|
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 $$ p_n(x) y^n + \cdots + p_1(x) y + p_0(x) = 0, $$ where the
$p_0(x), p_1(x), \ldots, p_n(x)$ 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.
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 $\ln x$ 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 $f(x_1,\ldots,x_n)$ on $A$ such that there is an $(n+m)$ -ary polynomial $p(x_1,\ldots,x_n,x_{n+1},\ldots, x_{n+m})$ on $A$ for some non-negative integer $m$ , and
elements $a_1,\ldots, a_m\in A$ such that $$f(x_1,\ldots,x_n) = p(x_1,\ldots,x_n,a_1,\ldots, a_m).$$
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 $A'$ 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
$A'$ (and vice versa).
For example, in a ring $R$ , a function $f$ on $R$ given by $f(x)=a_nx^n+\cdots + a_1x+a_0$ where $a_i\in R$ is a unary algebraic function on $R$ , as $f(x)=p(x,a_0,\ldots,a_n)$ , where $p$ is an $(n+2)$ -ary polynomial on $R$ given by $p(x,x_0,\ldots,x_n)=x_nx^n+\cdots + x_1x+x_0$ .
- 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).
|
