aarjaneiro/planetmath huggingface dataset:
factoring all-one polynomials using the grouping method

factoring all-one polynomials using the grouping method

The method of grouping terms can be used to factor

all-one polynomials

, i.e.

polynomials

of the form

∑

m

=

0

n

-

1

x

m

when

n

is

composite

. (When

n

is prime, these polynomials are

irreducible

, so there is nothing to do in that case.)

Let us consider a few examples:

n

=

4

:

1

+

x

+

x

2

+

x

3

=

(

1

+

x

)

+

(

x

2

+

x

3

)

=

(

1

+

x

)

+

x

2

⁢

(

1

+

x

)

=

(

1

+

x

)

⁢

(

1

+

x

2

)

n

=

6

:

1

+

x

+

x

2

+

x

3

+

x

4

+

x

5

=

(

1

+

x

+

x

2

)

+

(

x

3

+

x

4

+

x

5

)

=

(

1

+

x

+

x

2

)

+

x

3

⁢

(

1

+

x

+

x

2

)

=

(

1

+

x

3

)

⁢

(

1

+

x

+

x

2

)

n

=

8

:

1

+

x

+

x

2

+

x

3

+

x

4

+

x

5

+

x

6

+

x

7

=

(

1

+

x

+

x

2

+

x

3

)

+

(

x

4

+

(

x

5

+

x

6

+

x

7

)

=

(

1

+

x

+

x

2

+

x

3

)

+

x

4

⁢

(

1

+

x

+

x

2

+

x

3

)

=

(

1

+

x

4

)

⁢

(

1

+

x

+

x

2

+

x

3

)

Combining this result with the factorization we have for the case

n

=

4

, we obtain the following:

1

+

x

+

x

2

+

x

3

+

x

4

+

x

5

+

x

6

+

x

7

=

(

1

+

x

)

⁢

(

1

+

x

2

)

⁢

(

1

+

x

4

)

n

=

9

:

1

+

x

+

x

2

+

x

3

+

x

4

+

x

5

+

x

6

+

x

7

+

x

8

=

(

1

+

x

+

x

2

)

+

(

x

3

+

x

4

+

x

5

)

+

(

x

6

+

x

7

+

x

8

)

=

(

1

+

x

+

x

2

)

+

x

3

⁢

(

1

+

x

+

x

2

)

+

x

6

⁢

(

1

+

x

+

x

2

)

=

(

1

+

x

+

x

2

)

⁢

(

1

+

x

3

+

x

6

)

n

=

12

:

1

+

x

+

x

2

+

x

3

+

x

4

+

x

5

+

x

6

+

x

7

+

x

8

+

x

9

+

x

10

+

x

11

=

(

1

+

x

+

x

2

)

+

(

x

3

+

x

4

+

x

5

)

+

(

x

6

+

x

7

+

x

8

)

+

(

x

9

+

x

10

+

x

11

)

=

(

1

+

x

+

x

2

)

+

x

3

⁢

(

1

+

x

+

x

2

)

+

x

6

⁢

(

1

+

x

+

x

2

)

+

x

9

⁢

(

1

+

x

+

x

2

)

=

(

1

+

x

+

x

2

)

⁢

(

1

+

x

3

+

x

6

+

x

9

)

=

(

1

+

x

+

x

2

)

⁢

(

(

1

+

x

3

)

+

(

x

6

+

x

9

)

)

=

(

1

+

x

+

x

2

)

⁢

(

(

1

+

x

3

)

+

x

6

⁢

(

1

+

x

3

)

)

=

(

1

+

x

+

x

2

)

⁢

(

1

+

x

3

)

⁢

(

1

+

x

6

)

It might be worth pointing out that the polynomials produced by this factorization are not all irreducible. For instance,

1

+

x

3

=

(

1

+

x

)

⁢

(

1

-

x

+

x

2

)

.

However, to obtain this factorization, one needs to use some techique other than the

grouping method

. Likewise. the polynomial

1

+

x

6

is also

reducible

.

Title

factoring

all-one polynomials using the grouping method

Canonical name

FactoringAllonePolynomialsUsingTheGroupingMethod

Date of creation

2013-03-22 15:06:52

Last modified on

2013-03-22 15:06:52

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Example

Classification

msc 13P05

Related topic

AllOnePolynomial

Related topic

CyclotomicPolynomial

Generated on Fri Feb 9 17:35:04 2018 by

LaTeXML
grouping method for factoring polynomials

grouping method for factoring polynomials

Factoring

a given

polynomial

may in certain special cases by using the following

grouping method

:

1.

