gröbner bases in exterior algebra

Journal of Automated Reasoning 6: 233-250, 1990. 233 �9 1990 Kluwer Academic Publishers. Printed in the Netherlands.

Grrbner Bases in Exterior Algebra

T I M O T H Y STOKES Department of Mathematics, University of Tasmania, G.P.O. Box 252C, Hobart, Tasmania 7001, Australia

(Received: 23 November 1988; accepted: 27 July 1989)

Abstract. We show that the Buchberger algorithm for commutative polynomials over a field may be generalised to an algebraic structure which embeds such polymomials, the exterior polynomial algebra, and which is a natural domain for linear geometry. In particular, those finite sets of exterior polynomials which induce confluent reduction relations are characterised, and a means of algorithmically constructing them from a given set presented. A distinguished subset of such bases consists of the exterior algebra version of Grrbner bases. We charaeterise such bases and demonstrate how to construct them algorithmically from a given finite set of exterior polynomials.

Key words. Exterior algebra, reduction relation, left ideal, ideal, Grrbner basis.

1. Introduction

Exterior algebra is an important tool in linear algebra and geometry. In particular, it is a natural domain in which to state and prove many theorems in linear and affine geometry. The paper of White and McMillan [12] indicates the relationship between the Cayley algebra (derived from the corresponding exterior algebra) and projective geometry. There is a corresponding connection between the exterior algebra and affine geometry. For example, basic affine relations such as parallelism and collinearity may be expressed using exterior algebra; areas and mid-points are neatly dealt with. For a quite comprehensive introduction to the potential role of exterior or Grassmann algebras in automated theorem proving and geometric reasoning generally, see [5].

The role played by ideals in the exterior polynomial algebra in linear geometry theorems is very similar to that played by ideals in the standard polynomial algebra in the coordinatised versions of the same theorems. In particular, many theorems may be expressed using exterior algebra: the vanishing of several 'hypothesis' polynomials implies the vanishing of one or more 'conclusion' polynomials. The 'conclusion' polynomials are in the ideal of consequences of the set of 'hypothesis' polynomials. A sufficient condition for this is that the conclusion polynomials are in the ideal generated by the hypothesis polynomials.

All this makes exterior algebra a good candidate for an algorithmic treatment along similar lines to the Buchberger algorithm for multivariate polynomials over a field. We obtain an algorithm which, given a finite set of 'exterior polynomials', produces the equivalent in exterior algebra of a Gr/Sbner basis. (There are in fact

234 T. STOKES

two equally legitimate candidates for the title, due to the 'non-equational' definition of an exterior algebra.) The main advantage of exterior algebra in geometry is that the 'atoms' are not the scalar coordinates of points, with respect to a particular choice of coordinate axes, but rather vectors (or points). Consequently a more natural, intrinsic and efficient method of stating and proving theorems is possible.

The methods presented here involve a more complicated method of 'critical pair completion' or superposition than that which in essence recurs in such important algorithms as the Knuth-Bendix and Buchberger algorithms (see Huet and Oppen, 1980, [6], and Buchberger, 1976, [2], respectively) and in their various generalisations such as in Le Chenadec, 1986, [9], and Kandri-Rody and Wiespfenning, [8]. This is due to the existence in abundance of zero divisors in the exterior algebra polynomial structure which is the basis of what follows. Such zero divisors also lead to the existence of the two kind of Grrbner basis, a dichotomy not present in the Knuth-Bendix and Buchberger algorithms and their generalisations.

Previous variations of Buchberger's work include an 'algorithm' which semi- decides ideal membership within non-commutative polynomial rings over a field (Mora, 1986, [10]). Conceivably, one could employ the methods there in order to calculate 'Grrbner bases' for sets of exterior polynomials by incorporating each time as additional polynomials the various relations holding in the exterior polynomial algebra defined below. However, the 'algorithm' in [10] does not terminate in general and in any case would be drastically less efficient than the approach taken here, where the notion of compatible order applies to exterior algebra terms rather than to free products of non-commuting atoms, a method which is far more natural for the domain on which it operates. (A left ideal algorithm appearing in [10] does always terminate, but is unable to calculate the canonical forms of exterior algebra polynomials in general, so that calculations would not even correspond to calculations in the exterior polynomial algebra, which is a quotient of the free algebra obtained by factoring out a t w o - s i d e d ideal.)

2. Algebraic Preliminaries

Similarly to [5], a simple way to define the exterior algebra D associated with a vector space V of finite dimension n over a field F is as follows, i) is the associative ring with identity 1 generated by distinguished subsets ~= and V such that

(1) F is a field under the ring operations of Q, char F # 2. (2) V is an ~=-vector space of dimension n under the ring operations of t). (3) ~tx = xat for every ~t in 0= and x in V. (4) x y = - y x for all x, y in V. (5) For x~, x2 . . . . . Xk in V if x l x 2 " �9 �9 X k = 0 then xl, x2 . . . . . xk are linearly

dependent over H=. Elements of U= are called scalars, elements of V vectors. We note, as a consequence

of (4), that x 2 = 0 for any x in V.

GROBNER BASES IN EXTERIOR ALGEBRA 235

We define the order of f~ to be equal to the dimension of V. The exterior algebra of order n over any given field H: exists and is unique up to isomorphism.

It is easy to calculate using exterior algebra. For instance, for any vectors x, y and z, and scalars ~, 8, ~ and 6, we have

(otxy + flx)(Tz + 6zy)

= (~xy)(~z) + (~xy)(6zy) + (#x)(~,z) + (#x)(azy)

= ~7xyz - a6xyZz + ~yxz - ~6xyz

= ~,xyz - f l fxyz + fl~xz.

