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# algebraic function

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

$p_{n}(x)y^{n}+\cdots+p_{1}(x)y+p_{0}(x)=0,$ |

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

$f(x_{1},\ldots,x_{n})=p(x_{1},\ldots,x_{n},a_{1},\ldots,a_{m}).$ |

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 $A^{{\prime}}$ 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 $A^{{\prime}}$ (and vice versa).

For example, in a ring $R$, a function $f$ on $R$ given by $f(x)=a_{n}x^{n}+\cdots+a_{1}x+a_{0}$ 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 $p(x,x_{0},\ldots,x_{n})=x_{n}x^{n}+\cdots+x_{1}x+x_{0}$.

# References

- 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).

## Mathematics Subject Classification

08A40*no label found*26A09

*no label found*

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