.

2.

Factorize the separately.

3.

The whole polynomial may then possibly be written in form of a

product

.

Examples

a)

x

3

-

x

2

-

x

+

1

=

{

x

3

-

x

2

}

+

{

-

x

+

1

}

=

x

2

⁢

(

x

-

1

)

-

1

⁢

(

x

-

1

)

=

(

x

-

1

)

⁢

(

x

2

-

1

)

=

(

x

-

1

)

2

⁢

(

x

+

1

)

b)

x

4

+

3

⁢

x

3

-

3

⁢

x

-

1

=

{

x

4

-

1

}

+

{

3

⁢

x

3

-

3

⁢

x

}

=

(

x

2

+

1

)

⁢

(

x

2

-

1

)

+

3

⁢

x

⁢

(

x

2

-

1

)

=

(

x

2

-

1

)

⁢

(

x

2

+

1

+

3

⁢

x

)

=

(

x

-

1

)

⁢

(

x

+

1

)

⁢

(

x

2

+

3

⁢

x

+

1

)

c)

x

4

+

4

=

{

x

4

+

4

⁢

x

2

+

4

}

-

4

⁢

x

2

=

(

x

2

+

2

)

2

-

(

2

⁢

x

)

2

=

(

x

2

+

2

+

2

⁢

x

)

⁢

(

x

2

+

2

-

2

⁢

x

)

=

(

x

2

+

2

⁢

x

+

2

)

⁢

(

x

2

-

2

⁢

x

+

2

)

d)

x

4

+

x

2

+

1

=

{

x

4

+

2

⁢

x

2

+

1

}

-

x

2

=

(

x

2

+

1

)

2

-

x

2

=

(

x

2

+

1

+

x

)

⁢

(

x

2

+

1

-

x

)

=

(

x

2

+

x

+

1

)

⁢

(

x

2

-

x

+

1

)

The

trinomials

x

2

+

3

⁢

x

+

1

,

x

2

±

2

⁢

x

+

2

and

x

2

±

x

+

1

are

irreducible polynomials

.

Title

grouping method for factoring polynomials

Canonical name

GroupingMethodForFactoringPolynomials

Date of creation

2013-03-22 15:06:49

Last modified on

2013-03-22 15:06:49

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Algorithm

Classification

msc 13P05

Related topic

DifferenceOfSquares

Related topic

ExampleOfGcd

Related topic

ZeroRuleOfProduct

Defines

grouping method

Generated on Fri Feb 9 17:35:11 2018 by

LaTeXML
polynomial ring over integral domain

polynomial ring over integral domain

Theorem

.

If the

coefficient ring

R

is an

integral domain

, then so is also its

polynomial ring

R

⁢

[

X

]

.

Proof.

Let

f

⁢

(

X

)

and

g

⁢

(

X

)

be two non-zero

polynomials

in

R

⁢

[

X

]

and let

a

f

and

b

g

be their leading coefficients, respectively.  Thus

a

f

≠

0

,

b

g

≠

0

,  and because

R

has no

zero divisors

,

a

f

⁢

b

g

≠

0

.  But the

product

a

f

⁢

b

g

is the leading coefficient of

f

⁢

(

X

)

⁢

g

⁢

(

X

)

and so

f

⁢

(

X

)

⁢

g

⁢

(

X

)

cannot be the

zero polynomial

.  Consequently,

R

⁢

[

X

]

has no zero divisors, Q.E.D.

Remark.

The theorem may by induction be generalized for the polynomial ring

R

⁢

[

X

1

,

X

2

,

…

,

X

n

]

.

Title

polynomial ring over integral domain

Canonical name

PolynomialRingOverIntegralDomain

Date of creation

2013-03-22 15:10:06

Last modified on

2013-03-22 15:10:06

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13P05

Related topic

RingAdjunction

Related topic

FormalPowerSeries

Related topic

ZeroPolynomial2

Related topic

PolynomialRingOverFieldIsEuclideanDomain

Defines

coefficient ring

Generated on Fri Feb 9 17:39:13 2018 by

LaTeXML
polynomial ring which is PID

polynomial ring which is PID

Theorem.

