The Block Structure of Three Dixon Resultants and Their ... - CiteSeerX

The Block Structure of Three Dixon Resultants and Their Accompanying Transformation Matrices Eng-Wee Chionh [email protected] Department of Information Systems and Computer Science National University of Singapore Kent Ridge, Singapore 119260 Ming Zhang Ronald N. Goldman [email protected] [email protected] Department of Computer Science Rice University Houston, Texas 77005 Abstract

Dixon [1908] introduces three distinct determinant formulations for the resultant of three bivariate polynomials of bidegree (m; n). The rst technique applies Sylvester's dialytic method to construct the resultant as the determinant of a matrix of order 6mn. The second approach uses Cayley's determinant device to form a more compact representation for the resultant as the determinant of a matrix of order 2mn. The third method employs a combination of Cayley's determinant device with Sylvester's dialytic method to build the resultant as the determinant of a matrix of order 3mn. Here relations between these three resultant formulations are derived and the structure of the transformations between these resultant matrices is investigated. In particular, it is shown that these transformation matrices all have similar, simple, upper triangular, block symmetric structures and the blocks themselves have elegant symmetry properties. Elementary entry formulas for the transformation matrices are also provided. In light of these results, the three Dixon resultant matrices are reexamined and shown to have natural block structures compatible with the block structures of the transformation matrices. These block structures are analyzed here and applied along with the block structures of the transformation matrices to simplify the calculation of the entries of the Dixon resultants of order 2mn and 3mn and to make these calculations more ecient by removing redundant computations.

1 Introduction Dixon [1908] describes three distinct determinant expressions for the resultant of three bivariate polynomials of bidegree (m; n). The rst formulation is the determinant of a 6mn  6mn matrix generated by Sylvester's dialytic method. We shall refer to this determinant as the Sylvester resultant. The second expression is the determinant of a 2mn  2mn matrix found by using an extension of Cayley's determinant device for generating the Bezout resultant for two univariate polynomials. This 2mn  2mn matrix has come to be known as the Dixon resultant [Sederberg, 1983; Sederberg et al 1984; Chionh, 1990; Manocha and Canny, 1992]. However, to distinguish this determinant expression from Dixon's other two formulations for the resultant, we shall call this determinant the Cayley resultant. The third representation is the determinant of a 3mn  3mn matrix generated by combining Cayley's determinant device with Sylvester's dialytic method. We shall refer to this determinant as the mixed Cayley-Sylvester resultant. 1

Dixon introduces these three representations for the resultant independently, without attempting to relate one to the other. The purpose of this paper is to derive connections between these three distinct representations for the resultant. In particular, we will prove that the polynomials represented by the columns of the Cayley and the mixed Cayley-Sylvester resultants are linear combinations of the polynomials represented by the columns of the Sylvester resultant. Thus there are matrices relating the Sylvester resultant to the Cayley and mixed Cayley-Sylvester resultants. We shall show that these transformation matrices all have similar, simple, upper triangular, block symmetric structures and the blocks themselves are either Sylvester-like or symmetic. In addition, we shall provide straightforward formulas for the entries of these matrices. Reexamining the three Dixon resultant matrices, we nd that they too have simple block structures compatible with the block structures of the transformation matrices. Indeed, we shall see that the blocks for the transformation matrices are essentially transposes of the blocks for these resultant matrices, although the arrangements of the blocks within these matrices dier. We shall also show that the blocks of the Sylvester and mixed Cayley-Sylvester matrices are related to the Sylvester and Bezout resultants of certain univariate polynomials. The entries of the Sylvester resultant are just the coecients of the original polynomials, but the entries of the Cayley and mixed Cayley-Sylvester resultants are more complicated expressions in these coecients. We shall show how to take advantage of the block structure of the resultant matrices together with the block structure of the transformation matrices to simplify the calculation of the entries of the Cayley and mixed Cayley-Sylvester resultants and to remove redundant computations. These techniques for the ecient computation of the entries of the Cayley and mixed Cayley-Sylvester resultants are extensions to the bivariate setting of similar results on the ecient computation of the entries of the Bezout resultant of two univariate polynomials [Chionh et al 1998]. In a follow up paper [Zhang et al 1998] we plan to derive a new class of representations for the bivariate resultant in terms of determinants of hybrid matrices formed by mixing and matching appropriate columns from the Sylvester, Cayley, or mixed Cayley-Sylvester matrices. The derivation of these hybrid resultants is also based on the block structure of the Dixon resultants and the accompanying transformation matrices derived in this paper. Again these new hybrid resultants are generalizations to the bivariate setting of known hybrids between the Sylvester and Bezout resultants for two univariate polynomials [Weyman and Zelevinsky 1994; Sederberg et al 1997; Chionh et al 1998]. This paper is structured in the following manner. We begin in Section 2 by establishing our notation and reviewing Dixon's formulations of the Sylvester, Cayley, and mixed Cayley-Sylvester resultants. Next, in Section 3, we describe the truncated formal power series technique that is essential for the derivation of the matrices relating these three Dixon resultants. We go on in Sections 4 and 5 to derive explicit formulas for the entries of the matrices relating the Sylvester resultant to the Cayley and mixed CayleySylvester resultants and to discuss the block structure and symmetry properties of these transformation matrices. We devote Section 6 to deriving similar results for the transformation relating the mixed Cayley-Sylvester and Cayley resultants. We close in Section 7 by analyzing the block structure of the three Dixon resultants and showing how to take advantage of this block structure together with the block structure of the transformation matrices to simplify the calculation of the entries of the Cayley and mixed Cayley-Sylvester resultants and to make the calculations more ecient by removing redundant computations. We also derive some remarkable convolution identities relating the blocks of the Sylvester and the blocks of the mixed Cayley-Sylvester resultants.

2 The Three Dixon Resultants Consider three bivariate polynomials of bidegree (m; n): f(s; t) =

m X n X i=0 j =0

ai;j si tj ;

g(s; t) =

n m X X k=0 l=0

2

bk;l sk tl ;

h(s; t) =

n m X X p=0 q=0

cp;q sp tq :

Dixon outlines three methods for constructing a resultant for f; g; h [Dixon 1908]. In this section, we brie y review the construction of each of Dixon's resultants.

2.1 The Sylvester Resultant The Sylvester resultant for f(s; t); g(s; t); h(s; t) is constructed using Sylvester's dialytic method. Consider 6mn polynomials  f(s;the  fs t f; s t g; s t h j  = 0;


; 2m ? 1;  = 0;


; n ? 1g. Let L(s; t) = t) g(s; t) h(s; t) ; then this system of polynomials can be written in matrix notation as

2 1 66 .. . 6  L


s t L s t +1 L


s2m?1tn?1L  = 66 s t 66 s t +1 64 .. .

3T 77 77 77 Sm;n; 77 5

(1)

s3m?1 t2n?1 where the rows are indexed lexicographically with s > t. That is, the monomials are ordered as 1; t;


; t2n?1,


, s ; s t,


, s t2n?1,


; s3m?1 ; s3m?1t;


; s3m?1t2n?1. Notice that the coecient matrix Sm;n is a square matrix of order 6mn. The Sylvester resultant for f; g; h is simply jSm;n j.

