Dept. of Electrical Engineering, Linkoping University S-581 83 Linkoping, Sweden

Localization and Base Change Techniques in Computational Algebra

Krister Forsman

Dept. of Electrical Engineering, Linkoping University S-581 83 Linkoping, Sweden

email:

1993-02-04

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Localization and Base Change Techniques in Computational Algebra

Krister Forsman

Dept. of Electrical Engineering

Linkoping University, S-581 83 Linkoping, Sweden email:

1993-02-04

Abstract.

This report collects some useful results from commutative algebra and describes how they can be used to decrease the complexity in Grobner base computations. As a byproduct we see how Grobner bases relate to characteristic sets. The algorithms presented are well known but have, as far as the author knows not been collected in one place earlier.

Keywords:

Grobner bases, elimination, commutative algebra, localization, linear algebra, remainders, characteristic sets, zero-dimensional ideals.

1 Introduction and Terminology

The most important methods in elimination theory are resultants, Grobner bases (GB) and characteristic sets (CS).

Resultants are described in most classical textbooks on commutative algebra: 17, 20, 25, 32].

Some references for GB are 3, 10, 14, 15, 16, 29, 33].

Characteristic sets are discussed in e.g. 5, 6, 13, 18, 22, 31, 34].

Grobner bases are occasionally called standard bases, though this terminology does not seem to be historically correct the original reference for standard bases 19], treats a slightly dierent case.

It is the aim of this report to collect some useful results from commutative algebra and describe how they can be used to decrease complexity in GB computations. As a byproduct we will see how GB and CS are related. The reader is supposed to have some minimal background in commutative algebra and Grobner basis theory.

The following terminology and notation is used:

A

=



n ] where

is an arbitrary eld.

F

i =

(

n

i

n )

n

i ]. This makes

d an

-dimensional ring.

Capitals (

n ) are used for free variables, whereas lowercase letters denote vari-

ables subject to relations

i

.

The ideal generated by

m

is written

m

.

The leading monomial of a polynomial

is written lm

. If

=

_



 where

are

N

n -vectors we have

deg

= max





= 0

(1)

and lm

=

^p .

For an integral domain

its eld of fractions is written

(

). For example we have

F

n =

(

).

plex stands for purely lexicographic term-ordering.

The concept of ranking gives us a classication of all plex term-orderings on

.

Denition 1.1 A ranking of the variables

n is simply a permutation of these symbols. If

precedes

in the permutation we say that

has a lower rank than

, written

a<b

. If

and

are two sets of variables and any element of

has lower rank than all

elements of

we write

.

The ranking can be thought of as a reordering of the entries in the exponent vector if

is an element of the permutation group

n then

X

r

r

r

n

 =

_r ^

_r ^

_n

(2) Thus the \standard" ranking is

n .

Denition 1.2 We say that

is regular w.r.t.

i if lm



i ] using plex and ranking

X

i the highest.

Another word for regular is monic , but be careful not to confuse this with the property of having leading coecient 1. Consider

p

=

2

+ 5

1 (3)

using plex and the ranking

. Then lm

=

, so

is not regular w.r.t.

, but the leading coecient of

is 1.

Denition 1.3 If

Spec

then we dene the dimension of

to be dim

= trdeg k

(

)

2

Note that this denition covers the case

i too, since we have no requirements on

.

Denition 1.4 ^A

-dimensional ideal

Spec

is in weak noetherian position if

p\k



n

d

n ] = 0

p

is in noetherian position (n.p.) if it is in weak noetherian position and

is an integral

k



n

d

n ]-algebra.

To put an ideal in weak noetherian position one only has to renumber the variables. Any prime ideal can be put in noetherian position by a linear change of variables this is known as Noether's normalization lemma 1, 23, 28].

2 Some Facts about Grobner Bases

We will assume that Grobner bases are minimal, i.e. reduced. With this requirement and the convention that the elements of the GB have leading coecient 1 (the coecient ring is a eld) Grobner bases are unique 16]. We write

(

) for the Grobner base of the ideal

aA

, making an implicit agreement on the term-ordering and that GB are always reduced.

Theorem 2.1 Let

be an ideal in

. dim

= 0 i for all

there is a

(

) such that

is regular w.r.t.

i .

Proof. See 16] or 3, page 209].

This implies that

is an integral

-algebra. The theorem is true independently on what term ordering is used. Note that there may still be more than

polynomials in the GB:

Example 2.1 ^{The set}

is a GB of a zero-dimensional ideal in



].