If a

polynomial ring

D

⁢

[

X

]

over an

integral domain

D

is a

principal ideal domain

, then

coefficient ring

D

is a field. (Cf. the corollary 4 in the entry polynomial ring over a field.)

Proof.

Let

a

be any non-zero element of

D

.  Then the ideal

(

a

,

X

)

of

D

⁢

[

X

]

is a

principal ideal

(

f

⁢

(

X

)

)

with

f

⁢

(

X

)

a non-zero

polynomial

(

http://planetmath.org/ZeroPolynomial2

).  Therefore,

a

=

f

⁢

(

X

)

⁢

g

⁢

(

X

)

,

X

=

f

⁢

(

X

)

⁢

h

⁢

(

X

)

with

g

⁢

(

X

)

and

h

⁢

(

X

)

certain polynomials in

D

⁢

[

X

]

.  From these

equations

one infers that

f

⁢

(

X

)

is a polynomial

c

and

h

⁢

(

X

)

is a first degree polynomial

b

0

+

b

1

⁢

X

(

b

1

≠

0

).  Thus we obtain the equation

c

⁢

b

0

+

c

⁢

b

1

⁢

X

=

X

,

which shows that

c

⁢

b

1

is the unity 1 of

D

.  Thus

c

=

f

⁢

(

X

)

is a unit of

D

, whence

(

a

,

X

)

=

(

f

⁢

(

X

)

)

=

(

1

)

=

D

⁢

[

X

]

.

So we can write

1

=

a

⋅

u

⁢

(

X

)

+

X

⋅

v

⁢

(

X

)

,

where

u

⁢

(

X

)

,

v

⁢

(

X

)

∈

D

⁢

[

X

]

.  This equation cannot be possible without that

a

times the constant term of

u

⁢

(

X

)

is the unity.  Accordingly,

a

has a

multiplicative inverse

in

D

.  Because

a

was arbitrary non-zero elenent of the integral domain

D

,

D

is a field.

References

1

David M. Burton

:

A first course in rings and ideals

. Addison-Wesley Publishing Company. Reading, Menlo Park, London, Don Mills (1970).

Title

polynomial ring which is PID

Canonical name

PolynomialRingWhichIsPID

Date of creation

2013-03-22 17:53:04

Last modified on

2013-03-22 17:53:04

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13P05

Related topic

PolynomialRingOverFieldIsEuclideanDomain

Generated on Fri Feb 9 17:39:15 2018 by

LaTeXML
topics on polynomials

topics on polynomials

Definitions

1.

polynomial ring

2.

derivative of polynomial

3.

polynomial ring over integral domain

4.

polynomial ring over a field

5.

polynomial ring which is PID

6.

skew polynomial ring

7.

q

skew polynomial ring

8.

factor theorem

9.

order and degree of polynomial

10.

length of a

polynomial

11.

polynomial growth

Polynomials over

ℝ

1.

polynomial long division

2.

polynomial interpolation

,

Lagrange interpolation formula

3.

Weierstrass approximation theorem

4.

Sturm’s

theorem

5.

properties of quadratic equation

6.

cubic formula

7.

quartic formula

8.

Abel’s theorem

9.

algebraic equation

10.

square root of polynomial

11.

casus irreducibilis

12.

conditional congruence

Roots of polynomials

1.

zero of polynomial

2.

fundamental theorem of algebra

3.

continuous

dependence on

coefficients

4.

Tchirnhaus transformations

5.

rational root theorem

6.

polynomial equation of odd degree

7.

Viete formulas

8.

Descartes’ rule of signs

9.

Eisenstein criterion

10.

zeros of polynomial derivative

11.

homogeneous equation

Polynomials with special properties

1.

zero polynomial

2.

opposite polynomial

3.

symmetric polynomial

4.

monic polynomial

5.

minimal polynomial

6.

all one polynomial

7.

prime factors

of

x

n

-

1

(

http://planetmath.org/PrimeFactorsOfXn1

)

8.

primitive polynomial

9.

homogeneous polynomial

10.

weighted homogeneous polynomial

11.

irreducible polynomial

12.

irreducibility of binomials with unity coefficients

13.

reciprocal polynomial

14.

orthogonal polynomials

Classical orthogonal polynomials

1.

Chebyshev polynomials

2.

Hermite polynomials

3.

