Reduction among Bracket Polynomials - ACM Digital Library

Reduction among Bracket Polynomials Hongbo Li, Changpeng Shao, Lei Huang, Yue Liu KLMM, AMSS, Chinese Academy of Sciences, Beijing 100190, China [email protected], [email protected], [email protected],

ABSTRACT In this paper, we propose an SL(n)-invariant division of SL(n)-invariant polynomials by establishing an admissible order among the invariant polynomials in normal form. The invariant division leads to an invariant Gr¨ obner basis theory. The invariant division is closely related to multivariate coordinate polynomial division. This feature leads to a proof of the result that if f1 , . . . , fk are SL(n, K)-invariant where K is an arbitrary field, possibly of positive characteristic, then the invariant ideal generated by them is the intersection of the ideal generated by the fi in the polynomial ring of coordinates with the algebra of invariants.

Categories and Subject Descriptors I.1.1 [Computing Methodologies]: SYMBOLIC AND ALGEBRAIC MANIPULATION

General Terms Theory

Keywords Classical invariant theory; invariant division; invariant Gr¨ obner basis; invariant ideal membership; bracket algebra.

1. INTRODUCTION In history, geometric computing originated with the use of geometric invariants such as distances and angles, but culminated with the invention of coordinates, due to the complexity of algebraic dependencies among geometric invariants. With the development of computing machinery, symbolic geometric computing based on invariants rejuvenated in the 1980s and has found important applications in computer vision, robotics, and automated reasoning [5]-[7], [9]–[15].

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Despite the benefits of succinct representations of geometric constraints by invariants, algebraic manipulations among classical invariants are still incomplete. For one thing, if the division of one polynomial of geometric invariants by another is done in coordinate form, the traditional multivariate division of coordinate polynomials fails to produce invariant results. For example, let vi = (xi , yi )T for i = 1..4 be the homogeneous coordinates of four points on the projective line, and let f g

= (y1 x3 − y3 x1 )(y2 x4 − y4 x2 ), = −y1 x4 + y4 x1 ,

(1.1)

then f, g are SL(2)-invariants. By the order of variables y1  y2  y3  y4  x1  x2  x3  x4 , the division of f by g under the (degree) lexicographical order of monomials results in f = g(−y2 x3 )+(y1 y4 x1 x2 −y1 y4 x2 x3 −y2 y3 x1 x4 +y2 y4 x1 x3 ). (1.2) The quotient −y2 x3 is obviously not SL(2)-invariant, nor is the remainder. With the development of Gr¨ obner basis theory, defining invariant division becomes possible. To make things clear we provide a formal definition of the term “invariant division”: Definition 1. Let f1 , ..., fk be G-invariant polynomials in variables {xij |i = 1..m, j = 1..n}, where G is a Lie subgroup of GL(n), and for 1 ≤ i ≤ m, (xi1 , ..., xin ) is a vector variable in an n-dimensional vector space over a field K. Let g be another G-invariant polynomial. The G-invariant division of g by f1 , ..., fk refers to the procedure of finding G-invariant polynomials h1 , . . . , hk , r satisfying g = h1 f1 + . . . + hk fk + r, where the hi are called quotients and r is called the remainder, such that (1) r is obtained by top reduction, (2) for any g-invariant polynomials p, q, if p − q is in the G-invariant ideal generated by f1 , ..., fk , i.e., if there exist G-invariant polynomials s1 , . . . , sk such that p − q = s 1 f1 + . . . + sk f k ,

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then the remainders of p, q relative to f1 , ..., fk are identical. In classical invariant theory, any invariant is a polynomial of basic invariants. For example, when G = SL(n), the basic invariants are the determinants Bα of homogeneous coordinates of vector variables vi , say 1 ≤ i ≤ m where m > n. The algebraic relations among basic invariants can be described by a finitely generated ideal IB in the polynomial

ring K[{Bα }]. The normal form of an invariant f ∈ K[{Bα }] is the result of reduction relative to a Gr¨ obner basis of IB . Given invariants g, f1 , . . . , fk ∈ K[{Bα }], an invariant division of g by the fi can be defined as the reduction of g with respect to a Gr¨ obner basis of the ideal IB , f1 , . . . , fk  obner basis defines an invariant diviin K[{Bα }]. Each Gr¨ sion. In application, however, the invariant division defined above obhas several defects. With the introduction of IB , the Gr¨ ner basis of IB , f1 , . . . , fk  can be very complicated. Even for the simplest case k = 1 and G = SL(n), it is not clear what the Gr¨ obner basis is like for a general invariant f1 , and there can be many elements partially generated by f1 in the Gr¨ obner basis. For G = SL(n), the number of basic invarin , which grows rapidly with the increase of number ants is Cm of vector variables m, causing the Gr¨ obner basis computing impractical due to too many indeterminates. In this paper, we propose an SL(n)-invariant division based on an admissible order among the normal forms of SL(n)invariants. When k = 1, instead of computing a Gr¨ obner basis in K[{Bα }], we do top reduction to any normalized inobner variant g by f1 directly. When k > 1, we compute a Gr¨ basis of the ideal generated by f1 , . . . , fk in the quotient ring K[{Bα }]/IB , and use it to do reduction to g. For example, the invariant division of f by g in (1.1) is as following:

2.