2.2 The Cayley Resultant The Cayley resultant for f; g; h can be derived from the Cayley expression f(s; t) g(s; t) h(s; t) f(; t) g(; t) h(; t) f(; ) g(; ) h(; ) :
m;n (s; t; ; ) = ( ? s)( ? t)

(2)

Since the numerator vanishes when  = s or = t, the numerator is divisible by ( ? s)( ? t). Hence
m;n (s; t; ; ) is a polynomial in s; t; ; , so
m;n (s; t; ; ) = In matrix form,

X XX X

2m?1 n?1 m?1 2n?1

u=0 v=0 =0  =0

2 1 66 . 66 s..t
m;n (s; t; ; ) = 66 s t +1 66 .. 4 .

c;;u;v s t u v :

3T 2 1 77 66 . 77 66 u.. v 77 Cm;n 66 u v+1 77 66 .. 5 4 .

3 77 77 77 : 77 5

(3)

sm?1 t2n?1 2m?1 n?1 The rows and columns of Cm;n are indexed lexicographically by the pairs (s t ; u v ) with s > t,  > . Notice that the coecient matrix Cm;n is again a square matrix but of order 2mn. The Cayley resultant for f; g; h is jCm;nj.

2.3 The Mixed Cayley-Sylvester Resultant The mixed Cayley-Sylvester resultant can be derived by combining Cayley's determinant device with Sylvester's dialytic method. Consider the expression g(s; t) h(s; t) (4) (g; h) = g(s; ) h(s; ) =( ? t): 3

Since the numerator is always divisible by the denominator, (g; h) is a polynomial in s; t; , with degree 2m in s, n ? 1 in t, and n ? 1 in P .nCollecting the coecients of j , j = 0;


; n ? 1, we get n polyno? 1 mials f j (s; t) such that (g; h) = j =0 f j (s; t) j . Multiplying these n polynomials by the m monomials 1; s;


; sm?1 , we obtain mn polynomials si f j (s; t), 0  i  m ? 1, 0  j  n ? 1, in the 3mn monomials si tj , 0  i  3m ? 1, 0  j  n ? 1. Now do the same for







h(s; t) f(s; t) =( ? t); (h; f) = h(s; ) f(s; )



f(s; t) g(s; t) =( ? t): (f; g) = f(s; ) g(s; )

Altogether we get 3mn polynomials si f j ; sig j ; si hj , j = 0;


; n ? 1, i = 0;


; m ? 1, in 3mn monomials, where (g; h) =

nX ?1 j =0

f j (s; t) j ;

(h; f) =

nX ?1 j =0

gj (s; t) j ;

(f; g) =

P ?1 suu and L = [ f g h ]. Then Let (s; ) = mu=0 k k k k   (s; ) (g; h) (h; f) (f; g)

nX ?1 j =0

hj (s; t) j :

2 1 66 . 66 u.. v = [ L0


Ln?1


sm?1 L0


sm?1 Ln?1 ] 66 u v+1 66 .. 4 . m?1 n?1 2 1 3 2 1 3T 66 66 7 7 . . 66 u.. v 777 66 s..t 777 = 66 s t +1 77 Mm;n 66 u v+1 77 ; 66 66 77 77 .. .. 4 4 5 5 . .

3 77 77 77 77 5

(5)

m?1 n?1 s3m?1 tn?1 where Mm;n is the coecient matrix of these 3mn polynomials. Note that in this notation, Mm;n has 3mn rows and mn columns, but each entry of Mm;n is actually a 1  3 matrix. Thus the real size of Mm;n is 3mn  3mn; hence Mm;n is a square matrix. The mixed Cayley-Sylvester resultant is jMm;nj.

2.4 Notation We plan to derive explicit formulas for the entries of the conversion matrices between Sm;n , Cm;n and Mm;n , and then to apply these formulas to elucidate the structure and simplify the computation of the entries of the three Dixon resultants. For convenience, we adopt the following notation:

b Ak;l;p;q = b k;l p;q

a i;j ji; j; k; l; p; qj = ak;l ap;q c ck;l , B k;l k;l;p;q = c cp;q p;q

Li;j = [ai;j bi;j ci;j ];



bi;j ci;j bk;l ck;l , bp;q cp;q a b ak;l , C k;l k;l , k;l;p;q = a ap;q p;q bp;q Rk;l;p;q = [Ak;l;p;q Bk;l;p;q Ck;l;p;q ], 4

2 0 ?c b 3 p;q p;q Xp;q = 4 cp;q 0 ?ap;q 5. ?bp;q ap;q

0

2.5 Example: m = n = 1 2 a0;0 66 a0;1 = 666 aa1;0 4 10;1

b0;0 c0;0 0 0 0 3 2 L0;0 0 3 b0;1 c0;1 0 0 0 77 66 L0;1 0 77 b1;0 c1;0 a0;0 b0;0 c0;0 77 = 66 L1;0 L0;0 77 ; S1;1 b1;1 c1;1 a0;1 b0;1 c0;1 75 64 L1;1 L0;1 75 0 0 a1;0 b1;0 c1;0 0 L1;0 0 0 0 a1;1 b1;1 c1;1 0 L1;1  j0; 0; 1; 0;0;1j ?j1; 0; 0; 0; 1;1j  C1;1 = j0; 0; 1; 1;0;1j ?j1; 0; 0; 1; 1;1j ; 2 A 3 2 R 3 B0;0;0;1 C0;0;0;1 0;0;0;1 0;0;0;1 M1;1 = 4 A0;0;1;1 + A1;0;0;1 B0;0;1;1 + B1;0;0;1 C0;0;1;1 + C1;0;0;1 5 = 4 R0;0;1;1 + R1;0;0;1 5 : A1;0;1;1 B1;0;1;1 C1;0;1;1 R1;0;1;1

2.6 Review and Preview In this section, we have introduced the three Dixon determinants for the resultant of three bivariate polynomials of bidegree (m; n). In Sections 4, 5, 6, we will investigate relationships between the three Dixon resultant matrices Sm;n , Cm;n , and Mm;n . It follows by inspection from Equations (1), (3), (5) that the columns of Dixon's three resultant matrices represent bivariate polynomials in s, t. In particular, the columns of Sm;n , Mm;n , Cm;n are respectively bivariate polynomials of bidegree (3m ? 1; 2n ? 1), (3m ? 1; n ? 1), (m ? 1; 2n ? 1) . Since the columns of Sm;n are linearly independent, the 6mn polynomials represented by these columns span the space of bivariate polynomials of bidegree (3m ? 1; 2n ? 1). Therefore the polynomials represented by the columns of Mm;n and Cm;n must be linear combinations of the polynomials represented by the columns of Sm;n . Sections 4, 5 are devoted to deriving explicit formulas for the transformations from Sm;n to Mm;n and Cm;n compatible with the polynomial indexing of the rows and columns.

3 The Technique of Truncated Formal Power Series The key technique that we shall harness to derive explicit relationships between the three Dixon resultant matrices is the method of truncated formal power series. To illustrate the method, let us consider a simple example. If we write   1 Bi 1 = 1 1 ? B ?1 = X A?B A A Ai+1 i=0

we can turn the division into the multiplication



A2 ? B 2 A?B



A2 A1 + AB2 = A + B: This transformation from division to multiplication is permissible when we know a priori that the denominator exactly divides the numerator. In this case, the quotient has no negative powers, so the result of the division is simply a truncated nite sum of products. The product of a numerator term and a power 5

series term need be considered only if the product involves no negative powers. Only such products can survive when the rational expression is actually a polynomial. In Dixon's constructions, there are various rational expressions involving division by  ? s or ? t. Since these rational expressions are actually polynomials, we can apply the technique of truncated formal power series to change division into multiplication. If (x; y) is a polynomial in x and y, (x; y) =

m X i=0

ai (y)xi =

n X j =0

bj (x)yj ;

which is divisible by x ? y, then we have i?1 m X n j ?1 (x; y) = X i?1?k xk = ? X X bj (x)xj ?1?kyk : a (y)y i x ? y i=0 k=0 j =0 k=0

Note that when i or j is zero the summation range of k is vacuous. For brevity we usually treat a sum of vacuous range as zero and we will not adjust the summation range to be non-vacuous.