Laguerre polynomials

4.

Legendre polynomials

5.

Jacobi polynomials

6.

properties of orthogonal polynomials

Other polynomial families

1.

Bernoulli polynomials and numbers

2.

Bernoulli polynomials

3.

Bernstein polynomials

4.

http://planetmath.org/node/12200

Boubaker polynomials

5.

Euler polynomials

6.

Lamé polynomials

Miscellaneous applications

1.

Taylor polynomial

2.

Characteristic polynomial

in

linear algebra

(see also

Cayley-Hamilton theorem

)

Related concepts

1.

division algorithm

for polynomials

2.

since sets of polynomial are rings,

commutative algebra

concepts

like PID, UFD and

Euclidean domains

are heavily related

3.

Gröbner basis

Title

topics on polynomials

Canonical

name

TopicsOnPolynomials

Date of creation

2013-03-22 15:20:57

Last modified on

2013-03-22 15:20:57

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

31

Author

matte (1858)

Entry type

Topic

Classification

msc 13P05

Classification

msc 11C08

Classification

msc 12E05

Related topic

PropertiesOfOrthogonalPolynomials

Generated on Fri Feb 9 17:39:23 2018 by

LaTeXML
zero of polynomial

zero of polynomial

Let

R

be a

subring

of a

commutative ring

S

.  If

f

is a

polynomial

in

R

⁢

[

X

]

, it defines an

evaluation homomorphism

from

S

to

S

.  Any element

α

of

S

satisfying

f

⁢

(

α

)

=

0

is a

zero of the polynomial

f

.

If

R

also is equipped with a

non-zero unity

, then the polynomial

f

is in

S

⁢

[

X

]

divisible

by the binomial

X

-

α

(cf. the

factor theorem

).  In this case, if

f

is divisible by

(

X

-

α

)

n

but not by

(

X

-

α

)

n

+

1

, then

α

is a zero of the

order

n

of the polynomial

f

.  If this order is 1, then

α

is a

simple zero

of

f

.

For example, the

real number

2

(

∈

ℝ

) is a zero of the polynomial

X

2

-

2

of the polynomial ring

ℚ

⁢

[

X

]

.

Title

zero of polynomial

Canonical

name

ZeroOfPolynomial

Date of creation

2013-03-22 18:19:50

Last modified on

2013-03-22 18:19:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 13P05

Classification

msc 11C08

Classification

msc 12E05

Related topic

PolynomialFunction

Related topic

ZerosAndPolesOfRationalFunction

Defines

zero of polynomial

Defines

order of zero

Defines

order

Defines

simple zero

Defines

simple

Generated on Fri Feb 9 17:39:25 2018 by

LaTeXML
zero polynomial

zero polynomial

The

zero polynomial

in a ring

R

⁢

[

X

]

of

polynomials

over a ring

R

is the

identity element

0

of this

polynomial ring

:

f

+

𝟎

=

𝟎

+

f

=

f

∀

f

∈

R

⁢

[

X

]

So the zero polynomial is also the

absorbing element

for the multiplication of polynomials.

All

coefficients

of the zero polynomial are equal to 0, i.e.

𝟎

:=

(

0

,

0

,

0

,

…

)

.

Because always

f

⋅

𝟎

=

𝟎

and because in general

deg

⁡

(

f

⁢

g

)

=

deg

⁡

(

f

)

+

deg

⁡

(

g

)

when

R

has no

zero divisors

, one may define that that the zero polynomial has no degree (

http://planetmath.org/Polynomial

) at all, or alternatively that

deg

⁡

(

𝟎

)

=

-

∞

(see the

extended real numbers

).

Title

zero polynomial

Canonical name

ZeroPolynomial

Date of creation

2013-03-22 14:46:58

Last modified on

2013-03-22 14:46:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 13P05

Classification

msc 11C08

Classification

msc 12E05

Related topic

PolynomialRingOverIntegralDomain

Related topic

OrderAndDegreeOfPolynomial

Related topic

MinimalPolynomialEndomorphism

Generated on Fri Feb 9 17:39:27 2018 by

LaTeXML
derivation of Sylvester’s matrix for the resultant

derivation of Sylvester’s matrix for the resultant

Sylvester’s

matrix representation

of the

resultant

can be easily derived by thinking of

polynomials

as

linear equations

in powers of

x

. For ease of exposition, we shall consider the case where

p

is of

second order

and

q

is of third order but, once the basic idea has been grasped, it is

trivial

to extend it to polynomials of any orders whatsoever.