Indeed the equality of the first and last expressions is an identity for all exterior algebras, in the sense that, given an exterior algebra fl over a field 0:, the above identity holds for all substitutions of scalars in 0: for each of ~, fl, 7 and 6, and of vectors from the distinguished generating vector space V of ~ for each of x, y and z. This motivates the following

DEFINITION 2.1. Let A = {V~, V2 , . . . ,V ,} , the set of vector variables, and B = {~l, ~2, �9 � 9 am }, the set of scalar variables, be sets of finite cardinality n and m respectively. The exterior polynomial algebra of order (n, m) over the field U:, denoted by ~ . . . . is the ~:-ljnear associative algebra with identity element 1, generated by the elements of A and B subject to the relations:

(i) V i V j = - V j V , , l <~i,j<<.n

(ii) ~,~j =~j~,, 1 <<.i,j<<.m

(iii) V,~j=~jV,, l<~i<<.n,l<<.j<~m.

~n.,~ is in fact the algebraic formalism in which the equality of two polynomials expresses an identity involving up to n vectors and m scalars which holds for any exterior algebra over the field ~:. Conversely, any such identity may be expressed by the equality of two polynomials in ~n,r~-

We note that ~,,0 is a copy of the exterior algebra of order n over 0: with basis for the distinguished vector space being (for example) {V~, I/2 . . . . . V.}. Conse- quently it is possible to interpret some or all of the Vi as constants, thereby implicitly introducing a constant exterior algebra coefficient ring. Note also that ~o,,, ~ 0:[Xj, )(2 . . . . , Xm], the ring of commutative polynomials in m indetermi- nates over the field F. Consequently, the theory here developed may be considered a generalisation of the Buchberger theory for standard polynomials.

3. Noetherianness of ~ . ~ and Compatible Orders

We generalise the Hilbert Basis Theorem to ~n.m, n ~ 0. Recall that a ring R is left Noetherian if it has no infinite ascending chain of left ideals.

236 T. STOKES

LEMMA 3.1. Let S be an associative ring generated by a subring R and an element

V ~ R such that V 2 ~ R, and such that f o r any ~ ~ R there is a ~b E R such that

Vd? = ~ V. Then S is left Noetherian whenever R is.

Proo f Any 0 ~ S may be expressed as a sum of products of elements of R and V. Since V2~ R, all such products have the form r;, or r I V r 2 V . . . Vrk, or

V r l Vr2 . �9 �9 V r k , rl Vr2 V . . . r, V, or Vrl Vr2 V . . . Vr, V for some ri ~ R, i = 1, 2 , . . . , k. But since for any r; ~ R, there is an s; ~ R such that Vr, = si V, any such product may be written in the form r~, or IIk= ~ Si V m for some k, m; that is,

0 = r~ + r 2 V, with rl, r 2 ~ R. Consequently, S may be regarded as a two-generated R-module.

By Theorem 1.8 in [7], if R is left Noetherian, then the R-module S satisfies the ascending chain condition on R-submodules. Any subset of S which is an S- submodule is certainly an R-submodule. Consequently an infinite ascending chain of S-submodules is also one of R-submodules, of which, by assumption, there are none. Hence S satisfies the ascending chain condition on S-submodules, that is, it is left Noetherian. []

TH EOR EM 3.2. ~n.m is left Noetherian. Hence every left ideal has a f inite basis

over "~n,m" Proo f We proceed by induction on n. If n = 0, ~n,,, is the ring of standard

polynomials over the field ~=, which is known to be (left) Noetherian. Suppose ~k,m

is left Noetherian. We view ~k,m as being embedded in ~k + t,m" Let t ~ ~k,m consist of a single constant multiple of scalar and vector variables. Then, in ~k+ ~.m, Vk +l t = +--tVk +l, as follows easily from the constructive definition of ~n,m. We can express any ~b ~ #k,m as a sum of such terms t; suppose $ = t~ + t2 + �9 �9 �9 + t,. Then

Vk + lq~ = Vk + l(tl + t2 + . . . + t r )

= V k + l t I "~- V k + l t 2 + " ' " + Vk+ltr

= r l + t ~ 2 t 2 V k + l + " ' " + ~ r t r V k + l ( a l l t~j ~ { - - 1 , 1})

= ( ~ l t l + ~2t2 "1- " " " + ~ r t r ) V k + I"

Thus ~k + Lm is generated by #k,m and V k + i ~ #k,m with V 2 + 1 --- 0 ~ ~k,m, and is such that for any $ ~ #i ,m, there is a $ ~ #~k,m for which Vk+ ~c~ = $Vk+~.

Consequently, by Lemma 3.1, ~k + Lm is left Noetherian, and the result follows by induction. []

The proof of the second statement follows from a theorem which holds for all modules with ascending chain condition on submodules, as in [10].

We define a linear order on the variables in #..m as follows:

Vl < V2 < . . . < Vn < O~l < o~2 < . . . < O[m .

This allows the writing of any product of variables in a unique canonical form, with the variables written left to right, in increasing order, by applying the simple commuting and anticommuting rules used in the constructive definition of #~.,~


(which may introduce a minus sign, or perhaps even replace a product by zero).

Hence products Pl = ~3 Vl~3, P2 = ~l V2 V~, P3 = V3~2 I"3 have canonical forms of VI~32, -V1 V2~1 and 0 respectively.

DEFINITIONS 3.3, 3.4. A product of variables p is positive if the canonical form of p is also a product of variables; p is negative if its canonical form is the negative of a product of variables.

The set of terms of ~ . . . . denoted T,,,,, is the set of all non-zero positive products of variables in ~ . . . . together with the identity element 1 of ~,,m.

Thus Pl ~ T2.3, but P2, P3 ~ T2,3" However --P2 6 T2.3 since --P2 = 0tl VI V2, has canonical form V~ V2~. For a given non-zero product, p, of variables there corresponds a unique element of T,,m, denoted p. Thus p~ = p~,/~2 = -P2, and p~ is not defined.

DEFINITION 3.5. A total ordering, ~<, of T,,,. is compatible if, for all

s, t, u ~ Tn. m,

(i) 1 < t (ii) if s < t then us < ut, providing us v L 0 and ut v ~ O.