4 From Sylvester to Mixed Cayley-Sylvester Recall that the columns of Sm;n and Mm;n represent respectively bivariate polynomials of bidegree (3m ? 1; 2n ? 1) and (3m ? 1; n ? 1). In order to derive a 6mn  3mn conversion matrix Gm;n such that





Sm;n Gm;n = Mm;n (6) 0 ; that is, in order to insure that the 3mn zero rows are not interleaved with the 3mn non-zero rows, we have to abandon the default lexicographic orders s > t,  > and impose instead the lexicographic orders t > s, > . Matrices with this order of rows and columns will be indicated by an \" to dierentiate them from those matrices with the default lexicographic orders. The derivation below of the transformation matrix produces an explicit entry formula that reveals the simple elegant block structure of Gm;n.

4.1 The Conversion Matrix Gm;n To nd Gm;n , we expand the numerator on the right hand side of Equation (4) by cross-multiplying and writing g(s; ), h(s; ) as explicit sums of monomials to obtain: (g; h) =

n m X X p=0 q=0

(g(s; t)cp;q ? h(s; t)bp;q )sp q =( ? t):

Applying the technique of truncated formal power series with the numerator as a polynomial in yields: (g; h) =

n m X X

q?1 X

p=0 q=0

v=0

(g(s; t)cp;q ? h(s; t)bp;q )sp

tq?1?v v :

Rearranging the range of q and v results in: (g; h) =

n m X nX ?1 X v=0 p=0 q=v+1

(g(s; t)cp;q ? h(s; t)bp;q )sp tq?1?v v :

6





Substituting L = f g h leads to: (g; h) = Hence (s; )(g; h) = = Recalling that we have

n m X nX ?1 X v=0 p=0 q=v+1 mX ?1

2 0 3 L 4 cp;q 5 sp tq?1?v v : ?bp;q

su u (g; h);

u=0 n m X ?1 X mX ?1 nX

u=0 v=0 p=0 q=v+1

2 0 3 L 4 cp;q 5 sp+u tq?1?v u v : ?bp;q

2 0 ?c b 3 p;q p;q Xp;q = 4 cp;q 0 ?ap;q 5 ; ?bp;q ap;q



0

?1 nX ?1 X m X n  mX LXp;q sp+u tq?1?v u v :

(s; ) (g; h) (h; f) (f; g) =

u=0 v=0 p=0 q=v+1

(7)

Since 0  p + u  2m ? 1 and 0  q ? 1 ? v  n ? 1, this equation can be written as 2 1 3 2 LT 3T 77 66 .. .. 77 66 . . 6 7 6   T 6 v u 77   (8) (s; ) (g; h) (h; f) (f; g) = 666 ttss+1LLT 777 Gm;n 66 v u+1 77 ; 7 6 7 64 .. 75 64 .. 5 . . tn?1s2m?1 LT n?1 m?1 where Gm;n is the 2mn  mn matrix whose entries are the 3  3 matrices Xp;q . Thus the actual dimensions of Gm;n are 6mn  3mn.  (c.f. Equation (1) with the new lexicographic order) that It follows by the de nition of Sm;n 2 2 1 3 3T 1 66 66 77 77 .. .. . . 6 6 7   6   7  G 66 v u 777 : (s; ) (g; h) (h; f) (f; g) = 66 ttss+1 77 Sm;n (9) m;n 6 v u+1 7 66 6 7 7 64 75 75 .. .. 4 . . t2n?1s3m?1 n?1 m?1  (c.f. Equation (5) with the new lexicographic order), we see immeTherefore, by the de nition of Mm;n diately that    
G = Mm;n Sm;n (10) m;n 0 :

4.2 The Entries of Gm;n By solving p + u = , q ? 1 ? v =  in Equation (7), we obtain p =  ? u, q =  + v + 1. Consequently the 3  3 entry Xp;q of Gm;n indexed by (t s ; v u ) is X?u; +v+1 (11) for 0   ? u  m and  + v + 1  n; the entries are zero everywhere else. 7

4.3 Properties of Gm;n The structure of the 6mn  3mn conversion matrix Gm;n is most revealing if we view it as a 2mn  mn  be the 2m  m submatrix of Gm;n indexed matrix whose entries are the 3  3 matrices Xp;q . Let B;v   v u  contains the matrices X?u; +v+1 such by (t s ;  ), 0    2m ? 1, 0  u  m ? 1. That is, B;v that 0    2m ? 1, 0  u  m ? 1. Using the entry formula (11), we readily observe the following properties.

4.3.1 Gm;n is Block Upper Triangular  along the lower-left/top-right diagonal have  + v = n ? 1. The blocks B;v  below this The blocks B;v diagonal have  + v + 1 > n and hence are zero by the entry formula (11).

4.3.2 Gm;n is Block Symmetric and Striped  = Bv;  and B+1;v?1 = B;v  = B?1;v+1 . Hence By the entry formula (11) it follows easily that B;v    Gm;n is block symmetric and G +v = B;v is well-de ned. Consequently,

2 6 Gm;n = 664

3

G0 G1.

.
Gn?1 77 G1 . . . . 75 : .. . . . . Gn?1

(12)

4.3.3 The Lower-Left and Upper-Right Corners of Gi are Zeros The (m ? 1)m=2 entries at the lower-left and the (m ? 1)m=2 entries at the upper-right corners of Gi are indexed by (; u) with  > u + m and u >  respectively; hence they are zero by the entry formula (11).

4.3.4 The Columns of Gi March Southeasterly By the entry formula (11) we have Gi [; u] = Gi [ + 1; u + 1]:

4.3.5 Gi is Sylvester-Like Combining the two previous properties, the block Gi can be expressed as

2 X0;i+1 66 ... 66 ?1;i+1 Gi = 66 XXmm;i +1 66 4

8

...




X0;i+1


X1;i+1 ...

.. .

Xm;i+1

3 77 77 77 : 77 5

4.4 Example: m = 1 and n = 2

2 3 ? R ? R 0;1;0;0 0;2;0;0 7 66 66 ?R1;1;0;0 + R1;0;0;1 ?R1;2;0;0 + R1;0;0;2 777 66 ?R1;1;1;0 ?R1;2;1;0 777 M1;2 = 66 ; 66 ?R0;2;0;0 ?R0;2;0;1 777 66 ?R1;2;0;0 + R1;0;0;2 ?R1;2;0;1 + R1;1;0;2 77 4 5 ?R1;2;1;0

S1;2 G1;2

2L 66 L01;;00 66 0 66 L0;1 66 L1;1 6 = 66 L 0 66 L0;2 66 1;02 66 0 64 0 0

?R1;2;1;1

3 77 L0;0 77 L1;0 77 2 0 L0;0 3 L0;1 L1;0 L0;0 77 X0;1 X0;2 L1;1 0 L1;0 77 66 X1;1 X1;2 77 0 L0;1 0 77 4 X0;2 05 L0;2 L1;1 L0;1 77 X1;2 0 L1;2 0 L1;1 77 0 L0;2 0 77 0 L L 5 0

