Gröbner basis

Gröbner basis

Definition

of monomial orderings and

support

:

Let

F

be a field, and let

S

be the set of

monomials

in

F

⁢

[

x

1

,

…

,

x

n

]

, the polynomial ring in

n

indeterminates

.
A

monomial ordering

is a

total ordering

≤

on

S

which satisfies

1.

a

≤

b

implies that

a

⁢

c

≤

b

⁢

c

for all

a

,

b

,

c

∈

S

.

2.

1

≤

a

for all

a

∈

S

.

In practice, for any

a

=

x

1

a

1

⁢

x

2

a

2

⁢

⋯

⁢

x

n

a

n

∈

F

⁢

[

x

1

,

…

,

x

n

]

, we

associate

to

a

the string

(

a

1

,

a

2

,

…

,

a

n

)

and compare monomials by looking at

orderings

on these

n

-tuples.

Example.

An extremely prevalent example of a monomial ordering is given by the standard lexicographical ordering of strings. Other examples include graded

lexicographic ordering

and graded reverse lexicographic ordering.

Henceforth, assume that we have fixed a monomial ordering. Define the

support

of

a

, denoted

supp

⁡

(

a

)

, to be the set of terms

x

i

with

a

i

≠

0

. Then define

M

⁢

(

a

)

=

max

⁡

(

supp

⁡

(

a

)

)

.

A

partial order

on

F

⁢

[

x

1

,

…

,

x

n

]

:

We can extend our monomial ordering to a partial ordering on

F

⁢

[

x

1

,

…

,

x

n

]

as follows:
Let

a

,

b

∈

F

⁢

[

x

1

,

…

,

x

n

]

. If

supp

⁡

(

a

)

≠

supp

⁡

(

b

)

, we say that

a

<

b

if

max

⁡

(

supp

⁡

(

a

)

-

supp

⁡

(

b

)

)

<

max

⁡

(

supp

⁡

(

b

)

-

supp

⁡

(

a

)

)

.

It can be shown that:

1.

The relation defined above is indeed a partial order on

F

⁢

[

x

1

,

…

,

x

n

]

2.

Every descending chain

p

1

⁢

(

x

1

,

…

,

x

n

)

>

p

2

⁢

(

x

1

,

…

,

x

n

)

>

…

with

p

i

∈

[

x

1

,

…

,

x

n

]

is finite.

A

division algorithm

for

F

⁢

[

x

1

,

…

,

x

n

]

:

We can then formulate a division algorithm for

F

⁢

[

x

1

,

…

,

x

n

]

:

Let

(

f

1

,

…

,

f

s

)

be an ordered

s

-tuple of polynomials, with

f

i

∈

F

⁢

[

x

1

,

…

,

x

n

]

. Then for each

f

∈

F

⁢

[

x

1

,

…

,

x

n

]

, there exist

a

1

,

…

,

a

s

,

r

∈

F

⁢

[

x

1

,

…

,

x

n

]

with

r

unique, such that

1.

f

=

a

1

⁢

f

1

+

⋯

+

a

s

⁢

f

s

+

r

2.

For each

i

=

1

,

…

,

s

,

M

⁢

(

a

i

)

does not divide any monomial in

supp

⁡

(

r

)

.

Furthermore, if

a

i

⁢

f

i

≠

0

for some

i

, then

M

⁢

(

a

i

⁢

f

i

)

≤

M

⁢

(

f

)

.

Definition of Gröbner basis

:

Let

I

be a nonzero ideal of

F

⁢

[

x

1

,

…

,

x

n

]

. A finite set

T

⊂

I

of polynomials is a

Gröbner basis

for

I

if for all

b

∈

I

with

b

≠

0

there exists

p

∈

T

such that

M

(

p

)

∣

M

(

b

)

.

Existence of Gröbner bases

:

Every ideal

I

⊂

k

⁢

[

x

1

,

…

,

x

n

]

other than the

zero ideal

has a Gröbner basis. Additionally, any Gröbner basis for

I

is also a basis of

I

.

Title

Gröbner basis

Canonical

name

GrobnerBasis

Date of creation

2013-03-22 13:03:47

Last modified on

2013-03-22 13:03:47

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

28

Author

mathcam (2727)

Entry type

Definition

Classification

msc 13P10

Defines

monomial ordering

Defines

Gröbner basis

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