BRACKET ALGEBRA

In classical invariant theory, let vi = (xi1 , ..., xin )T for i = 1..m be m vectors in an n-dimensional vector space over a field K, then the bracket [vi1 . . . vin ] refers to the determinant of the matrix composed of the coordinates of vi1 , . . . , vin , and any SL(n)-invariant polynomial in {xij | i = 1..m, j = 1..n} is a polynomial in the brackets of the vi . The bracket algebra refers to the ring of bracket polynomials. Another description of bracket algebra is coordinate-free [13]: Definition 2. Let n > 0, and let v1 , . . . , vm be m > n different letters. Denote by Bn [{vα }] the set of all n-tuples of the letters, and let K[Bn [{vα }]] be the polynomial ring with variables in Bn [{vα }]. The n-dimensional bracket algebra generated by the m letters is the quotient of K[Bn [{vα }]] modulo the ideal IB generated by the following two kinds of elements: 1. vi1 . . . vin , if vij = vik for some j = k, and vi1 . . . vin +vi1 . . . vij−1 vik vij+1 . . . vik−1 vij vik+1 . . . vin for any j < k, if the vil are pairwise different; n l 2. l=1 (−1) (vi1 . . . vin−1 vjl )(vj1 . . . vjl−1 vjl+1 . . . vjn+1 ), where the vil are pairwise different, so are the vjl .

f = g(−y2 x3 + y3 x2 ) + (y2 x1 − y1 x2 )(y4 x3 − y3 x4 ). (1.3)

The equivalence class of an n-tuple vi1 . . . vin , denoted by [vi1 . . . vin ], is called a bracket of length (or size) n. The bracket algebra is denoted by the same symbol Bn [{vα }].

The quotient and the remainder are both SL(2)-invariant, and the quotient is essentially obtained from that of (1.2) by complementing −y2 x3 with y3 x2 to make the result invariant. The SL(n)-invariant division we propose is closely related to multivariate coordinate polynomial division. The admissible order among the normal forms of SL(n)-invariants agrees with the degree lexicographical order of coordinate monomials by Laplace expansion, so the invariant division can be done with the aid of coordinate polynomial division. This agreement makes it possible to answer the following question of consistency of invariant ideal membership. Given G-invariant polynomials g1 , ..., gk , let their coordinate polynomial forms be g1c , ..., gkc respectively, when K[{Bα }] is taken as a subring of the polynomial ring of coordinates K[{xij }]. For any G-invariant polynomial f in the ideal generated by g1c , ..., gkc in K[{xij }], is f always a member of the invariant ideal generated by g1 , ..., gk in K[{Bα }]? The invariant Gr¨ obner basis theory established upon the invariant division leads to an affirmative answer for G = SL(n), although in the case where the characteristic of K is 0, the answer can be given by the Reynolds operator [12]. This paper is organized as follows. Section 2 gives brief introduction to bracket algebra. Section 3 provides an admissible order among straight bracket monomials. Section 4 establishes an SL(n)-invariant division and the corresponding invariant Gr¨ obner basis theory. Section 5 discusses the coordinate-normal form of bracket polynomials and confirms the consistency of SL(n)-invariant ideal membership. By limitation of space we can only provide sketches of some of the proofs while omitting all the rest. Our work on G-invariant division is already extended to classical invariant theories of n-dimensional affine group, orthogonal group, and Euclidean group.

The first kind of elements in IB defines the antisymmetry among the entries of a bracket. The second kind of elements is called a Grassmann-Pl¨ ucker (GP) polynomial, and the corresponding equality is called a GP relation. It is equivalent to the requirement that the exterior product of any n + 1 different letters is zero, so it defines the dimension n of the vector space spanned by the m letters as vectors. Throughout this paper we use n to denote the dimension of the bracket algebra. Let “≺” be a total order among the m letters. We always assume that the n-tuples of letters take the lexicographical order induced from “≺”. By antisymmetry, the normal form of a bracket is the following: if two entries of the bracket are identical, then the normal form is zero; otherwise the bracket is nonzero, and by permuting the entries to make the sequence ascending, the normal form is the ascending sequence multiplied by the sign of permutation. A monic bracket monomial is the product of finitely many nonzero brackets. A bracket monomial of k nonzero bracket factors is said to be of bracket degree k. A bracket polynomial is said to be homogeneous if the multiset of vector variables in each term is the same. Throughout this paper, by “bracket polynomial”, or “f ∈ Bn [{vα }]”, we always assume the homogeneity. The bracket degree of a bracket polynomial is that of its terms. The tableau form of a monic bracket monomial of bracket degree k is the table of k by n, where each row is the sequence of entries of a bracket factor, and the order of the rows follows that of the bracket factors. A tableau (or bracket monomial) is said to be straight, if lexicographically each row is ascending, while each column is non-descending. A column is said to be straight if it is non-descending. A nonzero bracket polynomial is said to be straight if so is its every term.