Because any two elements of T,,m either commute or anti-commute, any such order also satisfies

(iii) if s < t then su < tu provided su ~ O, tu v~ O.

Examples of such orders are easy to produce; for instance the 'total degree order' (cf. [3]) on T2,1 :

1 < Vl < II2 < ~1 < Vl V2 < Vl~ < V2~ < ~ < Vl V2[~l < VI ~2 < V2 0~2 < 0~3 < . - .

We shall use the notation LI(F) for the left ideal generated by a finite subset F of :~,.m. The ideal generated by the subset will be denoted by Ideal(F).

Let < be a fixed but arbitrary compatible order (the total degree order in all computed examples). With respect to that order, we shall employ the following abbreviations throughout:

coef(f, t) is the coefficient of the term t in f e ~n,m is the highest term with non-zero coefficient occurring inf(re la t ive Hte rm( f )

to < )

Hcoef ( f )

Examples: In

is coef(f, Hterm(f)) .

~a,o with the total degree order on T4,o,

coef(V1 + 2V3/I1 ,Vl V3) = - 2

Hterm(Vl II2+ VzV3+ V4V3) = VaV4

Hcoef(V, V2 + V2V3 + 114 V3) = coef(V~ V2 + V2V3 + II4 V3,V3 V4) = - 1.

PROPOSITION 3.6. Every compatible order on T,,m is a well order.

The proof is a straightforward generalisation of the commutative case.

238 T. STOKES

4. Reduction

Throughout the remainder of the article we shall consistently use the following

variables.

i, j, m, n, p, q, w natural numbers

d, e elements of the subset {0, 1 } of the natural numbers

f , g, h, k elements of ~n,m for fixed n, m

F, G finite subsets of ~,,m

r, s, t, U elements of T~,,,

a, b, c elements of the ground field ~:.

DEFINITION 4.1 (cf. [3]). g~j=h (g left reduces to h modulo F) if there are b, u,

a n d f �9 F with h = g - buf, and coef(g, u �9 H te rm ( f ) ) # 0,b = coef(g, u �9 H t e r m ( f ) ) H co e f ( f ) # 0. In such circumstances we say that g ~ and that g ~ h where f ,b,u t,f t = U �9 Hte rm( f ) . Here ~ e is the reduction relation for F.

Example: F = {f} , f = V 1 V3 - 2 V~ V2 , g = Vm V 2 V3 . Then g o F V 1 V 2 V 3 - - ( - - 1 ) V 2 ( V 1 V 3 - - 2 V 1 V 2 ) = O . Thus also g ~.-70 where

) - . t = Y l V 2 V 3 , and g f , _ l , V 2

DEFINITION 4.2 (cf. [3]). h is in normal form modulo F if there is no h' such that h--*Fh'. Here h is a normal form of g modulo F if there is a sequence of reductions

g =ko ~ e k l -'+Fk2 " ~ F ' ' " ~ e k m =h,

and h is in normal form modulo F.

LEMMA 4.3. The reduction relation -~ e is a Noetherian relation, that is, there is no

infinite sequence o f reductions o f any f �9 ~,,m : f ~ e f l ~ F f2 ~ r . . . . The proof is straightforward and is so close to proofs for the corresponding result

for standard polynomials that we omit it. It relies on use of Proposition 3.6.

LEMMA 4.4. The following algorithm yields a normal form N(g) o f a given polynomial

g, modulo F.

begin

N(g) ..=g

while exist f �9 F, b, u such that N(g) r.-~.~' do J , ,

choose f e F, u e T~ m, b �9 H: such that N(g) ----. , f,b,u

N(g) ,= N(g) - b.u.f.

end

Proof. Correctness is clear. There is considerable arbitrariness in the choice o f f in F. Termination follows from Lemma 4.3. []

G R O B N E R BASES IN EXTERIOR A L G E B R A 2 3 9

5. Griibner Left Bases

We turn our concent ra t ion to the not ion corresponding to a G r 6 b n e r basis in ~ , , , , .

There are two possible approaches to defining such bases. One is to require that

reduct ion modu lo such a basis is a confluent and terminat ing reduct ion relat ion (see

below); the o ther is to require that any element in the (left) ideal generated by the

basis be reduced to zero modu lo the basis. Fo r s tandard polynomials , these two

definitions are equivalent. However , for ~n,m with n ~ 0, they lead to two different not ions o f G r 6 b n e r basis, and consequent ly two different reduct ion relations.

DEFINITIONS 5.1-5 .6 (cf. [3]). ~--'F* is the equivalence relation by ~ F , that is, ~--~F*

is the reflexive symmetr ic transit ive closure o f -~e. Let -** be the reflexive transit ive

closure o f " " ~ F "

We say f - t r g ( f i s left-congruent to g modulo F) whenever f - g ~ LI(F) , and,

as usual, f - rg ( f is congruent to g modulo F) whenever f - g ~ Ideal(F) .

F is a GrObner left basis ( G L B ) if for all g, hi and h2, if hi and hE are normal forms of g modu lo F, then h~ = h2.

The least common multiple of s and t, Icm(s, t), is the smallest term which is a mult iple o f bo th s and t, that is, which m a y be expressed both as u �9 s and u ' �9 t.

Such a term always exists by Theo rem 3.6.

The S-polynomial corresponding to f~ and f2 is

SP(f l ,f2) = ( - 1) e ' ' H c o e f ( ~ ) �9 u l f l -- ( -- 1) e2" H c o e f ( f l ) �9 u2f2,

with el, e2, ul, uz satisfying ( - l )e , . u , . H t e r m ( f 1 ) = ul" H t e r m ( f l ) = l c m ( H t e r m ( f 1 ),

Hte rm(f2) ) . = ( - l) e2. uz" H t e r m ( f 2 ) = u2" Ute rm( fz ) . Examples:

lcm(V3 Vs~ 2, V3 V4ct2) -- V3 V4 V: t 2

SP(V 3 V:t~ - 2 V 2 V4, V 3 V4~2 + 3V~ I:2)

= ( - - 1) 1V4(V3 Vso~ 2 - - 2V 2 V4) - - ( - - 1)~ V4~ 2 + 3V~ V2)

= v 3 v . v s ~ , + o - v 3 v 4 v s ~ - 3 v l v 2 v ~ 2

= - 3 V , VzVs~2.