Let us start with the

polynomial equations

p

⁢

(

x

)

=

0

and

q

⁢

(

x

)

=

0

. Written out in full, they look like

a

0

⁢

x

2

+

a

1

⁢

x

+

a

2

=

0

b

0

⁢

x

3

+

b

1

⁢

x

2

+

b

2

⁢

x

+

b

3

=

0

These two

equations

can be combined into a single

matrix equation

:

(

0

a

0

a

1

a

2

b

0

b

1

b

2

b

3

)

⁢

(

x

3

x

2

x

1

)

=

(

0

0

0

0

)

Since

square matrices

enjoy properties which matrices of arbitrary size do not, we will add more equations so as to come up with a new matrix equation involving a square matrices. There is no harm in adding an equation of the forn

x

k

⁢

p

⁢

(

x

)

=

0

or

x

k

⁢

q

⁢

(

x

)

=

0

to the system

p

⁢

(

x

)

=

0

,

q

⁢

(

x

)

=

0

because the enlarged system will have exactly the same solutions as the original system of two equations. Consider the system

p

⁢

(

x

)

=

0

x

⁢

p

⁢

(

x

)

=

0

x

2

⁢

p

⁢

(

x

)

=

0

q

⁢

(

x

)

=

0

x

⁢

q

⁢

(

x

)

=

0

This system may be written as a matrix equation

(

0

0

a

0

a

1

a

2

0

a

0

a

1

a

2

0

a

0

a

1

a

2

0

0

0

b

0

b

1

b

2

b

3

b

0

b

1

b

2

b

3

0

)

⁢

(

x

4

x

3

x

2

x

1

)

=

(

0

0

0

0

0

)

Now we have a square matrix. One important property of matrix equations involving square matrices is that they only have non-trivial solutions when the

determinant

of the matrix vanishes. The system

p

⁢

(

x

)

=

0

,

q

⁢

(

x

)

=

0

only has a solution when

p

and

q

have a common root. Hence the determinant will vanish whenever

p

and

q

have a common root.

Note that, at this stage, we cannot jump to the

converse

conclusion

that

p

and

q

always have a common root when the determinant vanishes. All we can say is that, if the determinant vanishes, there will be some non-zero vector in the kernel of the matrix, but we cannot say that the vector will be of the special form

(

x

4

,

x

3

,

x

2

,

x

,

1

)

that appears in the system. To assert the converse conclusion, we need to first prove that the determinant indeed equals the resultant. For this proof, please see the entry proof that Sylvester’s determinant equals the resultant.

Title

derivation

of Sylvester’s matrix for the resultant

Canonical name

DerivationOfSylvestersMatrixForTheResultant

Date of creation

2013-03-22 14:36:47

Last modified on

2013-03-22 14:36:47

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Derivation

Classification

msc 13P10

Generated on Fri Feb 9 17:39:29 2018 by

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example of resultant (1)

example of resultant (1)

To illustrate the concept of resultant, consider a simple example. Let

p

⁢

(

x

)

=

x

2

-

1

=

(

x

+

1

)

⁢

(

x

-

1

)

q

⁢

(

x

)

=

x

3

-

4

⁢

x

=

(

x

+

2

)

⁢

x

⁢

(

x

-

2

)

Then, in the notation used in the main entry,

r

1

=

-

1

r

2

=

+

1

s

1

=

-

2

s

2

=

0

s

3

=

+

2

Hence,

R

⁢

(

p

,

q

)

=

(

-

1

-

(

-

2

)

)

⁢

(

-

1

-

0

)

⁢

(

-

1

-

2

)

⁢

(

1

-

(

-

2

)

)

⁢

(

1

-

0

)

⁢

(

1

-

2

)

=

1

×

(

-

1

)

×

(

-

3

)

×

3

×

1

×

(

-

1

)

=

-

9

In the notation of the main entry,

a

0

=

1

a

1

=

0

a

2

=

-

1

b

0

=

1

b

1

=

0

b

2

=

-

4

b

3

=

0

The

determinant

for computing the resultant is

|

1

0

-

1

0

0

0

1

0

-

1

0

0

0

1

0

-

1

1

0

-

4

0

0

0

1

0

-

4

0

|

Since the matrix is quite sparse, its determinant is easy to compute, especially if one first simplifies it by performing some

row operations

such as subtracting the first row from the fourth row and the second row form the fifth row to make it even sparser. The determinat works out to be

-

9

, in agreement with the earlier answer for the resultant.