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A classical theorem in bracket algebra [16] states that every bracket polynomial equals a unique straight bracket polynomial, called the normal form, or straight form. The procedure of deriving the normal form, called straightening, is by repeatedly applying the following shuffle formula, also known as the van der Waerden (VW) relation: let Ap Bn+1 Cn−p−1

= vi 1 , . . . , vi p , = vj1 , . . . , vjn+1 , = vl1 , . . . , vln−p−1

be three sequences of letters where no two are identical in the same sequence, and where 0 ≤ p ≤ n − 1, then    Bn+1(2) Ap = 0, (2.1) Bn+1(1) Cn−p−1 (p+1,n−p)Bn+1

where Bn+1(1) , Bn+1(2) are the Sweedler notation of two subsequences of length p+1, n−p obtained by bipartitioning Bn+1 , with the sign of permutation of the partition included in the notation. Below we check to see how the VW relations are used to straighten a bracket monomial. Suppose n = 3, and v1 ≺ . . . ≺ v6 . Then   v1 v 4 v 6 (2.2) f= v2 v3 v5 is not straight, or more explicitly, the latter two columns are not straight. If the straightening is done to the last column, then p = 2 in (2.1), A2 = v1 v4 , and C0 is empty. The corresponding VW relation, after moving term f to one side of the equality, is       v1 v3 v4 v1 v2 v4 v1 v4 v5 + − . f= v2 v3 v6 v2 v5 v6 v3 v5 v6 (2.3) In the result, all the columns are straight except for the second column in the first term. One more straightening of the non-straight column leads to the normal form of f . If the straightening is done to the middle column of (2.2), the result after applying the corresponding VW relation is different from (2.3). Further straightening leads to the same normal form of f . That the straightening result is independent of the choice of applicable VW relations is a property of top reduction by a Gr¨ obner basis. Definition 3. The row total sequence of a bracket monomial of bracket degree k is the sequence of kn entries obtained by scanning the tableau form of the bracket monomial row by row sequentially. The row order “≺row ” among bracket monomials, is the degree lexicographical order among the row total sequences of the bracket monomials. Under the row order, straightening by VW relation is a strict order-decreasing procedure. By [14], the VW polynomials contain a Gr¨ obner basis of the generating ideal IB of bracket algebra under the row order. Now that in bracket algebra, any bracket polynomial has a unique normal form, to make top reduction we need to find an admissible monomial order “≺tot ” among monic straight bracket monomials, just like the case of “standard” coordinate monomials where the coordinate indeterminates are permuted in non-ascending order. Such an order should satisfy the following: (i) It is a strict total order, i.e., for any two monic straight bracket monomials f = g, either f ≺tot g or f tot g.

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(ii) For any straight bracket monomial h = 0, 0 ≺tot h. (iii) For any two straight bracket monomials f ≺tot g, and any other straight bracket monomial h = 0, if Lu denotes the leading term of the normal form of a bracket polynomial u under the order “≺tot ”, then Lf h ≺tot Lgh . An admissible order among straight bracket monomials induces a partial order, still denoted by “≺tot ”, among bracket polynomials: for any two bracket polynomials f, g, f ≺tot g if and only if Lf ≺tot Lg . The above property (iii) of “≺tot ” guarantees that this order is preserved by the product of bracket polynomials. The row order “≺row ” of straight bracket monomials, is unfortunately not an admissible order. For example, when n = 3, let v1 ≺ . . . ≺ v9 , and let   v2 v4 v7 , f = v5 v6 v8   v2 v5 v6 g = , v4 v7 v8   v 1 v3 v9 . h = All the three bracket monomials are straight, and f ≺row g. However, after straightening f h and gh, we get that in their normal forms, ⎡ ⎡ ⎤ ⎤ v1 v 3 v 7 v1 v 3 v 6 Lf h = ⎣ v2 v4 v8 ⎦ , Lgh = ⎣ v2 v5 v8 ⎦ , (2.4) v5 v6 v9 v4 v7 v9 so f h row gh.

3.

ADMISSIBLE ORDER OF STRAIGHT BRACKET MONOMIALS

Definition 4. Let f be a tableau of k rows and n columns. The column non-descending permutation of f , denoted by f  , is the k × n tableau whose i-th column is the downward non-descending permutation of the i-th column of f , for all 1 ≤ i ≤ n. 

   f g , and Lgh = . h h It is easy to prove that for any bracket monomial f whose bracket factors are in normal form, f  is always nonzero and straight. For example, in (2.4), Lf h =

Definition 5. The column total sequence of a bracket monomial f of bracket degree k, denoted by f˜, is the sequence of kn entries obtained by scanning f column by column sequentially. The negative column order among monic bracket monomials, denoted by “≺”, is defined as follows: for any two bracket monomials f, g of bracket degree df , dg respectively, (i) if df < dg , then f ≺ g. In particular, 0 ≺ f for all f = 0. (ii) If df = dg > 0 and f˜ lex g˜, then f ≺ g. For example, let v1 ≺ v2 ≺ v3 ≺ v4 , then [v2 v1 ] ≺ −[v1 v2 ]. So straightening a single bracket is an orderincreasing procedure under the negative column order. As another example,       v 1 v2 v1 v 3 v1 v 4 ≺ − + . (3.1) v2 v3 v3 v4 v2 v4

