Algebraic function

In mathematics, a function ${\displaystyle f(x)}$ that satisfies a polynomial equation of the form $${\displaystyle a_{n}(x)f(x)^{n}+a_{n-1}(x)f(x)^{n-1}+\cdots +a_{1}(x)f(x)+a_{0}(x)=0}$$ where the ${\displaystyle a_{k}(x)}$ are polynomials (not all zero), is called an algebraic function. Basic examples of algebraic functions are polynomial functions, rational functions, the nth root function, and functions obtained from these by composition and algebraic operations (addition, multiplication, subtraction, and division). Thus an example of an algebraic function is the function ${\displaystyle f(x)={\sqrt {1-x^{2}}}}$ (for ${\displaystyle -1<x<1}$), whose graph is the top half of the standard unit circle. This function satisfies ${\displaystyle x^{2}+f(x)^{2}-1=0}$. Algebraic functions are contrasted with transcendental functions, such as the exponential function, logarithm, and the trigonometric functions.

Algebraic functions are usually treated more generally as multivalued functions. The example of ${\displaystyle x^{2}+y^{2}-1=0}$ illustrates this, since it includes both the top semicircle ${\displaystyle y={\sqrt {1-x^{2}}}}$ and bottom semicircle ${\displaystyle y=-{\sqrt {1-x^{2}}}}$ in the same package. Algebraic functions are most often studied over the complex numbers. Formally, an algebraic function over the complex numbers is defined to be a multivalued function ${\displaystyle y}$ satisfying a polynomial equation $${\displaystyle P(x,y)=0}$$ where ${\displaystyle P(x,y)}$ is an irreducible polynomial of two variables, having positive degree in ${\displaystyle y}$ and complex coefficients. The example of ${\displaystyle x^{2}+y^{2}-1=0}$ can be expressed as having the two single-valued branches, ${\displaystyle y={\sqrt {1-x^{2}}}}$ and ${\displaystyle y=-{\sqrt {1-x^{2}}}}$, with branch points where the two branches come together, at ${\displaystyle x=\pm 1}$. This particular function can be written using finitely many algebraic operations and extraction of roots, but this is not generally the case such as with the Bring radical. Over the complex numbers, algebraic functions have local holomorphic branches away from finitely many branch points and poles, and are naturally studied as meromorphic functions on compact Riemann surfaces.

More generally, over a field ${\displaystyle K}$, an algebraic function in one variable ${\displaystyle x}$ is defined algebraically as an element algebraic over the rational function field ${\displaystyle K(x)}$. Equivalently, it satisfies a polynomial equation of positive degree in ${\displaystyle y}$,

${\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0,}$

where the coefficients ${\displaystyle a_{i}(x)}$ are polynomials in ${\displaystyle x}$ with coefficients in ${\displaystyle K}$. If the irreducible defining polynomial has degree ${\displaystyle n}$ in ${\displaystyle y}$, the algebraic function is said to have degree ${\displaystyle n}$.

An algebraic function in ${\displaystyle m}$ variables over ${\displaystyle K}$ is an element algebraic over the field of rational functions ${\displaystyle K(x_{1},\ldots ,x_{m})}$. Equivalently, it satisfies a polynomial equation

${\displaystyle p(y,x_{1},x_{2},\dots ,x_{m})=0.}$

In one variable, algebraic functions are closely related to algebraic curves and their function fields; in the separable case, they may also be studied via finite or ramified covers of the projective line.