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 is algebraic if is a solution of an equation of the form
where the are polynomials in . A function that satisfies no such equation is said to be transcendental.
Any rational function is algebraic, since is a solution to .
Remark. There is also a version of an algebraic function defined on algebraic systems. Given an algebraic system , an -ary algebraic function on is an -ary operator on such that there is an -ary polynomial (http://planetmath.org/PolynomialsInAlgebraicSystems) on for some non-negative integer , and elements such that
Equivalently, given an algebraic system , if we associate each element of a corresponding symbol, also written , we may form an algebraic system from by adjoining every symbol to the type of considered as a unary operator symbol, and leaving everything else the same. Then an algebraic function on is just a polynomial on (and vice versa).
For example, in a ring , a function on given by where is a unary algebraic function on , as , where is an -ary polynomial on given by .
- 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).
|Date of creation||2013-03-22 15:19:24|
|Last modified on||2013-03-22 15:19:24|
|Last modified by||CWoo (3771)|