0

0 0 0

0 0 0 0

1;2

0;2

0 L1;2

2 3 ?R0;1;0;0 ?R0;2;0;0 66 ?R1;1;0;0 + R1;0;0;1 ?R1;2;0;0 + R1;0;0;2 77 66 ?R1;1;1;0 ?R1;2;1;0 77 66 ?R0;2;0;0 ?R0;2;0;1 77 66 ?R1;2;0;0 + R1;0;0;2 ?R1;2;0;1 + R1;1;0;2 77 6 ?R1;2;1;0 ?R1;2;1;1 77 : = 66 0 0 77 66 0 0 77 66 0 0 77 66 0 0 77 64 0 05 0

0

5 From Sylvester to Cayley In this section we revert to the default lexicographic orders s > t and  > . We shall now derive a 6mn  2mn conversion matrix Fm;n such that C  Sm;n Fm;n = m;n (13) 0 : This result is very satisfying because it extends the following well-known expansion by cofactors property of 3  3 determinants 3 2A k;l;p;q   ji; j; k; l; p; qj = ai;j bi;j ci;j 4 Bk;l;p;q 5 = Li;j RTk;l;p;q : Ck;l;p;q The derivation below produces an explicit entry formula that reveals the simple elegant block structure of Fm;n. 9

5.1 The Conversion Matrix Fm;n To nd Fm;n, we expand the numerator on the right hand side of Equation (2) by cofactors of the rst row to obtain g(; t) h(; t) 1 0 g(; ) h(; ) C B 1 +


C (14)
m;n = B A ( ? s) : @f(s; t) ( ? t) Expanding the 2  2 numerator determinant yields
m;n = f(s; t)

m X n X m X n A X k;l;p;q tl k+p q

( ? t)

k=0 l=0 p=0 q=0

!

+


( ?1 s) :

Applying the technique of truncated formal power series by regarding the numerator as a polynomial in leads to
m;n =

! q?1 X l + q ? 1 ? v v k + p t +


( ?1 s) : Ak;l;p;q  f(s; t) v=0 k=0 l=0 p=0 q=0 m X n X m X n X

Rearranging the range of q and v yields
m;n = f(s; t)

!

n ?1 X m nX n X m X X

k=0 l=0 p=0 v=0 q=v+1

Ak;l;p;q k+p tl+q?1?v v +




1 : ( ? s)

But the degree in t of the sum is at most n ? 1; hence the summation range of q can be made more precise:

1 0 +v?l) nX ?1 X m X n X m min(n;n X Ak;l;p;q k+p tl+q?1?v v +


A ( ?1 s) :
m;n = @f(s; t) q=v+1

v=0 k=0 l=0 p=0

Again applying the technique of truncated formal power series by regarding the numerator as a polynomial in  leads to
m;n =

m min(n;n n X m X nX ?1 X X+v?l) v=0 k=0 l=0 p=0

q=v+1

f(s; t)Ak;l;p;q tl+q?1?v v

k+X p?1 u=0

sk+p?1?uu +




Rearranging the range of p and u yields
m;n =

X

m X

X XXX

2m?1 n?1 m n

min(n;n+v?l)

u=0 v=0 k=0 l=0 p=max(0;u+1?k)

q=v+1

f(s; t)Ak;l;p;q sk+p?1?utl+q?1?v u v +


:

Replacing the \


" with the corresponding expressions for g(s; t) and h(s; t) produces
m;n =

m X

X XXX

2m?1 n?1 m n

u=0 v=0 k=0 l=0 p=max(0;u+1?k)





X

min(n;n+v?l)

(fAk;l;p;q +gBk;l;p;q +hCk;l;p;q )sk+p?1?u tl+q?1?v u v :

q=v+1

Writing L = f g h leads to
m;n =

X XXX

2m?1 n?1 m n

m X

X

min(n;n+v?l)

u=0 v=0 k=0 l=0 p=max(0;u+1?k)

q=v+1

10

sk+p?1?utl+q?1?v LRTk;l;p;q u v :

(15)

Since k  m, u + 1 ? k  p  m, u  0, v + 1  q  n + v ? l, we have 0  k + p ? 1 ? u  2m ? 1 and 0  l + q ? 1 ? v  n ? 1. Hence there is a 6mn  2mn matrix Fm;n such that
m;n can be written in matrix form as 2 3 2 LT 3T 1 66 77 .. .. 77 66 . . 6 6  T 7 6 u v 77
m;n = 666 ss tt+1LLT 777 Fm;n 66 u v+1 77 : (16) 6 7 75 64 .. 64 75 .. . . s2m?1 tn?1LT 2m?1 n?1 It follows by Equation (16) and the de nition of Sm;n (Equation (1)) that

2 1 66 . 66 s..t
m;n = 66 s t +1 66 .. 4 .

s3m?1 t2n?1

2 3T 1 66 77 . 66 u.. v 77 77 Sm;n Fm;n 66 u v+1 66 77 .. 4 5 .

2m?1 n?1

Thus by the de nition of Cm;n (Equation (3)), we see that



3 77 77 77 : 77 5

(17)



Sm;n Fm;n = Cm;n (18) 0 : Notice that 4mn rows of zeros must be appended after the rows of Cm;n because the row indices s of Sm;n run from 0 to 3m ? 1 but the row indices s of Cm;n run from 0 to m ? 1.

5.2 The Entries of Fm;n Solving k + p ? 1 ? u =  and l + q ? 1 ? v =  in Equation (15) gives p =  + 1 + u ? k; q =  + 1 + v ? l: The constraints 0 0 max(0; u + 1 ? k)  v+1

k l p q

 m;  n;  m;  min(n; n + v ? l);

lead to max(0;  + 1 + u ? m)  k  min(m;  + 1 + u); max(0;  + 1 + v ? n)  l  : Therefore the entry of the cofactor matrix Fm;n indexed by (s t ; u v ) is the sum

X

min(m;+1+u)

 X

k=max(0;+1+u?m) l=max(0; +1+v?n)

11

RTk;l;+1+u?k; +1+v?l ;

(19)

for 0  ; u  2m ? 1, 0  ; v  n ? 1. But observe that the above sum contains mutually cancelling cofactors of the form RTk;l;p;q +RTp;q;k;l = 0 because Ak;l;p;q +Ap;q;k;l = 0, Bk;l;p;q +Bp;q;k;l = 0, Ck;l;p;q + Cp;q;k;l = 0. To nd the cancelling partner of RTk;l;+1+u?k; +1+v?l , let k =  + 1 + u ? k0; l =  + 1 + v ? l0 ;  + 1 + u ? k = k0;  + 1 + v ? l = l0 : A quick check reveals that when v + 1  l; we have max(0;  + 1 + u ? m)  k0  min(m;  + 1 + u); max(0;  + 1 + v ? n)  l0  : That is, the range of l need not run beyond v because terms out of that range mutually cancel. Consequently, the sum for the entry formula (19) of Fm;n will be free of mutually cancelling terms RTk;l;p;q if it is written as min( min(m; X;v) X+1+u) RTk;l;+1+u?k; +1+v?l ; (20) k=max(0;+1+u?m) l=max(0; +1+v?n)

for 0  ; u  2m ? 1, 0  ; v  n ? 1.

5.3 Properties of Fm;n The structure of Fm;n is most revealing if we view this 6mn  2mn matrix as a square 2mn  2mn matrix by considering its entries as sums of the 3  1 matrices RTk;l;p;q .

5.3.1 Fm;n is Symmetric From the entry formula (20), we see that the entries indexed by (s t ; u v ) and (su tv ;   ) are the same. In other words, the matrix Fm;n is symmetric.