Note that, in general, S P ( f , , f z ) = + S P ( f z , f ~ ) . We shall require some general results on arbi t rary reduct ion relations.

DEFINITIONS 5.7-5.9. Let --. be a reduct ion relation on a set S. Let ~ * be the

reflexive transit ive closure o f ~ and ,--,* its reflexive symmetr ic transit ive closure. is confluent if for all x , y , z ~ S, whenever z--** x and z --** y, there is w ~ S

such that x ~ * w and y ~ * w.

is locally confluent if for all x, y, z ~ S, whenever z --*x and z ~ y , there is w ~ S s u c h t h a t x ~ * w a n d y ~ * w.

--+ is Church - R o s s e r if for all x, y ~ S such that x ~ * y, there is z ~ S such that x - * * z a n d y ~ * z.

240 T. STOKES

LEMMA 5.10. The following statements are equivalent for an arbitrary Noetherian reduction relation on a set S ('Noetherian" having the same meaning as for ~ F )

(1) --* is confluent (2) ~ is locally confluent (3) ~ is Church-Rosser

(4) For all x, y ~ S, x ~ * y i f and only i f there exists z ~ S in normal form (i.e. there is no w ~ S such that z --*w) such that x ~ * z and y ~ * z.

Proofs of all the above may be found in [4].

LEMMA 5.11. The following conditions are equivalent.

(1) F is a GLB

(2) f *-** g i f and only i f there is an h in normal form modulo F such that f ~ * h and g --** h.

Proof (1) =~ (2). I f f ~ * g, f --** h, and if g ~ * g~, h ~ * h~ where gt and hi are normal forms, t h e n f ~r* gl a n d f ~ * hi, and so gl = h~. Thus ""~r is confluent and so, by Lemma 5.10, one half of (2) follows. That the other half follows is clear.

(2) =, (1). By Lemma 5.10, "r is confluent, so i f f ~ * g , f ~ * h, then there is a k such that g --** k, h ~ * k, and so g and h cannot be normal forms of f unless they are equal. []

In what follows, we shall need the following results which are straightforward generalisations of results occurring in [2]. We shall omit the proofs since they are formally almost identical to those in [2].

LEMMA 5.12. (a) I f f ~ k g, with t = H t e r m ( f ) and k E F, H t e r m ( f ) > Hterm(h),

then f + h ~ F g + h (b) I f g ~ F h, and i f all t such that coef(f, t) # 0 satisfies t > Hterm(g), then

f + g ~ F f + h (c) I f f ~ F g , then for all h there is a k such that f + h ~ * k , g + h --** k

(d) I f f - g ~ * O, then there is a k such that f ~ * k, g ~ * k.

We now characterize GLBs in an algorithmically useful way.

T H E O R E M 5.13 (Characterization Theorem for GLBs). The following conditions are equivalent.

(1) F is a GLB.

(2) I f f l , f2 ~ F, and t ~ T,.m satisfies t �9 l cm(Hte rm(f l ) , Hterm(f2) ) # 0, then

t . S P ( f , , f 2 ) ~ * O. Proof. (1) =~ (2). Suppose F is a GLB, and that, for f l and f2 ~ F, t satisfies

t �9 lcm(Hterm(f~) , Hterm(f2) ) # 0. Then there are el, e2, ul, u2 such that S P ( f l , f 2 ) = ( - 1) e ' ' Hcoef ( f2) " ul f l - ( - 1) e2" H c o e f ( f t ) " u2fz, where ( - 1) ~1. ul �9 H t e r m ( f l ) = ( - 1) ~z" u2" Hterm(f2) = l cm(Hterm(f l ) , Hterm(fz)) .


By a s s u m p t i o n o n t,

( - - 1 ) e t ' t u l �9 H t e r m ( f l ) = ( - 1 ) e 2 . tu2 �9 H t e r m ( f 2 ) = t . l c m ( H t e r m ( f l ) ,

n t e r m ( f 2 ) ) ~ 0.

L e t g = ( - 1 ) e l n c o e f ( f 2 ) " t u l f l . T h e n H t e r m ( g ) = ( - 1) el �9 tu I �9 H t e r m ( f l ) , a n d

so g i + 0, w h e r e f = tUl f l s = H t e r m ( f ) = H t e r m ( g ) , n a m e l y s , f

g ,----ff g - [ c o e f ( g , H t e r m ( / u l f l ) ) / H c o e f ( t u l f l )] "tul f l ,

s ince

c o e f ( g , H t e r m ( t u ~ f l ) ) = c o e f ( g , H t e r m ( g ) ) ~ 0.

T h u s

g ~ e g - [ c o e f ( g , H t e r m ( g ) ) / H c o e f ( t u l f l ) ] " tUl f l

= g - - [( - - 1) e~. H c o e f ( f 2 ) " H c o e f ( t u l f l ) / H c o e f ( t u l f l ) ] �9 t u l f l

= g - - ( - - 1 ) et" H c o e f ( f 2 ) "tu~f~ = g - g = 0 .

