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 a solution 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 $\mathrm{ln}x$ 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 $p(x,{x}_{0},\mathrm{\dots},{x}_{n})={x}_{n}{x}^{n}+\mathrm{\cdots}+{x}_{1}x+{x}_{0}$.
References
- 1 G. Grätzer: Universal Algebra^{}, 2nd Edition, Springer, New York (1978).
- 2 S. Burris, H.P. Sankappanavar: A Course in Universal Algebra, Springer, New York (1981).
Title | algebraic function |
Canonical name | AlgebraicFunction |
Date of creation | 2013-03-22 15:19:24 |
Last modified on | 2013-03-22 15:19:24 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 15 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 08A40 |
Classification | msc 26A09 |
Related topic | ElementaryFunction |
Related topic | PropertiesOfEntireFunctions |
Related topic | PolynomialsInAlgebraicSystems |
Defines | transcendental function |
Defines | transcendental |
Defines | algebraic curve |