Title

example of resultant (1)

Canonical name

ExampleOfResultant1

Date of creation

2013-03-22 14:36:33

Last modified on

2013-03-22 14:36:33

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Example

Classification

msc 13P10

Generated on Fri Feb 9 17:39:31 2018 by

LaTeXML
affine variety

affine variety

An

affine variety

over an algebraically closed field

k

is a subset of some

affine space

k

n

over

k

which can be described as the

vanishing set

of finitely many

polynomials

in

n

variables with coefficients in

k

, and which cannot be written as the union of two smaller such sets.

For example, the locus described by

Y

-

X

2

=

0

as a subset of

ℂ

2

is an affine variety over the

complex numbers

. But the locus described by

Y

⁢

X

=

0

is not (as it is the union of the loci

X

=

0

and

Y

=

0

).

One can define a subset of affine space

k

n

or an affine variety in

k

n

to be

closed

if it is a subset defined by the vanishing set of finitely many polynomials in

n

variables with coefficients in

k

. The closed subsets then actually satisfy the requirements for closed sets in a topology, so this defines a topology on the affine variety known as the

Zariski topology

. The definition above has the extra condition that an affine variety not be the union of two closed subsets, i.e. it is required to an irreducible topological space in the Zariski topology. Anything then satisfying the definition without possibly the irreducibility is known as an (affine)

algebraic set

.

A

quasi-affine variety

is then an open set (in the Zariski topology) of an affine variety.

Note that some geometers do not require what they call a

variety

to be

irreducible

, that is they call algebraic sets varieties. The most prevalent definition however requires varieties to be irreducible.

References

1

Hartshorne, Robin.

Algebraic Geometry

, Graduate Texts in Mathematics, Number 52. Springer-Verlag, New York, NY. 1977

Title

affine variety

Canonical name

AffineVariety

Date of creation

2013-03-22 12:03:55

Last modified on

2013-03-22 12:03:55

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14-00

Related topic

ProjectiveVariety

Related topic

scheme

Related topic

AffineScheme

Related topic

AlgebraicGeometry

Related topic

AffineSpace3

Related topic

Variety

Defines

quasi-affine variety

Defines

closed

Generated on Fri Feb 9 17:55:42 2018 by

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analytic algebraic function

analytic algebraic function

Let

k

be a field, and let

k

⁢

{

x

1

,

…

,

x

n

}

be the ring of convergent
power series in

n

variables. An element in this ring can be thought of as
a function defined in a

neighbourhood

of the origin in

k

n

to

k

. The most common cases for

k

are

ℂ

or

ℝ

, where the convergence is with respect to the standard

euclidean metric

. These definitions can also be generalized to other fields.

Definition

.

A function

f

∈

k

⁢

{

x

1

,

…

,

x

n

}

is said to be

k

-analytic
algebraic

if there exists a nontrivial polynomial

p

∈

k

⁢

[

x

1

,

…

,

x

n

,

y

]

such that

p

⁢

(

x

,

f

⁢

(

x

)

)

≡

0

for all

x

in a
neighbourhood of the origin in

k

n

.
If

k

=

ℂ

then

f

is said to be

holomorphic algebraic

and if

k

=

ℝ

then

f

is said to be

real-analytic algebraic

or a

Nash function

.

The same definition applies near any other point other then the origin by just translation.

Definition

.

A mapping

f

:

U

⊂

k

n

→

k

m

where

U

is a neighbourhood of the origin is said to be

k

-analytic algebraic if each component function is analytic algebraic.

References

1

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.

Title

analytic algebraic function

Canonical name

AnalyticAlgebraicFunction

Date of creation

2013-03-22 15:36:05

Last modified on

2013-03-22 15:36:05

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 14-00

Classification

msc 14P20

Synonym

k

-analytic algebraic function

Synonym

analytic algebraic

Defines

holomorphic algebraic function

Defines

real-analytic algebraic function

Defines

Nash function

Defines

analytic algebraic mapping

Generated on Fri Feb 9 17:55:47 2018 by

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