Algebraic function

An algebraic function is a function whose values satisfy a nonzero polynomial equation with coefficients that are rational functions of its input. Its branches can be studied through polynomial equations, even when no single formula describes them globally.

Polynomial: An expression formed from constants and variables using addition, subtraction, and nonnegative integer powers. Algebraic functions are defined by polynomial equations in the function value.
Branch point: A point around which analytic continuation can carry a function value to a different branch. Roots of the defining polynomial can exchange when the input circles a branch point.
Abel–Ruffini theorem: The theorem that no general solution by radicals exists for polynomial equations of degree five or higher. Algebraic functions exist beyond those expressible by nested radicals.
Algebraic differential equation: A differential equation in which the unknown function and its derivatives satisfy a polynomial relation. Algebraic functions form a notable class of solutions to polynomial differential relations.
Transcendental function: A function that is not algebraic over the field of rational functions in its input. Exponential and trigonometric functions exemplify behavior no polynomial relation captures.
Rational function: A quotient of two polynomials, defined wherever its denominator is nonzero. Its coefficients in an algebraic function’s defining equation may depend rationally on the input.
Riemann surface: A space that locally resembles the complex plane and supports complex analytic functions. It unifies the multiple local branches of an algebraic function into one surface.
Évariste Galois: A French mathematician whose work connected polynomial solvability by radicals to groups of permutations. Galois theory explains when algebraic branches can be expressed using radicals.
Elliptic integral: An integral involving a rational function and the square root of a cubic or quartic polynomial. Its integrand is algebraic, while its antiderivative is generally not algebraic.
Elementary function: A function built from algebraic, exponential, logarithmic, and trigonometric functions using finite operations and composition. Some algebraic functions are not expressible by elementary formulas.

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Grothendieck–Katz p-curvature conjectureRelated: Finite monodromy is closely associated with algebraic solutions of regular-singular equations.