2

For prime ideals the situation is simpler, though:

Theorem 2.2 A plex GB for a prime zero-dimensional ideal in



n ] has

ele- ments.

Proof. See 16].

In 16] it is also proved that generically the

1 rst elements of the GB are linear in the leading variable. Thus the generic look of a plex-GB for a zero-dimensional prime ideal w.r.t.

n

is

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

: X

1

;p

1

(

n )

X

2

;p

2

(

n ) ...

X

n

n

(

n )

p

n (

n )

(4)

where

:

i



n ] and deg

n

deg

i for

= 1

1. Note that the way this theorem is stated in 16] the GB is not reduced.

An (arbitrary) ideal that has a GB of the type (4) is said to be in generic position or general position. Thus a zero-dimensional ideal is in generic position if for two dierent zeroes (

n ) and (

n ) we have

n

=

n . In 16] it is proved that almost all variable changes of the type

X

i

i

n

n + ⁿ

1

c

i

i (5)

where all

i

, puts a zero-dimensional prime ideal in generic position. In fact, 15] show

that any zero-dimensional radical ideal is put in generic position by a generic morphism of

the type (5). It is clear that with the above denition of genericity

a homogeneous ideal cannot be in generic position unless all elements of the GB are linear.

it depends on the ranking used whether an ideal is in generic position. E.g.

is in generic position, but

is not.

3 Change of Term Ordering Using Linear Algebra

The main disadvantage with Grobner bases is that their computational complexity is very high, in general. It has been showed that the complexity for computing a GB is lower for total degree orderings, e.g. revlex, than for a plex ordering 21, 24, 26, 27].

There is an approach to elimination theory that is also based on Grobner bases, but does not use the plex term ordering. This approach is primarily due to Boege, Gebauer and Kredel 2] it is also discussed in e.g. 3, 15]. The algorithm, let us call it the BGK algorithm, works for the special case when the ideal is zero-dimensional and it gives the univariate polynomial that generates the contraction of

to



i ] for some

i . We call this polynomial the minimal polynomial of

i , since we think of

(

) as an algebraic extension of

, even though

does not have to be prime, i.e. maximal. (In fact

does not even have to be zero-dimensional see below.) The idea is rst to compute a Grobner base G ^{w.r.t. a}

revlex ordering and then compute remainders

R

d = rem(

_di

G ) (6)

for successive

:s. Each

d is considered as an element in the vector space



n ] d . Sooner or later these expressions for

d will be linearly dependent over

, since the ideal has dimension zero, and this linear dependency relation gives us the minimal polynomial for

i . The reason for using the approach described above is thus that it is typically more ecient than computing an entire plex GB see 2]. The BGK algorithm is part of the Grobner base package in Maple 4], available under the name

.

It turns out that the ideas above can be generalized to the following result: using linear algebra techniques it is possible to nd a Grobner base w.r.t. an arbitrary term-ordering given a GB w.r.t. any other term-ordering. This is known as the Faugere-Gianni-Lazard- Mora algorithm 7].

If the ideal considered is in generic position the minimal polynomial obtained from the BGK algorithm tells us

1. how many zeroes there are, 2. the multiplicity of the zeroes, 3. whether the zeroes are real or not.

These questions can be of practical importance e.g. when dealing with problems in real algebraic geometry, see e.g. 11].

Via Bezout's theorem we get a sucient condition for an ideal to be in generic position:

Theorem 3.1 Let

be a zero-dimensional ideal in

. Furthermore, let

be the product

of the total degrees of the polynomials of a generating set for the ideal

(

depends on the

particular set of generators) and

the degree of the minimal polynomial for

n . If

=

then

is in generic position.

Proof. The number of complex zeroes, counting multiplicity, is bounded from above by

according to the ane version of Bezout's theorem 12, page 223]. This means that

with equality i the ideal has

zeroes, all of which dier in the coordinate

n , i.e.

is in

generic position.

Theorem 3.1 is often dicult to use directly, since Bezout's theorem only states an inequal- ity. If the original set of generators for the ideal is made (auto-)reduced, chances increase that equality occurs (remember that the number

depends on what generators that are chosen).

It should be noted that the BGK algorithm might terminate even if the ideal in question is not zero-dimensional. The condition



i ]

= 0 is necessary and sucient for termination

a simple example of this is



]. More involved, application oriented examples can be found in 11] or 9, chapter 6].