5.3.2 Fm;n is Block Upper Triangular Let the n  n submatrix of Fm;n indexed by (s t ; u v ), 0  ; v  n ? 1, be B;u . When +u > 2m ? 1 in the entry formula (20), we have

X

min(m;+u+1)

k=max(0;+u+1?m)

=

m X k=+u+1?m

;

(21)

and clearly this summation range is vacuous. Thus the sum is zero. Consequently the conversion matrix Fm;n is block upper triangular in the sense that the blocks B;u with  + u > 2m ? 1 are all zero.

5.3.3 Fm;n is Block Symmetric and Striped From the entry formula (20), we easily see that B?1;u+1 = B;u = B+1;u?1 : Therefore, the notation F+u = B;u is well-de ned, and 2 F0 F1


F2m?1 3 .. .. 77 : 6 Fm;n = 64 F. .1. .. . . . 5 F2m?1 12

(22)

5.3.4 Fk's are Symmetric Notice that  and v are symmetric in formula (20); thus T : B;u = B;u

That is, each of the blocks Fk is symmetric.

5.3.5 Three Levels of Symmetry In summary, Fm;n has the following symmetry properties:

 Fm;n is symmetric if we view Fm;n as a 2mn2mn square matrix, whose entries are 31 submatrices,

i.e. sums of the RTk;l;p;q ;  Fm;n is also symmetric if we view Fm;n as a 2m  2m square matrix, whose entries are 3n  n matrices, i.e. the Fk 's;  the submatrices Fk are symmetric if we view each Fk as an n  n matrix, whose entries are 3  1 submatrices, i.e. sums of the RTk;l;p;q .

5.3.6 Fk's are Bezoutian Let

P fi (t) = nj=0 ai;j tj ,

P hi (t) = nj=0 ci;j tj .

P gi(t) = nj=0 bi;j tj ,

Using the technique of truncated formal power series, we can write Equation (14) as
m;n

3 1 0m 2 m m m X X X X 1 = 4f(s; t) @ gi (t)i hj ( )j ? gi( )i hj (t)j A ? t +


5
 ?1 s i=0 j =0 i=0 j =0   i + j ? 1 m m XX X g (t)h ( ) ? g ( )h (t)

=

i=0 j =0 u=0

si+j ?1?u u f(s; t) i

j

?t

i

j

+




Therefore, the entries of the submatrix B;u ( = F+u ) come from the coecients of X gi(t)hj ( ) ? gi( )hj (t) ; ?t i+j ?1=+u X hi(t)fj ( ) ? hi( )fj (t) ; ?t i+j ?1=+u X fi(t)gj ( ) ? fi( )gj (t) : ?t i+j ?1=+u

(23)

Each term in these three sums is a Cayley expression for two univariate polynomials fgi; hj g, fhi ; fj g, and ffi ; gj g; hence each term generates a Bezout matrix when written in matrix form [De Montaudouin and Tiller 1984; Goldman et al 1984; Chionh et al 1998]. Therefore, each block F+u = B;u contains three summations of Bezout matrices, where the three matrices interleave row by row. In particular, if  + u = 2m ? 1, then i = j = m in expression (23), so F2m?1 is the matrix obtained by interleaving the rows of the three Bezout matrices: Bezout(gm ; hm ), Bezout(hm ; fm ), Bezout(fm ; gm ). 13

Explicit formulas for the entries of the Bezout matrix of two univariate polynomials are given in [Goldman et al 1984]. More ecient methods for computing the entries of a Bezout resultant or the entries of a sum of Bezout resultants are described in [Chionh et al 1998]. Since the blocks Fk are Bezoutian, we can adopt these methods here to provide ecient algorithm for computing the entries of Fk . Alternatively, we can use Equation (20) directly together with the methods described in Section 7 to develop an ecient algorithm for computing the entries of Fk . Fast computation of the Bezoutiants Fk is important because, as we shall see in Section 7, we can use the blocks Fk to speed up the computation of the entries of the Cayley matrix Cm;n .

5.4 Example: m = 1 and n = 2 2 j1; 0; 0; 1;0; 0j j1; 0; 0; 2;0;0j 66 6 j1; 0; 0; 2;0;0j + j1; 1; 0; 1; 0; 0j j1; 0; 0; 2;0;1j + j1; 1; 0; 2; 0; 0j C1;2 = 66 64 j1; 2; 0; 1;0;0j + j1; 1; 0; 2; 0; 0j j1; 1; 0; 2;0;1j + j1; 2; 0; 2; 0; 0j j1; 2; 0; 2;0; 0j

j1; 2; 0; 2;0;1j

3

?j1; 1; 1; 0;0;0j ?j1; 2; 1; 0;0; 0j 7 ?j1; 2; 1; 0; 0;0j? j1; 1; 1; 0; 0; 1j ?j1; 2; 1; 0;0;1j? j1; 2; 1; 1; 0;0j 777 ; ?j1; 2; 1; 0; 0;1j? j1; 1; 1; 0; 0; 2j ?j1; 2; 1; 1;0;1j? j1; 2; 1; 0; 0;2j 775 ?j1; 2; 1; 0;0;2j ?j1; 2; 1; 1;0; 2j and S1;2F1;2 = 2L 0 66 L00;;01 L0;0 66 L0;2 L0;1 66 0 L0;2 66 L1;0 0 66 L1;1 L1;0 66 L1;2 L1;1 66 0 L1;2 66 0 0 66 0 0 4 0 0 0 0

3 77 77 77 2 77 ?RT1;1;0;0 + RT1;0;0;1 ?RT1;2;0;0 + RT1;0;0;2 ?RT1;1;1;0 ?RT1;2;1;0 3 L0;0 L0;1 L0;0 77 66 ?RT1;2;0;0 + RT1;0;0;2 ?RT1;2;0;1 + RT1;1;0;2 ?RT1;2;1;0 ?RT1;2;1;1 77 : L0;2 L0;1 77 4 ?RT1;1;1;0 ?RT1;2;1;0 0 05 T T 7 0 L0;2 7 ?R1;2;1;0 ?R1;2;1;1 0 0 L1;0 0 77 L1;1 L1;0 77 L L 5 0 0 0 0

0 0 0 0 0

1;2

1;1

0 L1;2

Now it is straightforward to check that indeed





S1;2 F1;2 = C01;2 :

6 From Mixed Cayley-Sylvester to Cayley We have shown that the polynomials represented by the columns of the Cayley resultant and the mixed Cayley-Sylvester resultant are linear combinations of the polynomials represented by the columns of the Sylvester resultant. There is, however, no way to represent the polynomials represented by the columns of the Cayley resultant as linear combinations of the polynomials represented by the columns of the mixed 14

Cayley-Sylvester resultant because the Cayley columns represent polynomials of bidegree (m ? 1; 2n ? 1) whereas the Cayley-Sylvester columns represent polynomials of bidegree (3m ? 1; n ? 1). Thus while the Cayley matrix is smaller than the Cayley-Sylvester matrix, the polynomials in the column space of the Cayley matrix do not form a subspace of the polynomials in the column space of the Cayley-Sylvester matrix. On the other hand, by Equation (3), the rows of the Cayley matrix represent polynomials of bidegree (2m ? 1; n ? 1). Therefore since the columns of the Cayley-Sylvester matrix are linearly independent, the polynomials represented by the rows of the Cayley matrix can be expressed as linear combinations of the polynomials represented by the columns of the Cayley-Sylvester matrix. Below we T . will derive the conversion matrix Em;n that transforms Mm;n to Cm;n

6.1 The Conversion Matrix Em;n To nd Em;n, we expand the determinant on the right hand side of Equation (2) with respect to the rst row. Recall from Section 2.3 that g(; t) h(; t) ) h(; ) g(; ) h(; ) =( ? t) = g(; g(; t) h(; t) =(t ? ) nX ?1

