Involutive Divisions and Monomial Orderings I

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ISSN 0361-7688, Programming and Computer Software, 2007, Vol. 33, No. 3, pp. 139–146. © Pleiades Publishing, Ltd., 2007.Original Russian Text © A.S. Semenov, P.A. Zyuzikov, 2007, published in Programmirovanie, 2007, Vol. 33, No. 3.

139

1. INTRODUCTION

In papers [2–4], a general algorithmic approach toconstruction of involutive bases, which are Gröbnerbases of a special form, has been developed. Thisapproach became one of the most efficient and widelyused tools for determination of standard bases of poly-nomial ideals. In turn, the theory of polynomial involu-tive bases relies on the theory of involutive divisions onmonomial sets.

In the course of investigation of involutive divisions,it has been found that, in order that an algorithm forconstructing involutive bases operate efficiently, theinvolutive division should possess some special proper-ties, namely, continuity and constructivity [1, 3, 4].

In addition, the algorithmic efficiency of the determina-tion of the set of multiplicative variables and of the fastsearch for the involutive divisor is important. This isdiscussed in [1] (pairwise property of involutive divi-sions) and [5] (fast search for the Janet divisor with thehelp of a tree).

In paper [1], a class of pairwise involutive divisionswas introduced. Its study was continued in paper [6].A relationship between the pairwise property [1] andthe filter axiom was established. A procedure for con-struction of pairwise continuous involutive divisions bythe method of pairwise closure was proposed. Never-theless, the pairwise closure procedure covers only the

case of involutive divisions without embedded cones.This approach seems to be incomplete, because, in thecourse of calculation of the minimal involutive basis bythe TQ

algorithm [1, 4], the set of leading monomials of the elements of the involutive basis T

being formed isinvolutively autoreduced; hence, it makes sense to con-sider involutive divisions that may contain embeddedcones from the algorithmic point of view.

In this paper, we consider a class of

-divisions,which was introduced in [6], and introduce a new class

of involutive

-divisions, which extend the induced

divisions discussed in [1]. We prove continuity andconstructivity of the

-divisions and give a simpleexample of calculation of an involutive basis. A numberof properties of the

-divisions are established.

Taking into account that the Janet division is a

-division associated with the lexicographical order-ing, we consider its “antipode” in the class of

-divisions,namely, the

-division associated with same ordering.

2. THEORY OF INVOLUTIVE DIVISIONS

By

, we denote the set of nonnegative integers.

Then,

= { …

|

d

i

∈

} is the set of all possible

monomials in n

variables.By deg(

u

) and deg

i

(

u

), we denote the total degree of monomial u

and the degree of u

with respect to variable

x

i

, respectively. For the least common multiple and thegreatest common divisor of two monomials u

and v

, weuse the notation lcm(

u

, v

) and gcd(

u

, v

).

Consider an arbitrary finite set of monomials U

withpairwise different elements. We say that an involutivedivision

L

is specified on U

if, for any u

∈

U

, a sub-monoid L

(

u

, U

) is defined in

such that the followingaxioms hold [3]:

• If w

∈

L

(

u

, U

) and v

|

w

, then v

∈

L

(

u

, U

).

• If u

, v

∈

U

and uL

(

u

, U

) ∩

v

L

(

v

, U

) ≠

, then

u

∈

v

L

(

v

, U

) or v

∈

uL

(

u

, U

).

• If v

∈

U

and v

∈

uL

(

u

, U

), then L

(

v

, U

) ⊆

L

(

u

, U

).

• If U

⊆

V , then ∀u ∈ U L(u, V ) ⊆ L(u, U ).

The last axiom is called the filter axiom.

Elements of L(u, U ) are multiplicative for u. If w ∈uL(u, U ), then u is involutive divisor of w, which isdenoted as u | Lw. The monomial w is called an involutivemultiple of u. The monomial v = w / u is multiplicativefor u, and the equality w = uv is written as w = u × v .

x 1d 1

x nd n

0

Involutive Divisions and Monomial OrderingsA. S. Semenov and P. A. Zyuzikov

Department of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119992 Russiae-mail: [email protected]; [email protected]

Received June 30, 2006

Abstract—In the paper, two classes of involutive divisions related to admissible monomial orderings—-divi-sions and -divisions—are considered. The latter may be viewed as an improvement of the class of induceddivisions introduced in [1]. The continuity and constructivity of the -divisions, as well as a number of addi-tional properties of the -divisions, are proved. Taking into account that the Janet division is a -division asso-ciated with the lexicographical order, its “antipode”—a-division associated with the same order—is separatedin the class of the -divisions.