From the left of (3.1) to the right is a straightening procedure based on VW relation. On the right side, the leading term is the second term, and has higher negative column order than the input term. This example shows that straightening by VW relation is an order-increasing procedure under the negative column order. The row order and the negative column order are related as follows. Proposition 1. Let M be a multiset of kn entries. Denote by S the set of straight bracket monomials of bracket degree k, with M as the multiset of entries. If k = 2 or n = 2, then when restricted to the set S, the negative column order agrees with the row order. When one of k and n is > 2, the conclusion of Proposition 1 is no longer true. For example in (2.4), where v1 ≺ . . . ≺ v9 , both Lf h and Lgh are straight, and Lf h ≺ Lgh in the negative column order, but Lf h row Lgh . The following results disclose the relationship between the negative column order of straight tableaux and the column non-descending permutation of a tableau. Proposition 2. Let f, g be two tableaux of size k ×n and l × n respectively. The concatenation f g of f and g is the (k + l) × n tableau obtained by attaching g to f . If f, g, h are three straight tableaux of size k × n, k × n and l × n respectively, then in the negative column order, f  g if and only if (f h)  (g h) . Theorem 3. Let f be a bracket monomial where each bracket is nonzero and in normal form. After straightening f , the leading term of the normal form in the negative column order is f  . By Proposition 2 and Theorem 3, we get Corollary 4. The negative column order is an admissible order among straight bracket monomials. The proof of Theorem 3 is by induction on the bracket degree k ≥ 2. Although the base case k = 2 can be proved by straightening with the VW relation, it is simpler to use another commonly used relation, the so called multiple relation [15]: for three sequences of letters A = vi1 , . . . , vin , B = vj1 , . . . , vjk , and C = vl1 , . . . , vln−k ,      B A(2) BC . (3.2) = A(1) C A (k,n−k)A

4. INVARIANT DIVISION AND INVARIANT GRÖBNER BASIS Definition 6. Given two straight bracket monomials f, g of bracket degree k, l respectively, where k ≥ l > 0, in the tableau forms of f, g, if for any 1 ≤ i ≤ n, the i-th column of g is a subsequence of the i-th column of f , and when k = l, after removing the entries of g from f column by column, the remaining entries of f , called the complement of g in f , form a straight tableau, then g is called a divisor of f , and f is said to be top reducible by g. Let vj be an entry of the i-th column of g. Then in the i-th column of f , vj may occur more than once. By the

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straightness of f , all the copies of vj in the i-th column of f are in neighboring positions, so removing any of them leads to the same result. This proves the correctness of the above definition. For example, let v1 ≺ . . . ≺ v6 , and let ⎤ ⎡ v 1 v3 f = ⎣ v2 v4 ⎦ , g1 = [v1 v4 ], g2 = [v1 v6 ]. v5 v6 Then g1 is a divisor of f with complement [v2 v3 ][v5 v6 ], while g2 is not a divisor, because its complement [v2 v3 ][v5 v4 ] is not straight. Definition 7. For two bracket polynomials f, g of bracket degree k, l respectively, where k ≥ l > 0, the following procedure generates two straight bracket polynomials h, r such that (1) f = gh + r; (2) Lr  Lf in negative column order; (3) either r is zero or Lr is not a divisor of Lg : Step 1. Set h = 0 and r = f ; normalize g. Step 2. Normalize r. Step 3. If Lg is not a divisor of Lr , output h, r and exit. Step 4. Set h = h + cLg , where c is the complement of Lg in Lr , and set r = r − cLg ; go to Step 2. The above procedure is called the invariant division of f by g, and h, r are called the quotient, remainder, respectively. If S is a set of bracket polynomials, then f −→∗S r denotes that f is top reduced to r by S. Let M be a fixed multiset of kn entries. Denote by S the set of straight bracket monomials of bracket degree k, with M as the multiset of entries. The termination of the invariant division is due to the finiteness of the set S, to which every r in the division procedure belongs. The invariant division has the following basic property: Proposition 5. (1) For any bracket polynomials f, g, the remainder of f by g after invariant division is zero if and only if there exists a bracket polynomial h such that f = gh. (2) For any two bracket polynomials f1 , f2 , the remainders of f1 , f2 by g after invariant division are identical if and only if f1 − f2 = gh for some bracket polynomial h. With the invariant division at hand, we start to establish the Gr¨ obner basis theory of bracket polynomials. Definition 8. Let I be an ideal in bracket algebra, and let G be a subset of I. Then G is called a Gr¨ obner basis of I if every element of I is top reducible by G. Let G be a Gr¨ obner basis of an ideal I in bracket algebra. It is easy to prove that any f ∈ I satisfies f −→∗G 0, and for any bracket polynomial h, if h −→∗G r1 and h −→∗G r2 , such that r1 , r2 are no longer top reducible by G, then r1 = r2 . Given m pairwise different letters v1 , . . . , vm , they form n different nonzero brackets of length n < m. Let n!Cm K[Bn∗ [{vα }]] be the polynomial ring with the nonzero brackets as indeterminates. Given f1 , f2 , . . . ∈ K[Bn∗ [{vα }]], the ideal generated by the fi in the n-dimensional bracket algebra is just the ideal If of K[Bn∗ [{vα }]] generated by the fi and the generators of IB of bracket algebra Bn [{vα }] in Definition 2, with the exclusion of all elements of the form [vi1 . . . vin ] where vij = vik for some j = k. Since If is finitely generated in K[Bn∗ [{vα }]], so is the ideal generated by f1 , f2 , . . ., in Bn [{vα }]. From this we deduce

Proposition 6. For any ideal in bracket algebra, there always exists a finite Gr¨ obner basis. We proceed to extend Buchberger’s algorithm [1], [2], [4] to bracket algebra. Definition 9. Given two straight tableaux (or bracket monomials) f, g, a straight tableau c is said to be a common multiplier of them, if there exist straight tableaux f1 , g1 such that 



c = (f g1 ) = (g f1 ) .

