Algebraic Function

An Algebraic Function is a Mathematical Function that can be defined as a root solution of a polynomial equation with integer coefficients.

• AKA: Algebraic Mapping, Polynomial Root Function, Algebraic Expression Function.
• Context:
  • It can typically be expressed through Finite Algebraic Operations including addition, subtraction, multiplication, division, and rational exponentiation.
  • It can typically satisfy Polynomial Equations of the form [math]\displaystyle{ a_n(x)y^n + a_{n-1}(x)y^{n-1} + \cdots + a_0(x) = 0 }[/math].
  • It can typically generate Algebraic Numbers when evaluated at rational inputs.
  • It can typically be classified by its Algebraic Degree, corresponding to the minimal polynomial degree.
  • It can typically exhibit Multiple Branches due to the multivalued nature of root extraction.
  • ...
  • It can often be represented through Implicit Function Definitions when explicit formulas are unavailable.
  • It can often require Galois Theory to determine its solvability by radicals.
  • It can often be approximated using Taylor Series Expansions within convergence radiuses.
  • It can often be composed to create more Complex Algebraic Functions.
  • ...
  • It can range from being a Simple Algebraic Function to being a Complex Algebraic Function, depending on its algebraic complexity.
  • It can range from being a Univariate Algebraic Function to being a Multivariate Algebraic Function, depending on its variable count.
  • It can range from being an Explicit Algebraic Function to being an Implicit Algebraic Function, depending on its expression form.
  • It can range from being a Continuous Algebraic Function to being a Discontinuous Algebraic Function, depending on its domain characteristics.
  • It can range from being a Finite Algebraic Function to being an Infinite Algebraic Function, depending on its expression length.
  • ...
  • It can have Mathematical Properties such as:
    • Closure Under Composition with other algebraic functions.
    • Closure Under Arithmetic Operations.
    • Algebraic Dependency on its defining polynomial.
    • Branch Points where function continuity may be disrupted.
  • It can be distinguished from Transcendental Functions by its polynomial definability.
  • It can form Algebraic Function Fields under field operations.
  • ...
• Example(s):
  • Polynomial Functions, such as:
    • Constant Functions: [math]\displaystyle{ f(x) = c }[/math] (degree 0).
    • Linear Functions: [math]\displaystyle{ f(x) = ax + b }[/math] (degree 1).
    • Quadratic Functions: [math]\displaystyle{ f(x) = ax^2 + bx + c }[/math] (degree 2).
    • Cubic Functions: [math]\displaystyle{ f(x) = ax^3 + bx^2 + cx + d }[/math] (degree 3).
    • Quartic Functions: [math]\displaystyle{ f(x) = ax^4 + bx^3 + cx^2 + dx + e }[/math] (degree 4).
    • Quintic Functions: [math]\displaystyle{ f(x) = a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 }[/math] (degree 5).
  • Rational Functions, such as:
    • Simple Rational Functions: [math]\displaystyle{ f(x) = \frac{1}{x} }[/math].
    • Proper Rational Functions: [math]\displaystyle{ f(x) = \frac{x^2 + 1}{x^3 + x + 1} }[/math].
    • Partial Fractions: [math]\displaystyle{ f(x) = \frac{1}{x-1} + \frac{2}{x+1} }[/math].
  • Radical Functions, such as:
    • Square Root Functions: [math]\displaystyle{ f(x) = \sqrt{x} }[/math].
    • Cube Root Functions: [math]\displaystyle{ f(x) = \sqrt[3]{x} }[/math].
    • General Nth Root Functions: [math]\displaystyle{ f(x) = \sqrt[n]{P(x)} }[/math] where P(x) is a polynomial.
  • Composite Algebraic Functions, such as:
    • Nested Radical Functions: [math]\displaystyle{ f(x) = \sqrt{1 + \sqrt{x}} }[/math].
    • Rational Radical Functions: [math]\displaystyle{ f(x) = \frac{\sqrt{1+x^3}}{x^{3/7}-\sqrt{7}x^{1/3}} }[/math].
    • Algebraic Combinations: [math]\displaystyle{ f(x) = \frac{x + \sqrt{x^2 - 1}}{2} }[/math].
  • Multivariate Algebraic Functions, such as:
    • Two-Variable Polynomials: [math]\displaystyle{ f(x,y) = x^2 + xy + y^2 }[/math].
    • Vector-Input Polynomial Functions: [math]\displaystyle{ f(x,y,z) = xyz + x^2y - z^3 }[/math].
    • Implicit Circle Functions: [math]\displaystyle{ x^2 + y^2 = 1 }[/math] defining [math]\displaystyle{ y = \pm\sqrt{1-x^2} }[/math].
  • Special Algebraic Functions, such as:
    • Elliptic Functions defined by algebraic differential equations.
    • Hyperelliptic Functions arising from hyperelliptic curves.
    • Modular Functions with algebraic values at algebraic points.
  • ...
• Counter-Example(s):
  • Transcendental Functions, which cannot be expressed as polynomial equation solutions (e.g., [math]\displaystyle{ e^x, \sin(x), \log(x) }[/math]).
  • Vector-Output Functions, which produce vector values rather than scalar values.
  • Boolean Functions, which output discrete values rather than algebraic values.
  • Piecewise Functions defined by conditional expressions rather than algebraic equations.
  • Software Functions, which are computational procedures rather than mathematical mappings.
  • Distribution Functions, which may involve non-algebraic operations.
  • Special Functions like Gamma Functions or Bessel Functions that are transcendental.
• See: Polynomial Function, Rational Function, Radical Expression, Algebraic Equation, Galois Theory, Abel-Ruffini Theorem, Algebraic Closure, Function Field, Algebraic Geometry, Transcendental Function, Implicit Function Theorem.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/list_of_mathematical_functions#Algebraic_functions Retrieved:2015-6-14.
  • Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.
    • Polynomials: Can be generated by addition, multiplication, and exponentiation alone.
      • Constant function: polynomial of degree zero, graph is a horizontal straight line
      • Linear function: First degree polynomial, graph is a straight line.
      • Quadratic function: Second degree polynomial, graph is a parabola.
      • Cubic function: Third degree polynomial.
      • Quartic function: Fourth degree polynomial.
      • Quintic function: Fifth degree polynomial.
      • Sextic function: Sixth degree polynomial.
    • Rational functions: A ratio of two polynomials.
    • Nth root.
      • Square root: Yields a number whose square is the given one [math]\displaystyle{ x^{\frac{1}{2}} \!\ }[/math] .
      • Cube root: Yields a number whose cube is the given one [math]\displaystyle{ x^{\frac{1}{3}} \!\ }[/math] .