DOI: 10.1134/S0361768807030036



140

PROGRAMMING AND COMPUTER SOFTWARE Vol. 33 No. 3 2007

SEMENOV, ZYUZIKOV

If u is an ordinary divisor of w, but not involutive one,then this equality is written as w = u ·v . In this case, themonomial v is nonmultiplicative for u.

For any u in U , there exists a partition of the set of all variables into two disjoint sets, namely, multiplica-tive variables M L(u, U ) ⊂ L(u, U ) and nonmultiplicativevariables NM L(u, U ) ∉ L(u, U ). Conversely, if, for anyelement u in a finite set U , a partition of variables is

defined such that the corresponding monomial sub-monoids L(u, U ) composed of variables from M L(u, U )satisfy the axioms, then this partition of variables spec-ifies an involutive division.

The submonoids L(u, U ) have natural geometricinterpretation [7]. Consider a set uL(u, U ) and denote itas C L(u, U ). One can easily verify that the image of the

set C L(u, U ) under the bijective mapping from onto

n is a discrete cone. The first three axioms are equiva-

lent to the following two geometric facts:

• The set C L(u, U ) is a discrete cone.

• C L(u, U ) ∩ C L(v , U ) ≠ ⇒ C L(u, U ) ⊆ C L(v , U ) ∨

C L(v

, U ) ⊆ C L(u, U ).In addition, we introduce the notation C L(U ) =

(u, U ).

Definition 1. Let an involutive division L be given.A set of mutually different monomials U is called a disjoint set if ∀u, v ∈ U , v ∉ uL(u, U ). A division L iscalled a disjoint division if any set of different monomi-als is a disjoint set.

Below, we give some examples of the well-knowninvolutive divisions.

Example 1 (Thomas division). Consider a finite setU of different monomials. Variable x i is considered as

multiplicative for u ∈ U if degi(u) = max{degi(v )|v ∈ U }and nonmultiplicative otherwise.

Example 2 (Janet division J ). Consider a finite set U of pairwise different monomials. For any 1 ≤ i ≤ n, theset U can be partitioned into subsets labeled with non-negative integers d 1, …, d i as follows:

Variable x i is multiplicative for u ∈ U if i = 1 anddeg1(u) = max{deg1(v )|v ∈ U } or i > 1, u ∈ [d 1, …, d i – 1],and degi(u) = max{degi(v )|v ∈ [d 1, …, d i – 1]}.

The Janet and Thomas divisions are disjoint divi-sions.

The filter axiom admits the following reformulation.Let two finite monomial sets U and V and a partitionrule L be given. Then, ∀u ∈ U ∩ V ,

(1)

The logic of the algorithm for computation of invo-lutive bases presupposes that, the greater the size anddimension of the sets of involutive multiples, thesmaller the number of involutive prolongations thatshould be considered. Based on this idea, we can distin-

0

C Lu U ∈∪

d 1 … d i, ,[ ] u U d j∈ deg j u( ) 1 j i≤ ≤,={ }.=

M L u U , V ∪( ) M L u U ,( ) M L u V ,( ).∩⊆

guish the best involutive divisions among all involutivedivisions. These are divisions L that, ∀u ∈ U ∩ V , sat-isfy the equation

(2)

This class of divisions coincides [6] with the class of pairwise divisions introduced in [1].

Definition 2. An involutive division L is pairwise if,

∀U , ∀u ∈ U : U \{u} ≠ , the following conditionholds:

(3)

In paper [1], it is proved that the Thomas and Janetdivisions are pairwise involutive divisions.

Definition 3. A subset U = {u1, u2}, u1 ≠ u2, is calleda basic set.

Definition 4. Consider a rule specifying a partitionon all basic sets that satisfies the first three axioms. Thisrule is denoted by L2 and referred to as an involutive2-partition. In other words, if B is a basic set and u ∈ B,then L2 uniquely determines the cone uL2(u, B) speci-

fied by multiplicative variables (u, B).

