Transformations and Transitions from the Sylvester to the Bezout Resultant Eng-Wee Chionh
[email protected] School of Computing National University of Singapore Lower Kent Ridge Road, Singapore 119260 Ming Zhang Ronald N. Goldman
[email protected] [email protected] Department of Computer Science Rice University Houston, Texas 77005 Abstract A simple matrix transformation linking the resultant matrices of Sylvester and Bezout is derived. This transformation matrix is then applied to generate an explicit formula for each entry of the Bezout resultant, and this entry formula is used, in turn, to construct an ecient recursive algorithm for computing all the entries of the Bezout matrix. Hybrid resultant matrices consisting of some columns from the Sylvester matrix and some columns from the Bezout matrix provide natural transitions from the Sylvester to the Bezout resultant, and allow as well the Bezout construction to be generalized to two polynomials of di erent degrees. Such hybrid resultants are derived here, employing again the transformation matrix from the Sylvester to the Bezout resultant.
1 Introduction Resultants play an important role in algorithmic algebraic geometry [Cox et al 1997], computer aided geometric design [Goldman et al 1984; De Montaudouin and Tiller 1984; Sederberg and Parry 1986; Manocha and Canny 1992], and robotics [Canny 1987; Canny 1988]. For two univariate polynomials of the same degree, there are two standard matrix representations for the resultant: the Sylvester resultant [van der Waerden 1950] and the Bezout resultant [Goldman et al 1984; De Montaudouin and Tiller 1984]. Typically these two resultants are constructed in isolation, without any regard for how one matrix is related to the other. The purpose of this paper is to explore a variety of mathematical connections between the resultant matrices of Sylvester and Bezout. We shall derive a simple, block symmetric, block upper triangular matrix that transforms the Sylvester matrix into the Bezout matrix. This matrix transformation captures the essence of the mathematical relationships between these two resultant matrices. We shall then apply this transformation matrix to: 1. derive an explicit formula for each entry of the Bezout matrix; 2. develop an ecient recursive algorithm for computing all the entries of the Bezout matrix;
3. construct a sequence of hybrid resultant matrices that provide a natural transition from the Sylvester matrix to the Bezout matrix, each hybrid consisting of some columns from the Sylvester matrix and some columns from the Bezout matrix; 4. extend the Bezout construction to two univariate polynomials of di erent degrees. Except for the recursive algorithm, all of these results are already known [Goldman et al 1984; Sederberg et al 1997; Cox et al 1997]. What is new here in addition to the recursive algorithm is our approach: deriving all these properties in a uni ed fashion based on the structure of the transformation matrix. Equally important, these seemingly straightforward features of the resultants for two univariate polynomials of degree n anticipate similar, but much more intricate, properties and interrelationships between the resultants for three bivariate polynomials of bidegree (m; n) [Chionh et al 1998; Zhang et al 1998]. This paper is organized in the following fashion. In Section 2 we brie y review the standard constructions for the Sylvester and Bezout resultants. In Section 3 we introduce the method of truncated formal power series, and in Section 4 we apply this method to generate the matrix transformation from the Sylvester matrix to the Bezout matrix. Section 5 is devoted to developing a new, ecient, recursive algorithm for computing all the entries of the Bezout resultant. Hybrid resultants are discussed in Section 6. We close in Section 7 with a derivation of the Bezout resultant for two univariate polynomials of di erent degree.