N o w f r o m t h e a b o v e t h e r e is a d s u c h t h a t

H t e r m ( t u 2 f z ) = ( - 1) d" t �9 H t e r m ( u 2 f z ) = ( - 1) d" t . H t e r m ( u l f l )

= H t e r m ( t u l f l ) = ( - - 1 ) d" t - ( - 1)e2" uz �9 H t e r m ( f 2 )

= ( - 1 ) a - t ( - 1 ) e ~ , u l - H t e r m ( f t ) -

S i n c e c o e f ( g , H t e r m ( t u z f 2 ) ) = c o e f ( g , H t e r m ( t u l f l ) ) = H c o e f ( g ) 4= 0, t h e n l e t t i n g

h = tu2f2, r = H t e r m ( t u 2 f 2 ) = H t e r m ( t u l f l ) w e h a v e

g r.-Tf' g - - [ c o e f ( g , Hterm( tu 2 f=) ) /Hc o e f ( t u 2 f2 ) ] �9 tu2f2

= g - [ c o e f ( g , H t e r m ( t u l f l ) ) / c o e f ( t u z f 2 , Hterm( tu2 f2 ) ) ] �9 tuz f2

= g - [( - 1) e l . H c o e f ( f 2 ) " H c o e f ( t u l f l ) / c o e f ( t u 2 f 2 , ( -- 1) d§ e2

�9 tu2" H t e r m ( fz ) ) ] " tuz f2

= g - [ ( - 1)el . H c o e f ( f 2 ) �9 H c o e f ( t u l f l ) / c o e f ( t u 2 " H c o e f ( f 2 ) �9 H t e r m ( f 2 ) ,

( _ l ) d + e2. tu2" H t e r m ( f2) ) ] " tu2f2

= g - - [( - - 1) e l . H c o e f ( f 2 ) " H c o e f ( t u ~ f l ) / ( - 1) d+ e2. H c o e f ( f2))] " tu2f2

= g - ( - 1) d+el +ez" H c o e f ( t u l f l ) �9 tu2fz

= g _ ( _ 1)d+ eL +e2. c o e f ( t u l " H c o e f ( f l ) ' H t e r m ( f l ) , ( - 1 ) d + e l

�9 tu l" H t e r m ( f l ) ) " t u z f z

= g _ ( _ 1)d+ e, +e2 . ( __ 1 )a+e , . H c o e f ( f l ) " tu2f2

= g - - ( - - 1) e2. H c o e f ( f l ) " tu2f2

= ( - - 1) -1 . H c o e f ( f 2 ) " tu~f~ -- ( -- 1) e2- H c o e f ( f ~ ) - tu2f2

= t . S P ( f l , f 2 ) -

242 T. STOKES

Thus t ' S P ( f i , f 2 ) * - , * O , as g- - - ,F0 and g - - , e t ' S P ( f l , f 2 ) . By L e m m a 5.10 t �9 S P ( f l , f 2 ) has normal fo rm zero.

(2) =:, (1). The p r o o f is essentially a generalisation of the corresponding p r o o f for

~0,n as presented in [1]. By L e m m a s 5.10 and 5.11, F is a G L B if and only if - - ' r is locally confluent.

Hence, we shall show that if (2) holds then for all f , g and h, i f f - - ' r g and f - , F h then there is a k such that g ~ * k and h --** k.

Suppose f----* g and f - - - - , h and w.l.o.g, s ~< t under the chosen compat ib le order t , f l s , f 2

o n Tn.ra. I f S < t, then the p r o o f is formal ly the same as the corresponding p r o o f in [1];

it relies on L e m m a 5.12(a), (b) and (d). We consider only the case where s = t.

Then f - - - ~ g and ft---~2h. Let t l , tE, d l , d2 satisfy ( - - 1 ) d ~ ' t l ' H t e r m ( f l ) = t0rl

Hterm(t l f l ) = t = Hterm(t2 f2) = ( - 1) a2" t2" H te rm( f2)- Then g = f - [coef(f , t ) /

Hcoef ( f l ) ] �9 t l f l , h = f - [ c o e f ( f , / ) / n c o e f ( f 2 ) ] �9 t2f2.

Let el, e2, Ul, u2 satisfy

( -- 1) e2" U 2 " H t e r m ( f 2 ) = ( -- l) el" U 1 " H t e r m ( f l )

= l cm(Hte rm( f2 ) , H t e r m ( f l )).

Thus SP( f~, f2) = ( - 1) e2" H c o e f ( f ~ ) . uEf2 - ( - 1) e~" Hcoef( f2 ) " Ul f l . Let t ' , d satisfy t = ( - 1) a . t ' . l c m ( H t e r m ( f l ) , H t e r m ( f 2 ) ) , so that t = ( - 1) a . t ' . ( - 1) e~.

H t e r m ( f l ) = H t e r m ( t ~ f l ) = ( - - 1 ) at �9 tl �9 n t e r m ( f ~ ) = ( - 1 ) a . t ' . ( - 1 ) e2. u2 �9

H t e r m ( f 2 ) = Hterm(t2 f2) = ( - 1) a2' t2" H t e r m ( f 2 ) . Hence ( - 1) at . tl = ( - 1) a+ el �9 t 'u l , ( - 1 ) a 2 " t 2 = ( - - 1 ) d + e 2 " t ' u 2 , since for products o f variables Pl ,PE ,P3 , if

PiP3 = P2P3, then Pl = P2 providing PiP3 =# O, PEP3 =# O. Let �9 = coef( f , t ) / [ n c o e f ( f l ) " Hcoef( f2)]- Then

g - - h = cx[( - l ) d 2 . n c o e f ( f l ) " t 2 f 2 - ( - l ) d l " ncoe f ( f2 ) " tl f l ]

= ~[( _ 1)d+ ~2. H c o e f ( f l ) " t 'uEf2 -- ( -- 1) a+ e~. n c o e f ( f2 ) " t'Ul f l ]

= ( -- 1) a" 0it'[( - 1) e2" H c o e f ( f l ) " u2f2 -- ( -- 1) el" Hcoef( f2 ) " l'Ul f l ]

= ( - 1) a . ctt ' . SP( f2, f l ) , with t ' . l c m ( H t e r m ( f l ) , H t e r m ( f 2 ) ) # 0

and so by (2), g - h has normal fo rm zero, that is, g - h ~ * O. Hence, by L e m m a

5.12(d), there is a k such that g - - . *k and h --** k. []

The difference f rom the result for s tandard polynomials is the need to consider not only S-polynomia ls but also multiples of S-polynomia ls by appropr ia te terms. The reason is that, unlike s tandard polynomials where there are no zero divisors, it is not the case that f ~ * 0 implies t f --** 0 for all terms t.