= Therefore,
m;n(s; t; ; ) = ?

v=0

f v (; )tv :

! nX ?1 nX ?1 v v v f(s; t) f v (; )t + g(s; t) g v (; )t + h(s; t) hv (; )t s ?1  : v=0 v=0 v=0 nX ?1

(24)

Using the notation of Section 2.3, write Lv = [ f v gv hv ]. Since the numerator is divisible by the denominator in Equation (24), we can apply the technique of truncated formal power series by treating the numerator as a polynomial in s to write
m;n(s; t; ; ) as
m;n(s; t; ; ) = ?

m X n X i?1 nX ?1 X i=0 j =0 =0 v=0

2a 3 i;j Lv
4 bi;j 5 i?1? s tj +v ci;j

= [L0


uLv uLv+1

2 1 66 . 66 s..t


m?1 Ln?1 ] Em;n 66 s t +1 66 .. 4 .

sm?1 t2n?1

for some coecient matrix Em;n. Hence by Equation (5),

2 1 66 . 66 u.. v
m;n (s; t; ; ) = 66 u v+1 66 .. 4 .

3m?1 n?1

2 1 3T 66 77 . 66 s..t 77 77 Mm;nEm;n 66 s t +1 66 77 .. 4 5 .

sm?1 t2n?1

15

3 77 77 77 : 77 5

3 77 77 77 ; 77 5

(25)

On the other hand, by Equation (3),

2 1 3T 2 3 1 66 66 7 7 . . 66 s..t 777 66 u.. v 777
m;n (s; t; ; ) = 66 s t +1 77 Cm;n 66 u v+1 77 66 66 77 77 .. .. 4 4 5 5 . . sm?1 t2n?1 2m?1 n?1 3T 2 2 1 3 1 7 66 7 66 . . 66 u.. v 777 66 s..t 777 T 6 = 66 u v+1 77 Cm;n 66 s t +1 777 7 66 75 75 64 .. .. 4 . . 2m?1 n?1 sm?1 t2n?1 2 2 1 3T 1 66 6 7 . . 66 u.. v 777  C T  666 s..t = 66 u v+1 77 0 m;n 66 s t +1 66 77 mn2mn 66 .. .. 4 4 5 . .

Therefore,

3m?1 n?1

 CT  : Mm;n
Em;n = 0mnm;n 2mn

sm?1 t2n?1

3 77 77 77 : 77 5 (26)

6.2 The Entries of Em;n From Equation (25), the 3  1 submatrix of Em;n indexed by (i?1? Lv ; s tj +v ) is ?LTi;j . By solving u = i ? 1 ? ;  = j + v; in Equation (25), we get i =  + u + 1; j =  ? v: Therefore the 3  1 submatrix of Em;n indexed by (u Lv ; s t ) is ?LT+u+1; ?v ; (27) when  + u + 1  m, 0   ? v  n, and zero otherwise.

6.3 Properties of Em;n From the above entry formula (27), arguments similar to those of Section 4.3 or 5.3 show that Em;n has the following striped block symmetric structure: 2 E0 E1


Em?1 3 77 66 E1 . . . .. . Em;n = 64 . 75 ; . .. .. Em?1 16

where

2 ?LT ?LT ;1 66 i0+1;0 ?LiTi+1 +1;0 Ei = 64 .. .. . .




?LTi+1;n?1 ?LTi+1;n 0





?LTi+1;n?2 ?LTi+1;n?1 ?LTi+1;n


...

.. .

0 0


?LTi+1;0 is of size n  2n with each entry a 3  1 submatrix.

.. .

?LTi+1;1

.. .

...

0 0 .. .

?LTi+1;2


?LTi+1;n

3 77 75 ;

(28)

6.4 Example: m = 1 and n = 2 M1;2 Em;n

2 ?R0;1;0;0 3 ?R0;2;0;0 ?R0;2;0;1 66 ?R0;2;0;0 77  T  ? R + R ? R + R 6 77 ?L1;0 ?LTT1;1 ?LTT1;2 0T 1 ; 1 ; 0 ; 0 1 ; 0 ; 0 ; 1 1 ; 2 ; 0 ; 0 1 ; 0 ; 0 ; 2 = 66 ?R 4 1;?2;0R;01;+1;1R;01;0;0;2 R1;1?;0;R2 1?;2R;1;10;2;0;1 75 0 ?L1;0 ?L1;1 ?L1;2 ?R1;2;1;1  C T?R1;2;1;0 =

1;2

0

:

7 The Block Structure of the Three Dixon Resultants The natural block structures of the transformation matrices prompt us to reexamine once again the three Dixon resultant matrices to seek natural block structures compatible with the block structures of the transformation matrices. Below we shall: 1. describe these block structures; 2. analyze how these blocks are related to resultant matrices for univariate polynomials; 3. reveal how the blocks of the transformation matrices are related to the blocks of the resultant matrices; 4. simplify the calculation of the entries of the Cayley and mixed Cayley-Sylvester resultants by taking advantage of these block structures; and 5. derive identities relating the blocks of the Sylvester and the mixed Cayley-Sylvester resultants.

7.1 The Block Structure of the Sylvester Resultant The block structure of Sm;n is very natural and easy to describe. As in Section 5.3.6, we set

P fi (t) = nj=0 ai;j tj ,

P gi(t) = nj=0 bi;j tj ,

P hi (t) = nj=0 ci;j tj .

Now let Si be the 2n  3n coecient matrix for the polynomials (tv fi ; tv gi ; tv hi), v = 0;


; n ? 1. Note that the matrix Si is Sylvester-like in the sense that if we drop the fi -columns from Si , then we get the Sylvester matrix of gi and hi ; dropping the gi -columns yields the Sylvester matrix of hi and fi ; dropping

17

the hi-columns yields the Sylvester matrix of fi and gi . Adopting the usual notation Li;j = [ai;j bi;j ci;j ], we can write 2 L 3 i;0 66 ... . . . 77 66 L 77 Si = 66 Li;n?1





LLi;0 77 ; (29) 66 i;n . i;. 1 77 . . .. 5 4 Li;n where the rows are indexed by the monomials 1;


; t2n?1, and the columns are indexed by the polynomials t0 [fi gi hi ],


, tn?1[fi gi hi ]. It follows from Equation (1) that 2 S0 3 66 ... . . . 77 66 Sm?1


S0 77 66 Sm


S1 S0 77 6 77 : .. . . . . . . ... Sm;n = 66 (30) . 77 66 Sm Sm?1


S0 77 66 Sm


S1 77 64 . . . ... 5 Sm Notice from Equations (28) and (29) that Ei = ?(Si+1 )T ; Si+1 = ?(Ei )T ; 0  i  m ? 1: (31) Similar results hold when we impose the order t > s. De ne

P fj (s) = mi=0 ai;j si ,

P gj (s) = mi=0 bi;j si ,

P hj (s) = mi=0 ci;j si .

Now let Sj be the 3m  6m coecient matrix for the polynomials (sv fj ; sv gj ; sv hj ), v = 0;


; 2m ? 1. Then Sj can be written as

2 L0;j 66 ... 66 L 66 Lm?1;j 66 m;j  Sj = 66 66 66 64

...




L0;j


L1;j ...

L0;j .. ... . Lm;j Lm?1;j


Lm;j


... .. .

3 77 77 77 77 77 ; L0;j 77 L1;j 77 .. 75 .