Any involutive division restricted to basic sets spec-ifies the corresponding 2-partition.

Definition 5 (pairwise closure). Consider a 2-parti-tion L2 on the basic sets. For any set U and ∀u ∈ U , wedefine multiplicative variables by the formula

The result of this procedure (as well as the proce-dure itself) is referred to as the pairwise closure of the

involutive 2-partition.As shown in [6], the pairwise closure does not

always determine an involutive division. Nevertheless,the following theorem holds.

Theorem 1 [6]. Let a disjoint 2-partition L2 begiven. Then, the pairwise closure L of the 2-partition L2

is a disjoint involutive division.

Involutive divisions are used for finding minimalinvolutive bases of polynomial ideals, the Gröbnerbases of special form. These bases are found by meansof the TQ algorithm [1, 4, 8].

Theorem 2. In the process of operation of the TQalgorithm, sets of leading monomials lm(T ) of all inter-

mediate sets T are disjoint sets.Proof. The assertion of the theorem follows from

the compactness of the polynomial set T on each step of the TQ algorithm [4]. Below is the direct proof of thisassertion, which repeats some arguments from theproof given in [4].

We assume that the set T is arranged in the ascend-ing order of leading monomials with respect to theordering . Let g1, g2 ∈ T and lm(g1) lm(g2). Then,by the algorithm construction, g1 has already belonged

M L u U , V ∪( ) M L u U ,( ) M L u V ,( ).∩=

0

M L u U ,( ) M L u u v ,{ },( ).v u v ,≠ U ∈

∩=

M L2

M L u U ,( ) M L2u u v ,{ },( ).

v U v , u≠∈∩=




INVOLUTIVE DIVISIONS AND MONOMIAL ORDERINGS 141

to T at some step, and g2 has been added to the set as h.In the course of the algorithm operation, the set T variesdynamically. Let T 0, T 1, … denote the sequence of itsstates.

Let, starting from some step i, the set T i containembedded cones. Let the set of multiples of the elementlm( f 1) include the set of multiples of the elementlm( f 2). Next, consider the step j of the algorithm and the

set T j in the case where f 2 plays the role of h and hasbeen added to the set T j – 1 for the last time before weobtain T i (any polynomial may go from T to Q and back several times). Suppose that T j ≠ T i. This implies that,after f 2, polynomials with leading monomials greaterthan lm( f 2) were added, and T j ⊂ T i.

Then, it follows from the filter axiom that lm( f 1)involutively divides lm( f 2) and the nesting is impossi-ble since the leading term of f 2 will be reduced.

TQ algorithm:

Input: a finite polynomial set F ∈ \{0}, involutivedivision L, admissible monomial ordering .

Output: T , minimal involutive basis of Id (F ).

1: select f ∈ F with minimal lm( f )

in accordance with

2: T := { f }; Q := F \ T

3: do

4: h := 0

5: while Q ≠ and h ≠ 0 do

6: select p ∈ Q with lm( p)

without proper divisors

in {lm(q)|q ∈ Q \{ p}}

7: Q := Q \{ p}

8: h := NF L( p, T )

9: end of loop10: if h ≠ 0 then

11: T := T ∪ {h}

12: for all {t ∈ T |lm(t ) lm(h)} do

13: Q := Q ∪ {t }; T := T \{t }

14: end of loop

15: Q := Q ∪ { NF J (t · x , T )|t ∈ T ,

x ∈ NM L(t , T ), NF L(t · x , T ) ≠ 0}

16: end of if

17: end of loop while Q ≠

The sufficient conditions for the existence of the

minimal involutive basis and for the correct operationof the TQ algorithm are continuity and constructivity of the involutive division L [3, 4].

Let a finite set U of different monomials and aninvolutive division L be given. A chain is a finitesequence of monomials {ui} (1 ≤ i ≤ k ) from U suchthat, ∀i < k ∃ x j ∈ NM L(ui, U ) and [ui + 1| Lui · x j], is calleda chain.

Definition 6 [3, 4]. An involutive division L is con-tinuous if, for any finite set of U of different elements

0

0

and any chain of elements from U , the inequality u j ≠ ui

holds ∀i ≠ j.