In what follows, N ( F , f ) denotes an a lgor i thm which calculates normal forms modu lo F. Recall that A denotes the set o f vector variables; given t, V(t) denotes the set o f vector variables occurring in t. Given a set V o f vector variables, T(V) is the set o f all terms all o f whose variables are. vector variables in V.


T H E O R E M 5.14. Given F, the following algorithm constructs a GLB G such that if

f ~-~* g then f ~-~* g, and if f ~-~* g then f =- tFg.

begin G:=F

B = {{fl, fz} If,, A e G; Z r A}

while B ~ 0 do

begin

{ f l , f2 } "= an element of B

B,=B -- {{f,, f2}}

g .'= SP( f , , f2)

comment: g is the S-polynomial of the pair f~, fz in G

V:=A - V(lcm(Hterm(f~), Hterm(f2)) )

comment: V consists of all vector variables not occurring in the least common multiple of Hterm(f0 and Hterm(f2)

end

end

r = T(V)u{1}

while Tyro do

begin

t := an element of T

r : = r - {t}

h:=t .g

h',= N(G, h)

if h' r O then

S ' .=Bu{{g ,h ' i lg ~ G}

a,=a~{h'}.

comment: if the normal form of t times the S-polynomial is non-zero then it is added to G and is used to produce more S-polynomials

end

244 T. STOKES

Proof. (a) Termination. Let Gi be the set of elements in G after the ith extension, and P, = {Hterm(f ) [f e G, }. If G, = G,_1 u {h'}, then P; = P i - i u {Hterm(h')}, and since h' is in normal form modulo G;_ l, it follows that Hterm(h') is not a multiple of any element of P,_1, and so L I ( P , _ 1 ) ~ LI(P;). Consequently, the

inclusions in the ascending chain of left ideals LI(PI) c LI(P2) c �9 . . are proper, so by Theorem 3.2 the sequence is finite. Hence the number of extensions of G is finite, so that T and B must become empty after finitely many calculations.

(b) Correctness. The set G obtained by applying the algorithm is easily seen to satisfy condition (2) of Theorem 5.13, so that G is a GLB. Since F~_G, clearly

f~--~*g implies f~---~*g. I f f~ , f2~ G~_1, then S P ( f l , f 2 ) e LI(G;_I) and so also t . SP(f~,f2) ~ LI(G~_ 1). Thus LI(G;) = LI(G~_ 1) for all i, so that LI(F) = LI(G). Since f*--~pg ( f ~ r g or g ~ F f ) implies f = teg, it follows that f~--**g implies f = teg (by induction on the number of *--~-steps involved). Hence iff~--**g then

f = ~g and so f = ~g, since LI(F) = LI(G). []

There are a number of ways in which the above algorithm may be made faster, in particular by employing simple checks on the terms used to premultiply the S-polynomials in accordance with calculations made earlier. However, the aim here is simply to demonstrate the existence of such an algorithm.

Example. In ~6.o we employ the total degree order on / '6.0 with

V 1 < V 2 < " ' ' < V 6 . Let F = { f~, f2 }, f~ = II5116 - V2113, f2 = II4 II5 - VI V3. Using the GLB algorithm to 'complete' F, we obtain F'={f~,f2,f3,f4}, where f3 = - V21"3 V4 + V1113 Vr, f4 = - V1 II2 V3 V6. The corresponding reduction relation ~F. will, by Lemma 5.11, produce unique normal forms of polynomials. However,

we note that although V2V3V4V6=V6(-V2V3V4)-~F, V6(V1V3V6)=O, hence II2113 V4V6 ~*. 0 by Lemma 5.11, it can be demonstrated by induction that

V2 V3 V4 V6 ~ * 0: if coef(p, 1121"3 V4 I"6) S0, then also coef(q, I"2 V3 I"4 Vr) whenever P ~ r q or q ~rP. Furthermore, V2 V3 II6 = - (II5 V6 - II2 V3) ~ LI(F) = LI(F'), but 1"21131"6 ~ * 0, so that ~ r . does not decide left ideal membership.

6. Gr6bner Left Ideal Bases

We have characterized those finite sets G which induce reduction relations ~ a which are confluent, and have shown how to construct such a G from a given set F in such a way that f,--** g implies f*--~* g, so that f a n d g certainly have the same normal form modulo G i f f ~--~* g. For 9~o.m (commutative polynomials over a field), and a subset F of ~o.,,, it is the case that = F and ~--~* are equal as relations, as is shown in [1]. As the above example demonstrates, that is not the case for 9~n.m in general.

DEFINITION 6.1. F is a Gri~bner left ideal basis (GLIB) i f f e LI(F) implies that

f--** 0.


LEMMA 6.2. Every GLIB is a GLB. Proof. I f f ~ * g, f ~ * h where g and h are normal form modulo F, then g ~--~* h

so that g =tFh, which follows because fl ~Ff2 implies f l - - f 2 E LI(F). Conse- quently, g - h = tF 0. But F is a GLB, so g - h ~ * 0, and since g - h is in normal form (as g and h are), it follows that g - h = 0. []

LEMMA 6.3. The following statements are equivalent.

( l ) F is a GLIB. (2) f = ~ g if and only i f f and g have equal normal forms. Proof. (1) ~ (2). Suppose f = tFg. Then f - - g ~ LI(F), so that f - g ~ * O.

Hence, by Lemma 5.12(d), f~--~* g. Since F is a GLIB, it is a GLB by Lemma 6.2 and s o f a n d g have equal normal forms by Lemma 5.11. Conversely, if f and g have equal normal forms, then since F is a GLB, Lemma 5.11 implies thatf,--~* g, so that

f =-tFg. (2) =~ (1). I f f ~ LI(F), then f - t F 0, SO that f has normal form zero. []

LEMMA 6.4. Let F be such that for f " ~ F and for all t, t . f ' ~ * O. Then f +--~* g if and only if f =- I F g.