Lm;j and it follows from Equation (1) (with the lexicographical order t > s) that 2 S 3 0 66 ... . . . 77 66  7 7 S


S  n ? 1 0 6 Sm;n = 6 S 


S  77 : 66 n . .1 77 . . .. 5 4 Sn 18

(32)

7.2 The Block Structure of the Mixed Cayley-Sylvester Resultant The block structure of Mm;n is also easy to describe, although the blocks themselves are more complicated. Recall from Section 2.3 that Lv = [f v gv hv ], v = 0;


; n ? 1, where f v ; gv ; hv are polynomials of degree 2m in s and degree n ? 1 in t. Let Mi , i = 0;


; 3m ? 1, be the n  3n coecient matrix of the monomials si [1;


; tn?1] of the polynomials L0;


; Ln?1. Note that Mi = 0 when i > 2m because every Lv is of degree 2m in s. Since the columns of Mm;n are indexed by si [L0;


; Ln?1], i = 0;


; m ? 1, and the rows by si [1;


; tn?1], i = 0;


; 3m ? 1, it is easy to see from Equation (5) that Mm;n has the following Sylvester-like (shifted block) structure: 2 M0 3 66 ... . . . 7 66 Mm?1


M0 777 6 .. 77 : .. Mm;n = 66 ... . 66 M2m


Mm. +1 777 64 .. 75 ... . M2m Using the notation fi ; gi; hi introduced in Section 5.3.6, we can derive explicit formulas for the blocks of Mm;n . To begin, write (g; h); (h; f); (f; g) as m X m g (t)h ( ) ? g ( )h (t) X i j i j si+j ; (g; h) = ?t i=0 j =0 m m X X hi(t)fj ( ) ? hi( )fj (t) i+j s ; (h; f) = ?t i=0 j =0 m X m f (t)g ( ) ? f ( )g (t) X i j i j si+j : (f; g) = ? t i=0 j =0 Now notice that the block Mk consists of the coecients of sk in these expressions. That is, Mk contains three matrices with their columns interleaved. Just like the expressions in Equation (23), the coecients of sk are sums of Cayley expressions. Hence each of these three matrices is the sum of the Bezout matrices of the polynomials fgi; hj g, fhi ; fj g, ffi ; gj g, for i + j = k. In particular, M0 contains the three interleaved Bezout matrices: Bezout(g0; h0), Bezout(h0; f0 ), Bezout(f0 ; g0); similarly, M2m contains three interleaved Bezout matrices: Bezout(gm ; hm ), Bezout(hm ; fm ), Bezout(fm ; gm ). Notice that Mk and Fk?1 [Section 5.3.6], k = 1;


; 2m, contain the same three Bezoutian matrices; however, Mk is the n  3n matrix where the three Bezoutian matrices interleave column by column, whereas Fk?1 is the 3n  n matrix where the three Bezoutian matrices interleave row by row. Since each Bezout matrix is symmetric [Goldman et al 1984], it is easy to see that Fk?1 = (Mk )T ; Mk = (Fk?1)T : (33) Therefore, we can apply the method of Section 5.3.6 to calculate the entries of either Mk or Fk?1, and obtain the entries of the other matrix without any additional computation.  has a dierent block structure. Recall from Equation (32) and Equation (12) that The matrix Mm;n   Sm;n and Gm;n can be written as 2 S 3 0 66 ... . . . 77 66 S


S 77  Sm;n = 66 Sn? 1


S0 77 ; 66 n . .1 77 . . .. 5 4 Sn 19

G0





Gn?1 3 .. 77 . ... ... Gm;n 75 ; .. . . . . Gn?1 where each Si is of size 3m  6m, and each Gj is of size 6m  3m. But by Equation (10),

2 6 = 664

S

m;n

so

2  =6 Mm;n 4


G

m;n =

M  m;n

0

;

3 2 G0





Gn?1 3 .. . . . . . . 77 . 75 : .. ..

S0 75
666 .. . . . . 4 Sn ?1


S0

2  =6 Mm;n 4

.

Gn?1

.

(34)

K0;0


K0;n?1 3 .. .. .. 75 ; . . . Kn?1;0


Kn?1;n?1  is of size 3m  3m. By construction, each column of blocks where each block Ki;j Write

2 64

K0;j 3 .. 75 . Kn?1;j

consists of the coecients of the polynomials [1;


; sm?1]
Lj , j = 0;


; n ? 1. Now by Equation (34),  = Ki;j

X

min(i;n?1?j )

l=0

Si?l
Gj +l ;

0  i; j  n ? 1:

(35)

Notice, however, that Equation (35) involves a lot of redundant calculations along the diagonals i + j = l for l = 1;


; 2n ? 3. In fact,  = S 
G + K  Ki;j (36) i j i?1;j +1:  in the following way: To eliminate these redundancies, we can compute the entries of Mm;n 1. Initialization:

that is,

2     ?M 
= 6 S0
.. G0
..

S0
G.. n?1 m;n init 4 . . .

Sn ?1
G0


Sn ?1
Gn?1

?K 
= S 
G  ; i j i;j init

3 75 ;

0  i; j  n ? 1:

2. Recursion: for i = 1 to n ? 1 for j = n ? 2 to 0 (step ?1)   + Ki?1;j +1: Ki;j Ki;j That is, we add along the diagonals from upper right to lower left, so schematically, 20

2 S0
G0 S0
G1 66 . . 6  . . Mm;n = 66 . ... 4 . . .

.



.
S0
Gn?2 S0
Gn?1 3 .. . S1
Gn?1 77 . . .. .. ...

77 : 7 Sn ?2
Gn?1 5 .. .

...

. .

Sn ?1
Gn?1

 with G generate the following n interesting identities: The products of the lower 3mn rows of Sm;n m;n n?X 1?j l=0

Sn ?l
Gj +l = 0;

0  j  n ? 1:

(37)

Therefore by Equation (35) and Equation (37) Kn?1;j =

n? X1?j l=0

Sn ?1?l
Gj +l = ?Sn
Gj ?1;

1  j  n ? 2;

so along the bottom row of blocks, we can initialize Kn?1;j = ?Sn
Gj ?1;

j = 1;


; n ? 2;

and remove the extraneous computations Kn?1;j

Kn?1;j + Kn?2;j +1;

j = 1;


; n ? 2:

7.3 The Block Structure of the Cayley Resultant The block structure of Cm;n is not so obvious as that of Sm;n or Mm;n , since the entry formulas for Cm;n are much more complicated [Chionh 1997]. However, informed by the transformation matrices Fm;n and T . Em;n derived in Sections 5 and 6, we can impose an interesting block structure on Cm;n and Cm;n

7.3.1 The Block Structure of Cm;n First recall from Equation (22) and Equation (33) that the conversion matrix Fm;n can be written as 2 M1T





M2Tm 3 6 .. . . . . 77 Fm;n = 664 .. . . . (38) 75 ; .. . . M2Tm where each block MjT is of size 3n  n. Since by Equation (18),

C  m;n

0

= Sm;n
Fm;n ;

it follows from Equation (30) and Equation (38) that 2 S0 3 2 T T T 3 66 S1 S0 77 6 M..1
.
.
M.m.
.
.
M2m 7 Cm;n = 4 .. 5: ... ... 5
4 .T . . T . . Mm


M2m Sm?1 Sm?2


S0 21

(39)

2 Cm;n = 64

Write

3

B0;0


B0;2m?1 75 ; .. .. .. . . . Bm?1;0


Bm?1;2m?1 where each block Bi;j is of size 2n  n. By construction, each column of blocks 2 B 3 64 0.. ;j 75 ; . Bm?1;j j = 0;


; 2m ? 1 consists of the coecients of the polynomials in s; t that represent the coecients of j [1;


; n?1] in Cayley's determinant expression on the right hand side of Equation (2). Now by Equation (39), Bi;j =

X

min(i;2m?1?j )

l=0

Si?l
MjT+l+1 ;

0  i  m ? 1; 0  j  2m ? 1:

(40)

Again, we observe that Bi;j = Bi?1;j +1 + Si
MjT+1; 1  i  m ? 1; 0  j  2m ? 2: which leads to a faster way for computing the entries of Cm;n: 1. Initialization:

2 S
MT 0 1 6 .. (Cm;n)init = 4 .

That is,




.. .