Definition 7 [3, 4]. A continuous involutive division L is called constructive if, for any finite set U of differ-ent monomials and any u ∈ U , x i ∈ NM L(u, U ) such thatu · x i has no involutive divisions in U and

the condition

holds.

3. INVOLUTIVE DIVISIONSAND MONOMIAL ORDERINGS

The description of involutive divisions that may beconsidered optimal from the algorithmic standpoint isan important theoretical problem. There exist many

approaches to solving this problem (see, e.g., [2]).In this paper, the optimality is meant to be maximalityof possible sets of multiplicative variables for all ele-ments of U . For example, since all multiplicative vari-ables for the Thomas division are Janet multiplicativevariables, but not all Janet multiplicative variables areThomas multiplicative ones, the Janet division seemsmore preferable.

The use of the pairwise property minimizes restric-tions on the number of multiplicative variables imposedby the filter axiom for sets consisting of three or moredifferent elements. Hence, it will suffice to define invo-lutive division on sets consisting of one or two ele-ments.

If a set U consists of one element, U = {u}, then

NM L(u, {u}) = { } is the best selection for the involu-tive division. Hence, to describe pairwise involutivedivisions with maximum number of multiplicative vari-ables, it is required to select optimal 2-partitions.

If a set U consists of two different elements and nei-ther of them divides the other in the ordinary sense, theoptimal division seems to be that where one elementhas an involutive cone of dimension n and the other, acone of dimension (n – 1); in other words, only one ele-ment has one nonmultiplicative variable.

Let one element divide the other. If the division may

contain embedded cones, all variables for both ele-ments are reasonable to declare multiplicative. Other-wise, it is required to define one nonmultiplicative vari-able for one of the elements.

For the rules of variable partitioning on sets consist-ing of two elements, it is reasonable to preserve the fol-lowing properties of the Janet division:

• effective maximality, i.e., existence of at least onecone of dimension n for each set U ;

• continuity;

v ∀ U ∈( ) x j∀ NM L v U ,( )∈( ) v x j u x i⋅ ⋅ v , x j⋅(

≠ u x i )⋅ v ⇒ x j⋅ C L U ( ),∈

w∀ C L U ( ) u x i⋅ C L U w{ }∪( )∉[ ]∈

0







144


SEMENOV, ZYUZIKOV

Theorem 7. Let L be an involutive -division, U bean arbitrary finite monomial set of different elements,and u1, u2 be elements such that u1 · x ∈ U 2 L(u2, U ),where x is a nonmultiplicative variable for u1 and U .Then, u2 >lex() u1.

Proof. Consider monomials u1 and u2 from U andvariable x ∈ NM L(u, U ) such that u1 · x ∈ C L(u2, U ).To prove the theorem, it will suffice to consider the case

where u2 ≠ u1 · x . In accordance with the filter axiom andpairwise property, we have u1 · x ∈ u2 L(u2, {u1, u2}).

Consider the pair {u1, u2}. Then, since NM L(u1, {u1,

u2}) = x , we have u1 = u2 in accordance with

the pairwise property and u2w u2 x . Hence, w x .If w = 1, then it is evident that 1 <lex() x . Otherwise, let z be a variable such that z |w, w = zw1. It is evident that z x and z <lex() x . Then, this is true for any z, whichresults in w <lex() x .

From w <lex() x , it immediately follows that u2w

<lex() u2 x . Hence, u1 = <lex() u2.

Theorem 8. Let L be an involutive -division, U bean arbitrary finite monomial set of different elements,and u1, u2 be elements such that u1 · x ∈ u2 L(u2, U ),where x is a nonmultiplicative variable for u1 and U .Then, either u1 · x = u2 or u2 u1, where is deg()-order.

Proof. Consider monomials u1 and u2 from U andvariable x ∈ NM L(u, U ) such that u1 · x ∈ C L(u2, U ).To prove the theorem, it will suffice to consider the casewhere u2 ≠ u1 · x .

Consider the pair {u1, u2}. Then, by Lemma 3, we

have u1 = u2 in accordance with the pairwise

property and u2w u2 x . Hence, w x .

If deg(w) > deg( x ), then we have w x and u1 u2.If deg(w) = deg( x ), then w is a variable, and the relationw x holds by the definition of for variables w and x . This proves the theorem.