4 Some Theorems on Localization

An apparent disadvantage with the BGK algorithm is that a (sucient) condition for it to terminate is that the ideal in question be zero-dimensional. However, if the ideal is not zero-dimensional we can in some cases change the coecient eld by localization so that the ideal becomes zero-dimensional and then apply BGK. Before doing this we have to know something about what happens to ideals under localization.

Theorem 4.1 Let

be a commutative ring and

a multiplicative system in

. There is a one-to-one correspondence between the prime ideals of

that do not intersect

and the prime ideals of

(the localization of

at

).

Proof. See 1, proposition 3.11, iv] or 35, chapter IV,

10, theorem 15].

. Thus if we consider extension and contraction

we have that

8p2

Spec

:

=

^ec =

(7) In the applications considered here

will be

and

=



n

d

n ]

0

for some

. Thus

=

d . The reasons for extending ideals to

d are two:

computing in

d is cheaper than computing in

.

the ideals considered may become zero-dimensional in

so that the BGK algorithm can be used.

Theorem 4.2 ^Let

Spec

be

-dimensional and in weak noetherian position. If

^e is the extension of

to

d then dim

^e = 0 .

Proof.

^e

Spec

d according to theorem 4.1. But dim

=

implies that

n

d

are algebraic over



n

d

n ]. Thus

dim

^e = trdeg _k

_X

_:::X

(

d

e ) = 0 (8)

2

Many times the ideals involved are not prime, but we can still use the localization tech-

nique as long as we are a little careful. The following theorem explains exactly how the ideals

of

relate to those of

.

Theorem 4.3 Let

be an ideal in

and consider extension and contraction

. We have

a

ec =

s

S (

:

)

Proof. See 1, proposition 3.11, ii].

5 Characteristic Sets

The reason that there are two apparently dierent methods in elimination theory is that reduction can be made in dierent ways. Buchberger's algorithm uses ordinary polynomial division, whereas Ritt's algorithm for computing CS 31] relies on pseudodivision. For details, see the references mentioned in the introduction. It is sometimes claimed that characteristic sets are more ecient than GB from a computational point of view, but it is questionable if this is still true when the \tricks" described above (sections 3 and 4) are used before applying Buchberger's algorithm.

In fact, in this section we will see that in the case of a prime ideal the dierence between CS and GB is not very large. We will use the following theorem:

Theorem 5.1 Let

be a

-dimensional prime ideal and G a plex GB for

w.r.t. the ranking

n . If

is in n.p. then G =

n

d

where

i is regular w.r.t.

i

for all

.

Proof. This is proposition 5.9 in 16].

Furthermore we need a simple fact on reduction:

Lemma 5.1 If

and

is regular w.r.t. its leading variable then

is reduced w.r.t.

in the sense of Ritt i it is reduced w.r.t. g using the standard denition in Gr obner base theory.

Proof. Immediate from the denitions.

Theorem 5.2 Suppose that

is a

-dimensional prime ideal of

in noetherian position.

If G ⁼

⁽

⁾ w.r.t. a plex term-ordering then then G is a CS for

.

Proof. G is a subset of

satisfying the denition of CS according to theorem 5.1 and

lemma 5.1.

Theorem 5.3 Suppose that

is a

-dimensional prime ideal of

in weak noetherian position and that

^e is the extension of

to

_d . If G =

(

^e ) w.r.t. a plex term-ordering and G

is obtained from G by clearing denominators, then G

is a CS for

.

Proof. Since

^e is zero-dimensional (theorem 4.2) G =

n

d

and

i are regular w.r.t.

i . This means that the denominators are polynomials in



n

d

n ]. It is clear that there is no nonzero polynomial

in

such that deg(

i )

deg(

i

i ), so G

forms a CS for

.

Note that G

is not necessarily a GB for

:

Example 5.1 Let

=

be an ideal in



]. Clearly

is prime. We have that

F =

(

) =

(9) So

is in weak noetherian position. If

^e =

then

G =

(

^e ) =

X

3 X

2

 X 2

2

; X

2

3

X

4

g

(10)

so that

G

=

= F

2

In 5, chapter 5] it is explained how a GB can be obtained from a CS using the Rabinovich trick 30, 8].

Acknowledgement

This work was nancially supported by the Swedish Council for Technical Research (TFR).

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