Sm?1
M1T




(Bi;j )init = Si
MjT+1 ;

3

S0
MmT S0
MmT +1


S0
M2Tm 75 : .. .. .. .. . . . . Sm?1
MmT Sm?1
MmT +1


Sm?1
M2Tm 0  i  m ? 1; 0  j  2m ? 1:

2. Recursion: for i = 1 to m ? 1 for j = 2m ? 2 to 0 (step ?1) Bi;j Bi;j + Bi?1;j +1:

2 S0
M T 66 . 1 Cm;n = 666 . . . 4 .

3

S0
M2Tm S1
M2Tm 77 77 : .. . . .. .. . . .. . . .. . 7 . .. . . . Sm?2
M2Tm 5 . . . . Sm?1
M2Tm . .. . . Notice that this method for computing the entries of Cm;n eliminates lots of redundant calculations: after the initialization, we need only update the blocks along the southwest diagonals by one (block) addition per block. This approach is much faster than the standard method of calculating each entry independently [c.f. Sederberg 1983; Chionh 1997], especially when we compute the Bezoutiants MkT eciently [c.f. Section 5.3.6]. Indeed this ecient computation of the entries of the Cayley resultant mirrors quite closely the ecient computation of the entries of the Bezout resultant [Chionh et al 1998]. Finally notice that the products of the lower 4mn rows of Sm;n with Fm;n generate the following 2m interesting identities:

X



.
S0
MmT S0
MmT +1

.
.. .. . .

min(m;2m?1?j )

l=0

Sm?l
MjT+l+1 = 0; 22

0  j  2m ? 1:

(41)

As in the previous section, these identities can be applied to simplify the computation of the last row of blocks of Cm;n since by Equation (40) and Equation (41), Bm?1;j =

X

min(m?1;2m?1?j )

l=0

Sm?1?l
MjT+l+1 = ?Sm
MjT ;

1  j  2m ? 2:

Thus we can initialize Bm?1;j = ?Sm
MjT ;

1  j  2m ? 2;

and remove the extraneous computation Bm?1;j

Bm?1;j + Bm?2;j +1 ;

1  j  2m ? 2:

T and the Convolution Identities 7.3.2 The Block Structure of Cm;n T as By Equation (26) and Equation (31), we can write Cm;n

2 M0 66 ... 66 M T Cm;n = 666 m?1 66 Mm 4

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

3 77 2 77 ?S1T


?SmT 3 M0 7 6 .. 75 : 77
4 . . . . M1 7 ?SmT .. 75 .

(42)

Mm

T . Therefore, it follows from If Bi;j is the (i; j)th block of Cm;n, then (Bj;i )T is the (i; j)th block of Cm;n Equation (42) that min(i;m X?1?j) (Bj;i )T = ? Mi?l
SjT+1+l ; 0  j  m ? 1; 0  i  2m ? 1: (43) l=0

Hence Bi;j = so

min(j;m X?1?i) l=0

Si+1+l
MjT?l ; 0  i  m ? 1; 0  j  2m ? 1;

(44)

Bi;j = ?Si+1
BjT + Bi+1;j ?1: (45) Now using a method similar to that of Section 7.3.1, we can again compute the entries of Cm;n very eciently, except here after the initialization step we march to the northeast instead of to the southwest. Combining the results in this section and the Section 7.3.1, we can derive the following remarkable convolution identities: minX (k;m) Sl
MkT?l = 0; 0  k  3m: (46) l=max(0;k?2m) The identities in Equation (46) for 1  k  3m ? 1 are obtained by comparing the two entry formulas Equation (40) and Equation (44); for m + 1  k  3m, these identities are equivalent to Equation (41). Finally, the case of k = 0 follows from k = 3m. Observe that the matrices Sm and M2m depend only on the polynomials fm ; gm ; hm ; in the same way the matrices S0 and M0 depend only on the polynomials f0 ; g0; h0. Therefore, replacing the arbitrary polynomials fm ; gm ; hm in the identity Sm
M2Tm = 0 (k = 3m); 23

by the polynomials f0 ; g0; h0 yields the identity S0
M0T = 0 (k = 0): Parallel computation can be used to speed up the algorithms in this section and Section 7.2 even further. To begin, we can perform the initialization of the blocks in parallel. Moreover, the recursions along dierent diagonals are independent, so these steps can also be done in parallel. Finally, the identities in Equation (37) and Equation (41) can be used to simplify the computation of the last row of blocks in  ; similarly, Equation (44) can be applied to simplify the computation of the rst column Cm;n and Mm;n of blocks of Cm;n . Therefore the recursions can march simultaneously northeast and southwest if we choose the appropriate initialization.

Acknowledgments Eng-Wee Chionh is supported by the National University of Singapore for research at Rice University. He greatly appreciates the hospitality and facilities generously provided by Rice. Ming Zhang and Ron Goldman are partially supported by NSF grant CCR-9712345.

References [1] E. W. Chionh. Base Points, Resultants, and the Implicit Representation of Rational Surfaces. Ph.D. Thesis, University of Waterloo, 1990. [2] E. W. Chionh. Concise Parallel Dixon Resultant. Computer Aided Geometric Design, 14(6):561{570, August 1997. [3] E. W. Chionh, M. Zhang, R. N. Goldman. Transformations and Transitions from the Sylvester to the Bezout Resultant, in preparation, 1998. [4] A. L. Dixon. The Eliminant of Three Quantics in Two Independent Variables. Proc. London Math. Soc., 6:49{69,473{492, 1908. [5] R. N. Goldman, T. Sederberg, D. Anderson. Vector Elimination: A Technique for The Implicitization, Inversion, and Intersection of Planar Parametric Rational Polynomial Curves. Computer Aided Geometric Design, 1:327-356, 1984. [6] D. Manocha and J. F. Canny. Algorithms for Implicitizing Rational Parametric Surfaces. Computer Aided Geometric Design, 9(1):25{50, May 1992. [7] Y. De Montaudouin, W. Tiller. The Cayley Method in Computer Aided Geometric Design. Computer Aided Geometric Design, 1:309{326, 1984. [8] T. Sederberg. Implicit and Parametric Curves and Surfaces. Ph.D. Thesis, Purdue University, August 1983. [9] T. Sederberg, D. Anderson, and R. N. Goldman. Implicit Representation of Parametric Curves and Surfaces. CVGIP, 28:72{84, 1984. [10] T. Sederberg, R. N. Goldman and H. Du. Implicitizing Rational Curves by the Method of Moving Algebraic Curves. J. Symbolic Computation, 23:153{175, 1997. [11] J. Weyman, A. Zelevinsky. Determinantal Formulas for Multigraded Resultants. J. Algebraic Geometry, 3: 569-597, 1994. [12] M. Zhang, E. W. Chionh, R. N. Goldman. Hybrid Dixon Resultants, The Mathematics of Surfaces VIII, Edited by R. Cripps, Information Geometers Ltd., Winchester, UK, 193-212, 1998. 24