Algebraic Function

An algebraic function of variables is a function for which there is a nonzero polynomial with coefficients in the rational numbers such that

Equivalently, after clearing denominators, may be taken to have coefficients in the integers. Such a polynomial is called a defining polynomial of .

A defining polynomial need not determine a unique algebraic function. For example, defines both and on . A particular branch can be selected by an additional condition, such as for the principal square root, or by exact root isolation data.

Every expression obtained from polynomials by finitely many additions, subtractions, multiplications, divisions, and rational powers is an algebraic function. A defining polynomial for such an expression with radicals can be constructed recursively using resultants (Zippel 1993, Maaz and Strzeboński 2025). Not every algebraic function can be expressed using radicals, as follows from Abel's impossibility theorem. A function which is not algebraic is called a transcendental function.