(4.1)

A common multiplier c of f, g is said to be a least common multiplier (LCM), if there does not exist any common multiplier d of f, g such that c = (d h) for some non-empty straight tableau h. For example, let v1 ≺ . . . ≺ v6 , and let f = [v1 v5 ], g = [v2 v6 ]. Then ⎤ ⎡   v1 v3 v1 v5 (4.2) , ⎣ v2 v5 ⎦ v2 v6 v4 v6 are both LCMs of f, g. This example shows that the monic LCMs of two straight bracket monomials are not unique. Notice that in (4.1), in general c = f g1 = gf1 as bracket monomials. Instead, c = Lf g1 = Lgf1 . Proposition 7. When f = g and both are straight tableaux of size 1 × 2, say f = [va1 va2 ], g = [vb1 vb2 ], then their LCMs are (f g) , together with ⎤ ⎡ min(va1 , vb1 ) vc ⎣ max(va1 , vb1 ) min(va2 , vb2 ) ⎦ , (4.3) max(va2 , vb2 ) vd for all vector variables vc , vd satisfying min(va2 , vb2 )  vd vc  max(va1 , vb1 ). The following is an estimation (not sharp) of the maximal number of rows of the LCMs of two general straight tableaux. It gives a finite list of candidates for us to find the LCMs by trial and error. Proposition 8. For two straight tableaux of size k×n, l× n respectively, let a LCM of them have size w × n, then w ≤ (k + l)n + 2kl(n − 1). If one of k, l equals 1, then w ≤ n + max(k, l). The latter estimation is sharp.

s-polyr (pf, qg) = (s-polyt (f, g)) h + p0 f − q0 g,

Proposition 9. For two straight tableaux f, g, if every entry of f ≺ every entry of g, then their LCM is unique and is f g. Definition 10. For two straight bracket polynomials g1 , g2 , let h be a LCM of Lg1 , Lg2 , let c1 , c2 be respectively the complement of Lg1 , Lg2 in h, and let l1 , l2 be respectively the coefficient of Lg1 , Lg2 . Then (4.4)

Proposition 11. Let I be an ideal in bracket algebra, and let G be a set of straight bracket polynomials in I. Then G is a Gr¨ obner basis of I if and only if for any g1 , g2 ∈ G, for any LCM h of Lg1 , Lg2 , s-polyh (g1 , g2 ) −→∗G 0. Proposition 12. Buchberger’s algorithm terminates and leads to a Gr¨ obner basis for any ideal in bracket algebra. For example, let v1 ≺ . . . ≺ v5 , and let f = [v1 v5 ], obner g = [v2 v4 ]. The following procedure computes the Gr¨ basis G of f, g using Buchberger’s algorithm. Step⎡1. By Proposition 7, the LCMs of Lf and Lg are ⎤   v1 v 3 v1 v 4 . a2 generates a a1 = ⎣ v2 v4 ⎦ and a2 = v2 v5 v3 v5 zero S-polynomial, while a1 generates     v2 v 3 v 1 v3 h = s-polya1 (f, g) = f− g v3 v4 v3 v5 ⎡ ⎤ ⎡ ⎤ v1 v 3 v 1 v2 = − ⎣ v2 v3 ⎦ − ⎣ v3 v4 ⎦ , v4 v5 v3 v5 where the last equality is by straightening. Step 2. By Proposition 8, the height of any LCM of Lh and Lg is between 4 and 5. By enumeration, we get that ⎡ ⎤ v1 v 3 ⎢ v1 v3 ⎥ ⎥ the LCMs of Lh and Lg are b1 = ⎢ ⎣ v2 v4 ⎦ and b2 = v4 v5 ⎡ ⎤ v1 v 3 ⎢ v2 v3 ⎥ ⎢ ⎥ ⎣ v2 v4 ⎦. We have v4 v5 ⎡ ⎤ v1 v2 s-polyb1 (g, h) = ⎣ v3 v4 ⎦ f, v3 v4 ⎡ =

v1 ⎣ v3 v3

⎡

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⎤ v2 v4 ⎦ g. v5

Step 3. Again by ⎡enumeration, ⎡ the LCMs ⎤ we get that ⎤ of v1 v 3 v1 v 3 ⎢ v1 v3 ⎥ ⎢ v2 v3 ⎥ ⎢ ⎥ ⎥ Lh and Lf are c1 = ⎢ ⎣ v2 v5 ⎦ and c2 = ⎣ v2 v5 ⎦. We v4 v5 v4 v5 have ⎡ ⎤ v1 v2 s-polyc1 (f, h) = ⎣ v3 v4 ⎦ f, v3 v5

is called the S-polynomial of g1 , g2 relative to h. The S-polynomials of two straight bracket polynomials are not unique, but finite.

