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) = πx 3 + x¼ - 2/x
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 pn(x)y n + ... + p1(x)y + p0(x) = 0 where the p0(x), p1(x), ... , pn(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 3v + u 2v 2 + uv 3 + v 4 is a rational, integral, homogeneous function of the fourth degree in u and v.