algebraic function

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

+

⋯

+

p

1

⁢

(

x

)

⁢

y

+

p

0

⁢

(

x

)

=

0

,

where the

p

0

⁢

(

x

)

,

p

1

⁢

(

x

)

,

…

,

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

)

=

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

,

…

,

x

n

)

on

A

such that there is an

(

n

+

m

)

-ary polynomial (

http://planetmath.org/PolynomialsInAlgebraicSystems

)

p

⁢

(

x

1

,

…

,

x

n

,

x

n

+

1

,

…

,

x

n

+

m

)

on

A

for some non-negative integer

m

, and

elements

a

1

,

…

,

a

m

∈

A

such that

f

⁢

(

x

1

,

…

,

x

n

)

=

p

⁢

(

x

1

,

…

,

x

n

,

a

1

,

…

,

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

n

⁢

x

n

+

⋯

+

a

1

⁢

x

+

a

0

where

a

i

∈

R

is a unary algebraic function on

R

, as

f

⁢

(

x

)

=

p

⁢

(

x

,

a

0

,

…

,

a

n

)

, where

p

is an

(

n

+

2

)

-ary polynomial on

R

given by

p

⁢

(

x

,

x

0

,

…

,

x

n

)

=

x

n

⁢

x

n

+

⋯

+

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).

Title

algebraic function

Canonical name

AlgebraicFunction

Date of creation

2013-03-22 15:19:24

Last modified on

2013-03-22 15:19:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A40

Classification

msc 26A09

Related topic

ElementaryFunction

Related topic

PropertiesOfEntireFunctions

Related topic

PolynomialsInAlgebraicSystems

Defines

transcendental function

Defines

transcendental

Defines

algebraic curve

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