(4.5)

for some monic straight bracket monomial h and straight bracket polynomials p0 , q0 , where p0 ≺ p and q0 ≺ q.

s-polyb2 (g, h)

In some special cases, the LCMs can be very simple.

s-polyh (g1 , g2 ) := l2 c1 g1 − l1 c2 g2

Proposition 10. Let f, g be two straight bracket polynomials, and let p, q be two monic straight bracket monomials such that (p Lf ) = (q Lg ) = r. Then there exists a divisor t of r such that

s-polyc2 (f, h)

=

v1 ⎣ v3 v3

⎤ v2 v5 ⎦ g. v5

So G = {f, g, h}. If the order of vector variables is changed to v1 ≺ v2 ≺ v5 ≺ v4 ≺ v3 , then G = {f, g}. In contrast, in coordinate form, let vi = (xi , yi )T , then the coordinate polynomial forms of f, g are c

f = −y1 x5 + y5 x1 ,

Let {vα | α = 1..m} be vector variables generating the ndimensional bracket algebra Bn [{vα }]. Let (xα1 , . . . , xαn )T be the bracket homogeneous coordinates of vα with respect to a fixed basis {e1 , . . . , en } of the n-dimensional vector space, i.e., ˇj ] := (−1)j−1 [vα e1 . . . e ˇj . . . en ], (5.1) xαj = (−1)j−1 [vα e ˇj denotes that ej does not occur in the sequence where e e1 . . . en . The above coordinatization of vα is just the GP relation Jvα = xα1 e1 + . . . + xαn en , where J := [e1 . . . en ] = 0. ⎛

x α1 1 ⎜ . . . vαn ) = (e1 . . . en ) ⎝ ... xα 1 n

(5.2)

... .. . ...

⎞ x αn 1 .. ⎟ , . ⎠ x αn n (5.3)

we get det(xαi j )i,j=1..n = J n−1 [vα1 . . . vαn ].

(5.4)

By the First Fundamental Theorem in classical invariant theory, a homogeneous polynomial f ∈ K[{xαj }] is SL(n)invariant if and only if it can be written as a polynomial in K[{det(xαj )}], and every 1 ≤ α ≤ m occurs the same number of times as the first subscripts of the indeterminates x’s in different terms of f . By (5.4), if f is of homogeneous degree k in the variables {det(xαj )}, then f equals J k(n−1) multiplied with a bracket polynomial in Bn [{vα }]. A bracket in Bn [{vα , ej }] is up to coefficient a subdeterminant. For example, [v1 . . . vk ek+1 . . . en ] is the determinant of the first k coordinates of each of v1 , . . . , vk . Definition 11. Let v1 ≺ . . . ≺ vm , and let e1 ≺ . . . ≺ en . The following order (row by row) among the bracket homogeneous coordinates xαj is called the negative basis order: x1n  x1(n−1) .. .  x11

 x2n  x2(n−1) .. .  x21

 ...  ... .. .  ...

 xmn  xm(n−1) .. .  xm1 .

(5.5)

For example, if ei ≺ ej , then under the negative basis order, xαi ≺ xβj for any α, β ∈ {1, . . . , m}. If vα ≺ vβ , then xαi  xβi for all i = 1..n.

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(5.6)

a standard monic monomial in the xαi must be of the form

g = −y2 x4 + y4 x2 .

5. COORDINATE-NORMAL FORM AND INVARIANT IDEAL MEMBERSHIP

J(vα1

e 1 ≺ . . . ≺ en ≺ v1 ≺ . . . ≺ v m ,

c

With respect to the degree lexicographical order induced by the order of indeterminates y1  . . .  y5  x1  . . .  x5 , the Gr¨ obner basis of the ideal f c , g c  in K[xi , yi | i = 1..5], c is {f , g c }. No matter how the order of indeterminates is changed, the Gr¨ obner basis remains the same.

From

Under the order of variables

xin1 n xin2 n . . . xink n n xin−1 (n−1) xin−1 (n−1) . . . xin−1 1

2

kn−1

.. .

(n−1)

(5.7)

xi1 1 xi1 1 . . . xi1 1 , 1

2

k1

where for all 1 ≤ l ≤ n, kl ≥ 0, and il1≤ il2 ≤ . . . ≤ ilkl . n By (5.1), when multiplied with J l=1 kl , (5.7) equals a bracket monomial whose tableau form is up to sign the following: ⎤ ⎡ e1 e2 . . . en−2 en−1 vin1 ⎥ ⎢ e1 e2 . . . en−2 en−1 vin2 ⎥ ⎢ ... ... ⎥ ⎢ ... ... ... ... ⎥ ⎢ n ⎥ ⎢ e1 e2 . . . en−2 en−1 vikn ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ e1 e2 . . . en−2 en vin−1 ⎥ ⎥ ⎢ 1 ⎢ e1 e2 . . . en−2 en vin−1 ⎥ ⎥ ⎢ 2 ⎥ ⎢ ... ... ... ... ... ... ⎥ ⎢ (5.8) ⎥. ⎢ e e . . . e e v n−1 1 2 n−2 n ⎥ ⎢ ik ⎢ n−1 ⎥ ⎥ ⎢ . .. .. .. .. .. ⎥ ⎢ . . . . . . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ e vi1 ⎥ ⎢ 2 e3 . . . en−1 en 1 ⎥ ⎢ ⎥ ⎢ e2 e3 . . . en−1 en vi1 2 ⎥ ⎢ ⎦ ⎣ ... ... ... ... ... ... vi1 e2 e3 . . . en−1 en k1

