Definition:Algebraic Function

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Definition

Real Algebraic Function

Let $y$ be a solution to the polynomial equation:

$\map {p_0} x + \map {p_1} x y + \dotsb + \map {p_{n - 1} } x y^{n - 1} + \map {p_n} x y^n = 0$

where $\map {p_0} x \ne 0, \map {p_1} x, \dotsc, \map {p_n} x$ are real polynomial functions in $x$.


Then $y = \map f x$ is a (real) algebraic function.


Complex Algebraic Function

Let $w$ be a solution to the polynomial equation:

$\map {p_0} z + \map {p_1} z w + \dotsb + \map {p_{n - 1} } z w^{n - 1} + \map {p_n} z w^n = 0$

where $\map {p_0} z \ne 0, \map {p_1} z, \dotsc, \map {p_n} z$ are complex polynomial functions in $z$.


Then $w = \map f z$ is a (complex) algebraic function.


Examples

Square Root

Let $f: \C \to \C$ be the complex function:

$\forall z \in \C: \map f z = z^{1/2}$

Then $f$ is an algebraic function.


Also see

  • Results about algebraic functions can be found here.


Sources

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