Theorem 7 underlies an interesting fact regardingthat the Janet division may be considered optimal in theclass of -divisions associated with orderings forwhich x 1 x 2 … x n. Here, optimality is meant ineven finer sense, namely, as minimality of the numberof elements in the involutive basis of a monomial ideal.

In what follows, the statement ∀ p < q deg p(v ) ≥deg p(u) is briefly written as degq–(v ) ≥ degq–(u), and thestatement ∀ p < q deg p(v ) = deg p(u), as degq–(v ) = degq–(u).

Theorem 9. Let L be an involutive -division and U be a finite monomial set of different elements that isinvolutive with respect to L. Then, ∀u ∈ U ,

and

u2 w×

x ---------------

u2 w×

x ---------------

u2 w× x

---------------

uL u U ,( ) m ∈ u, v U ∈ v m,{ }lex ( )max={ }=

Proof. Let us assume the contrary. Suppose that

there exists U such that ∃m ∈ , u1 ∈ U , u1 = u ∈

U , u |m}, u1 m. Then, m contains a nonmultiplicative

variable x ∈ NM L(u1, U ). Since U is involutive, u1 · x hasan involutive divisor u2 ∈ U , u1 · x =u2 × w. By Theorem 7,

u1 <lex() u2, which contradicts the original assumption.

The second equation is verified similarly: if u1 · x has an involutive divisor u2 ∈ U , u1 · x = u2 × w, thenu1 u2.

Lemma 5. Let L be an involutive -division and U be a finite monomial set of different elements that isinvolutive with respect to L. Then, ∀u ∈ U ,

Proof. By the previous theorem, if x s ∈ NM L(u, U ),then ∃v >lex() u, u · x s ∈ v L(v , U ) and { x s} = NM L(u,

{u,v }). Hence, NM L(u, U ) ⊂ NM L(u, {(v ≥lex() u) ∧ (v u)}). The opposite inclusion follows from the pairwiseproperty.

Theorem 10. Let L be an involutive -division and U be a finite monomial set of different elements that is invo-lutive with respect to L, and let, for , x 1 x 2 … x n.Then, U is involutive with respect to the lex()-divi-sion, namely, the Janet division.

Proof. Let J denote the Janet division.

It is evident that

By virtue of the previous lemma, we need to prove

that

By the definitions of J and L, we have

hence,

To prove that the sets are equal, it will suffice to show

that, for any ∈ U such that NM J (u, {u, }) = { x s},

( u) ∧ ( >lex() u), there exists an element u0 suchthat u0 u, u0 >lex() u, and NM J (u, {u, u0}) = NM L(u,{u, u0}) = x s.

Since the variable x s is not multiplicative for u andfor the Janet division, we have

uL u U ,( ) m ∈ u, v U ∈ v m,{ }

max={ }.=

{lex ( )max

| L

NM L u U ,( )

= NM L u v U v ≥lex ( ) u( )∈ v u( )∧{ }},( ).

NM J u U ,( ) NM J u v U v ≥lex ( ) u∈{ },( ).=

NM J u v U v ≥lex ( ) u∈{ },( )

= NM L u v U v ≥lex ( ) u( )∈ v u( )∧{ },( ).

NM J u v U v ≥lex ( ) u( )∈ v u( )∧{ },( )

= NM L u v U v ≥lex ( ) u( )∈ v u( )∧{ },( );

NM J u v U v ≥lex ( ) u∈{ },( )

⊇ NM L u v U v ≥lex ( ) u( )∈ v u( )∧{ },( ).

u u

u u

degs– u( ) degs– u( ), degs u( ) degs u( ).<=




INVOLUTIVE DIVISIONS AND MONOMIAL ORDERINGS 145

Consider the monomial lcm(u, ). It satisfies therelations

Let lcm(u, ) ∈ u0 L(u0, U ). By Theorem 9, u <lex()

≤lex(

) u

0 and u u

0. Then, u

0 ≠ u, u

0 ≠ . Since

u0|lcm(u, ) and <lex() u0, we have degs–(u0) = degs–( )and degs(u0) > degs(u). Hence, NM L(u, {u, u0}) = x s,which proves the theorem.

