a polynomial on $A^{\prime }$ (and vice versa).

For example, in a ring $R$, a function $f$ on $R$ given by $f(x)=a_n{x}^n+\mathrm{\cdots }+a_{1}x+a_0$ where $a_i\in R$ is a unary algebraic function on $R$, as $f(x)=p(x,a_0,\mathrm{\dots },a_n)$, where $p$ is an $(n+2)$-ary polynomial on $R$ given by $p(x,x_0,\mathrm{\dots },x_n)=x_n{x}^n+\mathrm{\cdots }+x_{1}x+x_0$.