Proof I f f*--,* g then certainly f = ~ g. I f f = tFg, t h e n f - - g E LI(F). If F = {fl , f2 . . . . . fp}, t h e n f - g = Eq=l ajtjf) for

some al, a2 . . . . . aq and tl, t2, �9 �9 �9 tl. H e n c e f = g + Eq= 1 ajtjfj. We use induction on the number of summands q to show that f,---~* g.

If q = l , let f = g + a ~ t l f l for some i l e { 1 , 2 . . . . . p}. By assumption, al t~f l - -** O. Consequently f - g = a l t l f l ~ * O, so by Lemma 5.12(d), f4--~*g.

We assume the case q = w, namely if f = g + ET=lajt~f ~, then f +-~,* g. If f = g + E~'__+l I ajtjfj = g + a, t~fj + Ey~+z~ ajtjf? then by the induction assump-

tion, f ~--** g + a~ tlf, j, but as for q = 1 above, g + al tl f,i *--+* g. Hence f*-~* g and so by induction the result follows. []

T H E O R E M 6.5 (Characterization Theorem for GLIBs). The following statements are equivalent.

(1) F is a GLIB. (2) I f f 1 , f2 ~ F, then for any t, t "fl --** 0 and t �9 SP(f l , f2) ~ * 0. (3) I f f l , f z ~ F and t~ satisfies tt . Hterm(f l ) = 0 and t2 satisfies

t2' lcm(Hterm(f~), Hterm(f2)) ~ 0, then t~ .f~ --,* 0 and t2" SP(f l , f2) ~ * 0.

Proof (1) ~ (2). t .f~, t" SP(f~,f2) ~ LI(F), so t .f~ ~F* 0, t" SP(f l , f2) --** 0. (2) =~ (3). Obvious. (3) =~ (1). The two conditions onf~,f2, tl and tz imply that - t r a n d f +--~r* are the

same, and that F is a GLB (by Lemma 6.4 and Theorem 5.13 respectively; note that

if t~ �9 Hterm(f~) 4: 0, then t~ "ft ~ 0, where t = t~ �9 Hterm(f~)). By Lemma 5.11 it

follows that f -= ~rg if and only if f and g have equal normal forms. The result follows by Lemma 6.3. []

246 T. STOKES

T H E O R E M 6.6. Given F, the following algorithm constructs a GLIB G such that LI(F) = LI(G).

begin

G:=F

H , = F

B.'= {{A ,A} If, ,f2 �9 F,f. #f2}

comment: H plays two roles in what follows

while H ~ O do

comment: in the next procedure H is the subset of G which supplies polynomials that when multiplied by appropriate terms are to be included in G if they are not of normal form zero modulo

G

begin subroutine I

while H ~ 0 do

begin

f. .= an element of H

H ,= H - {f}

W:= V(Hterm(f ) )

s = { t It ~ T(a), V(O n w~r

comment: S is the set of all terms whose variables arc vectors that do not occur in H t e r m ( f )

while S # 0 do

begin

u := an element of S

k:=u " f

k'..= N(G, k)

i f k" ~ 0 then

H , = H u { k ' } ,

G:=Cu{k'},

B , = B u { { g , k'} [g �9 G}

GROBNER BASES IN EXTERIOR ALGEBRA

end

end

end

comment:

247

comment: if the normal form of u times f is not zero, it is added to both G and H, thereby enlarging the basis and providing more polynomials for multiplication by appropriate terms as above, and it is used to produce S-polynomials below

end

H is now empty; in the next procedure, H will consist of the additions to G which occur in the course of enlarging G into a GLB

while B ~ 0 do

begin

{ f~, f2 }"= an element of B

B,=B -- {{fl,fz}}

g ..= SP(f, ,)rE)

V.'= A - V(lcm(Hterm(fO, Hterm(f2)))

T,= T(V) w {1}

while T # O do

begin

t ,= an element of T

T , = T - { t }

h ,=t .g

h',= N(G, h)

if h' # O then

B . . = n u { { g , h ' } l g e G}

a,=au{h'}

H , = H u { h ' }

end

end

248 T. STOKES

Proof (a) Termination occurs for the same reason as in Theorem 5.14, since again the only polynomials that are used to extend G or H are in normal form modulo G.

(b) The first half of the algorithm converts G into a set to which Lemma 6.4 applies. The remainder of the algorithm ensures G satisfies (2) in Theorem 5.13 (and so is a GLB). The algorithm will not terminate unless G has these two properties as in (3) of Theorem 6.5, so that G is GLIB. Furthermore, since every polynomial used to extend the original set F in the initial run through the routine is either a left multiple of an element of F or a linear combination of such left multiples (a left multiple of an S-polynomial), and hence is in LI(F), it follows by induction that G c LI(F), and hence that LI(G) = LI(F), since F c G. []

Example. With F = {fl,f2} as before in the GLB example, we obtain, after performing subroutine I the first time, G = {fl , f2fs , f6, fTfs} where f5 = 1"2 V3 V6,f6 = - - V2 V3 Vs,f7 = - V1113 Vs,f8 = - Vl 113 II4. The first round of S- polynomial calculations brings inf3,f4, as before. Further computations produce no further polynomials, and so G = { f ~ , f z . . . . . f8} is a GLIB such that LI(F) = LI(G).

7. Homogeneous Polynomials

DEFINITIONS 7.1-7.5. f ~ ~.,m is vector homogeneous ofdegreep > 0 i f f e ~p,,., f r ~ p - l,m (where we are considering ~p,., to be a subset of ~., , . , p < n); f i s vector homogeneous of degree zero i f f e ~0,... (Henceforth we shorten 'vector homogeneous' to 'homogeneous'.)

