# algebraic function

An algebraic function is a function that may be expressed in a finite number of terms, involving only the elementary algebraic operations of addition, subtraction, multiplication, division, and root extraction. For instance,

*f*(*x*) = π*x*^{ 3} + *x*^{¼} - 2/*x*

is an algebraic function. They are distinguished from transcendental functions, such as the exponential, logarithmic, and trigonometric functions, which can be expressed only by an infinite series.

Another way of saying this is that the function *f*(*x*) is
algebraic if *y* = *f*(*x*) is a solution of an equation
of the form *p _{n}*(

*x*)

*y*

^{ n}+ ... +

*p*(

_{1}*x*)

*y*+

*p*(x) = 0 where the

_{0}*p*(x),

_{0}*p*(

_{1}*x*), ... ,

*p*(

_{n}*x*) are polynomials in

*x*. All polynomials are algebraic. A function that satisfies no such equation is transcendental.

There are various types of algebraic function. A **rational function** is one in which there are no fractional powers of the variable or variables. **Integral functions** do not include the operation of division
in any of their terms. A **homogeneous function** is one in
which the terms are all of the same degree – i.e., the sum of the
indices of the variables in each term is the same for every term. For example, *u *^{4} + *u*^{ 3}*v* + *u *^{2}*v*^{ 2} + *uv*^{ 3} + *v *^{4} is a rational, integral,
homogeneous function of the fourth degree in *u* and *v*.