The tableau is straight except for the last column. That tableau (5.8) is not straight is equivalent to that the corresponding bracket monomial is not in normal form under the row order of the bracket monomials in Bn [{vα , ej }]. Then under what order is the tableau in normal form? The answer is the column (lexicographical) order. Proposition 13. Let f be a bracket monomial in Bn [{vα , ej }] of bracket degree k, whose multiset of vector variables is composed of s entries of the vα , and kn−s entries of the ej . Then by the following formula: for any bipartition (ei1 , . . ., ein−h ; ej1 , . . . , ejh ) of e1 , . . . , en , where 2 ≤ h ≤ n, 

e1 . . . ei1 

... ... . . . ein−h

... v l1

en ...

ˇ jh . . . e n e1 . . . e (ei1 . . . ein−h ejh )↑

 v lh v l1

=

h 

(−1)t−1

t=1

v lt ˇ l t . . . vl h ... v

 ,

(5.9) where A↑ denotes the ascending permutation of sequence A, J s−k f can be changed into a bracket polynomial where in the tableau form of each term, the first n − 1 columns are straight and are composed of the ej . The result is called the coordinate-normal form of f . (5.9) is a procedure of strictly reducing the column order of the left side. The coordinate-normal form of a bracket polynomial is the sum of the coordinate-normal forms of its terms. Two bracket polynomials are equal if and only if they have identical coordinate-normal form.

For a bracket polynomial f ∈ Bn [{vα }] of bracket degree k, in its coordinate-normal form, each term has the tableau form (5.8) where k1 = . . . = kn = k. The coordinate-normal form can also be obtained from the Laplace expansion of the brackets in f . We see that there are two normal forms for a bracket polynomial f ∈ Bn [vα ] of bracket degree k: the straight form, and the coordinate-normal form. The straight form is obtained by top reduction of f with VW relation under the row order, and the coordinate-normal form is obtained by top reduction of J k(n−1) f with (5.9) under the column order. The two normal forms are equal up to factor J k(n−1) . From (5.5)–(5.8), we find out that the negative column order in bracket algebra Bn [{vα , ej }] among the bracket monomials where each bracket factor is of the form [eˇj vα ], agrees with the degree lexicographical order among the coordinate monomials in K[{xαj }]. Hence the multivariate polynomial division in K[{xαj }] under the degree lexicographical order is just the invariant division in Bn [{vα , ej }] under the negative column order. Compared with the invariant division in Bn [{vα }], the invariant division in coordinate-normal form is much easier, because normalizing the product of two bracket monomials of the form (5.8) is simply by reordering the bracket factors in the product. Lemma 14. Under the order (5.5), if k1 = . . . = kn = k and il1 ≤ il2 ≤ . . . ≤ ilkl for all 1 ≤ l ≤ n, then (5.8) is up to sign the leading term of the Laplace expansion of the following straight bracket monomial: ⎤ ⎡ vin1 vin−1 . . . vi11 1 ⎢ v n v n−1 . . . v 1 ⎥ ⎢ i2 i2 ⎥ i2 (5.10) ⎥. ⎢ ... ... ⎦ ⎣ ... ... vink vin−1 . . . vi1 k

k

Proposition 15. Under Laplace expansion, the degree lexicographical order among polynomials of the xαj induced by (5.5) agrees with the negative column order among straight bracket monomials of the vα . The above proposition suggests a coordinate approach to invariant division. Given f, g ∈ Bn [{vα }], let their coordinate forms be f c , g c respectively, with leading terms lt(f c ), lt(g c ) under the degree lexicographical order induced by the negative basis order of the coordinates {xαj }. The monomial division of lt(f c ) by lt(g c ) results in a quotient h that is up to coefficient of the form (5.7). By Lemma 14, h is the leading term of the Laplace expansion E of bracket monomial (5.10). Let T = E − h. Then f c = (h + T )g c + r, where r is SL(n)-invariant, and r ≺ f c . Continue to reduce r by g c with this strategy of “divide and complement”. The invariant division of f c by g c is thus realized. In Section 1, the invariant division (1.3) obtained from the usual division (1.2) is such an example. Proposition 16. Let f ∈ Bn [{vα }], and let its coordinate polynomial form be f c . Then f is factorizable in Bn [{vα }] if and only if so is f c in K[{xαj }]. In the end, we consider the consistency problem of the ideal membership in bracket algebra Bn [{vα }] and the corresponding coordinate polynomial ring K[{xαj }].

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Theorem 17. Let f, g1 , . . . , gs ∈ Bn [{vα }] be homogeneous bracket polynomials, and let their coordinate polynomial forms be f c , g1c , . . . , gsc respectively. Then f is in the ideal generated by the gi in Bn [{vα }] if and only if f c is in the ideal generated by the gic in K[{xαj }]. Only the “if” part needs proof. We need the following framework. K[y]

Definition 12. Let Bn [{vα , ej }] be the n-dimensional bracket module over the coefficient ring K[y] and generated by the vα and the ej . The basis bracket module Bn [y, {vα , ej }] K[y] is the quotient of Bn [{vα , ej }] modulo the ideal generated by Jy − 1, i.e., [e1 . . . en ]y = 1.