Let MB L(U ) and MB J (U ) denote minimal involutivebases for the set U for the -division L (if x 1 x 2 …

x n takes place) and the Janet lex()-division, respec-tively. Then,

This proves that the minimal involutive basis inaccordance with the -division with x 1 x 2 … x nis an involutive set for the Janet division and, therefore,

is a Janet basis. However, this basis cannot be minimalfor the lex()-ordering. Thus, computation of the min-imal Janet basis seems preferable for -divisions withproperties x 1 x 2 … x n.

These results cannot be extended to the case of -divisions.

Example 8. Suppose that we have two variables x and y, u1 = x and u2 = y2. Lexicographically, y2 < x ; how-ever, with respect to deglex, x < y2, where x is greaterthan y.

Consider thedeglex-division. For this division, y2 · x = x × y2, and { x , y2} is the minimal involutive basis.

At the same time, for the

lex()-division (“Janetantipode”), the minimal involutive basis is { x , xy, y2}.

This example demonstrates that the “Janet anti-pode” is not a division with the minimum number of elements in the involutive basis compared to all -divi-sions for which x 1 x 2 … x n.

5. CALCULATION EXAMPLES

In this section, we give simplest examples of calcu-lations with the use of the “Janet antipode.”

As an example of variable partition, we consider afinite monomial set U and find multiplicative variables

for the Janet division and “Janet antipode”:

The results are presented in the table.

As can be seen, for the “Janet antipode,” any mono-mial has at least one multiplicative variable, which is apotential computational advantage for a number of polynomial systems.

u

degs– u( ) degs– lcm u u,( )( ),=

degs u( ) degs lcm u u,( )( ).<

u

u u u

u u u

MB J U ( ) MB L U ( ).⊆

U x 5 y

5 z

5 x

3 z

5 x

2 y

7 x

6 z

2 x

2 y

3,, , , ,{=

xy5 z

7 x

2 y

4 z x

2 y

3 z }., ,

The next example presents minimal bases withrespect to the Janet division and “Janet antipode” for anideal of three variables with monomial ordering degrev-lex, where x is greater than y and y is greater than z:

The basis for the Janet division:

The basis for the “Janet antipode”:

f 1 x 3 yz xz

2, f 2– x

2 y

2 z

2,–= =

f 3 xy2 z xy z.–=

j1 x 2 y

2 z

2,–=

j2 xy2 z xyz,–=

j3 x 2 yz z

3,–=

j4 xyz2

xz2,–=

j5 yz3

z3,–=

j6 x 2 z2 z4,–=

j7 xz3

xz2,–=

j8 z5

z4.–=

a1 xz4

xz2,–=

a2 yz4

z4,–=

a3 z5

z4,–=

a4 x 2 y2 z2,–=

a5 x 2 yz z

3,–=

a6 x 2 z

2 z

4,–=

a7 xy2 z xyz,–=

a8 xyz2

xz2,–=

Table

u ∈ U w.r.t. lex-order M J (u, U ) M A(u, U )

x 6 z2 x , y, z x

x 5 y5 z5 y, z x , z

x 4 y4 z4 y, z x

x 3 z5 y, z x , z

x 2 y7 y, z x , y x 2 y4 z z x , z

x 2 y3 z z x , z

x 2 y3 – x , z

xy5 z7 y, z x , y, z



146


SEMENOV, ZYUZIKOV

In this example, the Janet basis turned out smallerthan the “Janet antipode” basis. Nevertheless, this doesnot imply the advantage of the former over the latter.Comparison of these divisions will be the subject of

further studies.

6. CONCLUSIONS

In this paper, the class of involutive -divisions,which are generalizations of the induced divisions con-sidered in [1] and are “antipodes” of the -divisions[6], has been introduced. The continuity and construc-tivity of these divisions are proven, and an example of a minimal involutive basis for the “Janet antipode” divi-sion is given.

ACKNOWLEDGMENTS

The authors are grateful to their scientific advisorE.V. Pankrat’ev, as well as to V.P. Gerdt andYu.A. Blinkov, for the help, remarks, and useful ideaswhich influenced the work.

This work was supported in part by the Russian Foun-dation for Basic Research, project no. 05-01-00671.

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Involutive Divisions and Monomial Orderings I