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algebraic function (Definition)

A function of one variable is said to be algebraic if it satisfies a polynomial equation whose coefficients are polynomials in the same variable. Namely, the function $ f(x)$ is algebraic if $ y=f(x)$ is a solution of an equation of the form

$\displaystyle p_n(x) y^n + \cdots + p_1(x) y + p_0(x) = 0, $
where the $ p_0(x), p_1(x), \ldots, p_n(x)$ are polynomials in $ x$. A function that satisfies no such equation is said to be transcendental.

The graph of an algebraic function is an algebraic curve, which is, loosely speaking, the zero set of a polynomial in two variables.

Examples

Any rational function $ f(x) = P(x)/Q(x)$ is algebraic, since $ y=f(x)$ is a solution to $ Q(x)y - P(x) = 0$.

The function $ f(x)=\sqrt{x}$ is algebraic, since $ y=f(x)$ is a solution to $ y^2 - x = 0$. The same is true for any power function $ x^{n/m}$, with $ n$ and $ m$ integers, it satisfies the equation $ y^m-x^n=0$.

It is known that the functions $ e^x$ and $ \ln x$ are transcendental. Many special functions, such as Bessel functions, elliptic integrals, and others are known to be transcendental.

Remark. There is also a version of an algebraic function defined on algebraic systems. Given an algebraic system $ A$, an $ n$-ary algebraic function on $ A$ is an $ n$-ary operator $ f(x_1,\ldots,x_n)$ on $ A$ such that there is an $ (n+m)$-ary polynomial $ p(x_1,\ldots,x_n,x_{n+1},\ldots, x_{n+m})$ on $ A$ for some non-negative integer $ m$, and elements $ a_1,\ldots, a_m\in A$ such that

$\displaystyle f(x_1,\ldots,x_n) = p(x_1,\ldots,x_n,a_1,\ldots, a_m).$

Equivalently, given an algebraic system $ A$, if we associate each element $ a$ of $ A$ a corresponding symbol, also written $ a$, we may form an algebraic system $ A'$ from $ A$ by adjoining every symbol $ a$ to the type of $ A$ considered as a unary operator symbol, and leaving everything else the same. Then an algebraic function on $ A$ is just a polynomial on $ A'$ (and vice versa).

For example, in a ring $ R$, a function $ f$ on $ R$ given by $ f(x)=a_nx^n+\cdots + a_1x+a_0$ where $ a_i\in R$ is a unary algebraic function on $ R$, as $ f(x)=p(x,a_0,\ldots,a_n)$, where $ p$ is an $ (n+2)$-ary polynomial on $ R$ given by $ p(x,x_0,\ldots,x_n)=x_nx^n+\cdots + x_1x+x_0$.

Bibliography

1
G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
2
S. Burris, H.P. Sankappanavar: A Course in Universal Algebra, Springer, New York (1981).



"algebraic function" is owned by CWoo. [ full author list (2) | owner history (1) ]
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See Also: elementary function, properties of entire functions, polynomials in algebraic systems

Also defines:  transcendental function, transcendental
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Cross-references: ring, operator symbol, unary, type, associate, operator, algebraic systems, elliptic integrals, Bessel functions, integers, power function, rational function, zero set, curve, graph, solution, coefficients, equation, polynomial, variable, function
There are 38 references to this entry.

This is version 11 of algebraic function, born on 2005-05-31, modified 2008-09-22.
Object id is 7131, canonical name is AlgebraicFunction.
Accessed 7616 times total.

Classification:
AMS MSC: 26A09 (Real functions :: Functions of one variable :: Elementary functions)
 08A40 (General algebraic systems :: Algebraic structures :: Operations, polynomials, primal algebras)
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