(5.11)

K[y]

Write every xαj as a bracket in Bn The hypothesis is

[{vα , ej }] by (5.1).

f = h1 g1 + . . . + hs gs + hy (yJ − 1)

(5.12)

K[y]

for some hi ∈ Bn [{vα , ej }]. The conclusion is that there exist r1 , . . . , rs ∈ Bn [{vα }] such that f = r1 g1 + . . . + rs gs . We need to eliminate the ej from the right side of (5.12). Set the order of variables as y ≺ v1 ≺ . . . ≺ v m ≺ e 1 ≺ . . . ≺ e n .

(5.13)

Let the ideal g1 , . . . , gs , yJ − 1BK[y] [{v ,e }] have a reduced α j n Gr¨ obner basis p1 , . . . , pw , where each bracket polynomial pi is in straight form, and the leading terms of p1 , . . . , pu do not involve y and the ej , but not so for the leading terms of pu+1 , . . . , pw . By reducedness we mean the leading terms of any two different pi , pj are irreducible with respect to each other. By Proposition 9, it can be proved that all terms of p1 , . . . , pu do not involve y and the ej , and p1 , . . . , pu B[{vα }] = g1 , . . . , gs , yJ−1BK[y] [{v n

α ,ej }]

∩B[{vα }].

(5.14) After this, it only remains to prove p1 , . . . , pu B[{vα }] ⊆ g1 , . . . , gs B[{vα }] . This inclusion is nontrivial, as we only know that for all 1 ≤ l ≤ u, 1 ≤ i ≤ s, pl =

s 

hli gi ,

i=1

gi =

u 

fij pj

(5.15)

j=1

for some hli = hli (y, {vα , ea }) and fij = fij ({vα }). The inclusion is proved by permuting {pi , 1 ≤ i ≤ u} by the bracket degree to a non-descending sequence, and using induction on the bracket degree to prove that every gi can obner basis. replace some pl in the reduced Gr¨

6.

CONCLUSION

As is well known, coordinate polynomial division and factorization can be used to do geometric theorem proving [3], [8]. When it comes to geometric conclusion discovering, however, such division leads to non-geometric results. In projective geometry, although invariant division and invariant factorization cannot be used to prove more theorems than their coordinate counterparts, they are theoretically more suitable to do geometric computing since they always lead to geometric results. Our future work will include geometric applications.

7. REFERENCES [1] Adams, W. and Loustaunau, P. An Introduction to Gr¨ obner Basis. Graduate Studies in Mathematics 3, American Mathematical Society, Providence, 1994. [2] Becker, T. and Weispfenning, V. Gr¨ obner basis. A computational approach to commutative algebra. GTM 141, Springer, New York, 1993. [3] Buchberger, B. Application of Gr¨ obner basis in non-linear computational geometry, in: Scientific Software, J. Rice (ed.), Springer, New York, 1988. [4] Cox, D., Little, J., and O’shea, D. Ideals, Varieties, and Algorithms, 3rd Edition. Springer, New York, 2007. [5] Crapo, H. and Richter-Gebert, J. Automatic proving of geometric theorems, in: Invariant Methods in Discrete and Computational Geometry, N. White (ed.), Kluwer Academic Publishers, 1994, 107–139. [6] Dress, A. and Wenzel, W. Grassmann-Pl¨ ucker relations and matroids with coefficients. Advances in Mathematics, 1991, 86, 68–110. [7] Hestenes, D. and Sobczyk, G. Clifford Algebra to Geometric Calculus. D. Reidel, Dordrecht, Boston, 1984. [8] Kutzler, B. and Stifter, S. On the application of Buchberger’s algorithm to automated geometry theorem proving, J. Symb. Comp., 1986, 2: 389–398. [9] Li, H. Invariant Algebras and Geometric Reasoning. World Scientific, Singapore, 2008. [10] Mourrain, B. New aspects of geometric calculus with invariants. MEGA’91, 1991. [11] Olver, P.J. Classical Invariant Theory. Cambridge University Press, Cambridge, 1999. [12] Rota, G.-C. Reynolds operators. Proc. Sympos. Appl. Math., Vol. XVI, Providence, R.I.: Amer. Math. Soc., pp. 70-83, 1964. [13] Rota, G.-C. and Stein, J. Symbolic method in invariant theory. Proc. Nat. Acad. Sci. 83: 844-847, 1986. [14] Sturmfels, B. and White, N. Gr¨ obner bases and invariant theory. Advances in Math. 76: 245-259, 1989. [15] White, N. Implementation of the straightening algorithm of classical invariant theory. In: Invariant Theory and Tableaux, D. Stanton (ed.), pp. 36-45, Springer, New York, 1990. [16] Young, A. The Collected Papers of Alfred Young, 1873-1940. Univ. of Toronto Press, 1977. This paper is supported partially by China 973 project 2011CB302404, and NSFC 10871195, 60821002/F02.

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