The set of homogeneous polynomials of degree p in ~., , . , p < n, will be denoted by --.,m'~(P) Thus --n,m'~(P) =~i~nm\~p_l,m, p , = 1,2, . . . , n, and ~.,,.'~(~ : ~ 0 , m "

We define T(P) = Tn m {') 6J~(P) - - n , m , v n , m �9

Clearly ~ . , . = v . ~(P) so that every ~b ~ ~ . m may be uniquely expressed as , g a p ~ 0 ' J n , m ,

a sum of its homogeneous components. A homogeneous left ideal I of enm is a left ideal for which I = ]Ep= 0 (IcT~(r)~ , n , m / � 9

We note that ~.,, . is a graded ring with respect to the grading determined by the degree of homogeneity. The above definition of 'homogeneous' is consistent with that used for graded rings generally. For elaboration of these basic ideas, see [11], in which the next result appears in more general form.

PROPOSITION 7.6. I f I is a left ideal o f the graded ring R, then the following statements are equivalent

(1) I is a homogeneous left ideal. (2) I f i ~ I, then all homogeneous components of i are in L (3) I can be generated as a left ideal by homogeneous elements.

LEMMA 7.7. I f s ~ T<P) t ~ T(q) then st = ( - - 1) pq" ts. Indeed i f f e ~(P) - - n , m , ~ n , m , n , m ,

g ~ -cliO(q) then fg = ( - - 1 ) p q " g f . ~ n , m ,


Proof. The first statement follows easily from the anti-commutativity property of vector variables. The second may be proved easily by repeated application of the first. []

In the exterior algebraic formulation of a geometry theorem, it is homogeneous polynomials which feature, as they are semantically the important polynomials in geometry. Thus if F is a finite set of 'hypothesis polynomials', F will consist entirely of homogeneous polynomials. The previous lemma demonstrates that, for such F, the 'left-sidedness' of the definition of reduction is merely a matter of convenience, and that in fact one could post-multiply elements o f f by terms and the definition would be effectively the same because --*F would be the same relation. Indeed allowing 'two-sided' reductions would not alter the effective possible reductions modulo F. Thus, a homogeneous GLB is really a 'two-sided' basis. We can say even more if we have a GLIB generating a homogenous left ideal, by virtue of

THEOREM 7.8. In ~ . . . . a homogeneous left ideal is an ideal.

n ( I c ~ ( p ) ~ then for any f = Z , 7 = o f ( O e ~ n m and any Proof. If I = Xp = o, ~.,~,, g = X~'=og(J)e I (each expressed as a sum of its homogeneous components), we have gO) e I for all j, by Proposition 7.6. Then we have

g.f= ~ g(,,. ~ fo,= ~ ~ g,S,.f(o= ~ ~ (_l)Uf<,,g(.O J = O l = 0 i = O j = 0 / = 0 j = O

by Lemma 7.7. But each summand is in L so that g . f ~ L Hence I is also a right ideal and therefore an ideal of ~,m- []

If the elements of F are homogeneous, then LI(F) is a homogeneous left ideal by Proposition 7.6. Consequently, the above theorem shows that for the polynomials of greatest interest in ~n.m, the vector homogeneous polynomials, Theorem 6.5 and

~ .~(o). do in fact the associated algorithm actually apply to ideals. Since ~0.m = ~" . . . . we have a generalisation of Buchberger's results for polynomials over a field.

Acknowledgement The author is grateful to Desmond Fearnley-Sander for two reasons: first for introducing to the author the exterior algebra structure and demonstrating its great advantages as a formalism in which to express and solve linear geometry problems; and secondly for the idea of employing equational reasoning in exterior algebra, especially of the sort employed by Buchberger in his work on Grrbner bases. In addition to these particular words of thanks, the author is indebted to Desmond Fearnley-Sander for many discussions, not only concerning the above topics but concerning automated theorem proving in general. I would also like to thank him for spending time in assisting with the correction of the manuscript for this paper.

250 T. STOKES

References

1. Bachmair, L and Buchberger, B., 'A Simplified Proof of the Characterization Theorem for Gr6bner Bases', ACM-SIGSAM Bulletin 14(4) (1980) 29-34.

2. Buchberger, B., 'A Theoretical Basis for the Reduction of Polynomials to Canonical Forms', ACM-SIGSAM Bulletin 10(3) (1976) 19-27.

3. Buchberger, B., 'Gr6bner Bases: An Algorithmic Method in Polynomial Ideal Theory', in Multidi- mensional Systems Theory (ed. N. K. Bose), D. Reidel, pp. 184-232 (1985).

4. Buchberger, B. and Loos, R., 'Algebraic Simplification', in Computer Algebra. Symbolic and Algebraic Computation (ed. B. Buchberger, B. G. Collins, R. Loos), Springer-Verlag, New York, pp. 11-43 (1983).

5. Fearnley-Sander, D., 'The Idea of a Diagram', in Resolution of Equations in Algebraic Structures, (eds. Hassan Ait-Ka~i and Maurice Nivat), Academic Press (1989).

6. Huet, G. and Oppen, D. C., Equations and Rewrite Rules: A Survey, Technical Report no. CSL-I 1, SRI International (1980).

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Type', Technische Berichte der Fakult~t fiir Mathematik und Informatik Universitfit Passau, MIP-8807 (1988).

9. Le Chenadec, P., Canonical Forms in Finitely Presented Algebras, Pitman, London (1986). 10. Mora, F., 'Groebner Bases for Non-Commutative Polynomial Rings', in 3rd International Confer-

ence, Algebraic Algorithms and Error Correcting Codes, Grenoble, France, July 15-29, 1985, Proceedings (ed. Jacques Calme0, Lecture Notes in Computer Science 229, Springer-Vedag, pp. 353-362 (1986).

l 1. Northcott, D. G.., Lessons on Rings, Modules and Multiplicities, Cambridge University Press (1968). 12. White, N. L. and McMillan, T., 'Cayley Factorization', Institute for Mathematics and its Applica-

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gröbner bases in exterior algebra

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