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/algebraic_function Retrieved:2014-11-23.
  • In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions can be expressed using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power:  :[math]\displaystyle{ f(x)=1/x, f(x)=\sqrt{x}, f(x)=\frac{ \sqrt{1+x^3}}{x^{3/7}-\sqrt{7} x^{1/3}} }[/math] are typical examples.

    However, some algebraic functions cannot be expressed by such finite expressions (as proven by Galois and Niels Abel), as it is for example the case of the function defined by  : [math]\displaystyle{ f(x)^5+f(x)^4+x=0 }[/math].

    In more precise terms, an algebraic function of degree n in one variable x is a function [math]\displaystyle{ y = f(x) }[/math] that satisfies a polynomial equation  : [math]\displaystyle{ a_n(x)y^n+a_{n-1}(x)y^{n-1}+\cdots+a_0(x)=0 }[/math] where the coefficients ai(x) are polynomial functions of x, with coefficients belonging to a set S.

    Quite often, [math]\displaystyle{ S=\mathbb Q }[/math], and one then talks about "function algebraic over [math]\displaystyle{ \mathbb Q }[/math]", and the evaluation at a given rational value of such an algebraic function gives an algebraic number.

    A function which is not algebraic is called a transcendental function, as it is for example the case of [math]\displaystyle{ \exp(x), \tan(x), \ln(x), \Gamma(x) }[/math]. A composition of transcendental functions can give an algebraic function: [math]\displaystyle{ f(x)=\cos (\arcsin(x)) = \sqrt{1-x^2} }[/math].

    As an equation of degree n has n roots, a polynomial equation does not implicitly define a single function, but n

    functions, sometimes also called branches. Consider for example the equation of the unit circle:

    [math]\displaystyle{ y^2+x^2=1.\, }[/math]

    This determines y, except only up to an overall sign; accordingly, it has two branches:

    [math]\displaystyle{ y=\pm \sqrt{1-x^2}.\, }[/math]

    An algebraic function in m variables is similarly defined as a function y which solves a polynomial equation in m + 1 variables:  :[math]\displaystyle{ p(y,x_1,x_2,\dots,x_m)=0.\, }[/math]

    It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem.

    Formally, an algebraic function in m variables over the field K is an element of the algebraic closure of the field of rational functions K(x₁,...,x_m).

2013

• http://www.mathworks.com/help/optim/ug/writing-objective-functions.html
  • Many Optimization Toolbox™ solvers minimize a scalar function of a multidimensional vector. The objective function is the function the solvers attempt to minimize. Several solvers accept vector-valued objective functions, and some solvers use objective functions you specify by vectors or matrices.

2009

• http://en.wiktionary.org/wiki/algebraic_function
  • (Algebra) Any function that only uses the operations of addition, subtraction, multiplication, division and raising to a rational power.

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