2 The Sylvester and Bezout Resultants P
P
Let f(t) = ni=0 aiti , g(t) = nj=0 bj tj be two polynomials of degree n, and let L(t) = [ f(t) g(t) ]. The determinant of the 2n 2n coecient matrix of the 2n polynomials t L, 0 n ? 1, is known as the Sylvester resultant of f and g. To x the order of the rows and columns of the Sylvester matrix Sn , we de ne L tL tn?1L = 1 t2n?1 S : (1) n Let Li = [ ai bi ]; then 3 2 L0 77 66 ... . . . 7 66 . . . L 77 6 L 0 7: Sn = 66 n?1 . 66 Ln . . L1 777 64 . . . ... 75 Ln Note that we have adopted a somewhat nonstandard form for the Sylvester matrix by grouping together the polynomials f and g (compare to [van der Waerden 1950]). This grouping will simplify our work later on and will change at most the sign of the Sylvester determinant. Observe too that with this ordering of the columns the top (bottom) half of the Sylvester matrix is striped, block lower (upper) triangular, and block symmetric with respect to the northeast-southwest diagonal. To obtain the Bezout matrix for f and g, we consider the Cayley expression f(t) g(t) n (t; ) = f( ) g( ) =( ? t): (2) Since the denominator exactly divides the numerator in n(t; ), this rational Cayley expression is really a degree n ? 1 polynomial in t and in , so we can write n(t; ) =
nX ?1 v=0
2
Dv (t) v
(3)
where Dv (t), 0 v n ? 1, are polynomials of degree n ? 1 in t. The determinant of the n n coecient matrix of the n polynomials D0 (t); ; Dn?1(t) is known as the Bezout resultant of f and g. To x the order of the rows and columns of the Bezout matrix Bn , we de ne 2 1 3 nX ?1 (4) Dv (t) v = 1 tn?1 Bn 64 ... 75 : v=0 n?1 Thus the rows of Bn are indexed by 1; t; ; tn?1 and the columns by 1; ; ; n?1. It is well-known that the Bezout matrix Bn is symmetric [Goldman et al 1984]. Here we provide an elementary, high level, proof of this fact without computing entry formulas for Bn . Since f(t) g(t) f( ) g( ) n (t; ) = f( ) g( ) =( ? t) = f(t) g(t) =(t ? ) = n( ; t); (5) we have 2 1 31T 0 2 1 3 . [1 n?1 ] BnT 4 .. 5 = @[ 1 n?1 ] BnT 4 ... 5A n?1 tn?1 2 1 t3 = [1 tn?1 ]Bn 4 ... 5 n?1 = n(t; ) = n( ; t) [by Equation (5)] 2 1 3 = [1 n?1 ] Bn 4 ... 5 : tn?1 Hence BnT = Bn , so Bn is symmetric. The columns of the Sylvester matrix Sn represent polynomials of degree 2n ? 1, whereas the columns of the Bezout matrix Bn represent polynomials of degree n ? 1. Since, in general, jSn j = 6 0, the columns of Sn are linearly independent. Hence the 2n polynomials represented by these columns span the space of polynomials of degree 2n ? 1. In particular, we can express the polynomials represented by the columns of the Bezout matrix as linear combinations of the polynomials represented by the columns of the Sylvester matrix. Thus there must be a matrix that transforms the Sylvester matrix into the Bezout matrix. In the next section we introduce a mathematical technique that will help us to nd explicit formulas for the entries of this transformation matrix.
3 Exact Division by Truncated Formal Power Series P
Let (x; y) = mi=0 ai (y)xi be a polynomialin x and y. If the rational expression (x; y)=(x?y) is actually a polynomial| that is, if x ? y exactly divides (x; y)| wePcan convert the division to multiplication by replacing each monomial xi in the numerator by the sum ik?=01 yi?1?k xk . That is, m X i=0
where the vacuous sum formally we have
ai (y)xi =(x ? y) =
m X i=0
ai (y)
i?1 X
k=0
yi?1?k xk ;
(6)
P?1 is taken to be zero. This identity is motivated by the observation that k=0 1 yk X 1 x ? y = k=0 xk+1 : 3
Thus xi
1 yk X k=0 x
k+1
i?1 yk 1 k X iX y + x k+1 k+1 k=0 x k=i x i?1 X i?1?k k
= xi =
k=0
y
x + terms involving negative powers of x:
Since, by assumption, the quotient on the left hand side of Equation (6) is a polynomial, terms involving negative powers of x must cancel; therefore, these terms can be ignored. To illustrate this division by truncated formal power series technique, consider the example: 2 ? 4y ? 7y2 (x; y)=(x ? y) = 4x + 7xx ? : y P Using thePmethod, we replace the monomial x by the sum 1k?=01 y1?1?k xk = 1 and the monomial x2 by the sum 2k?=01 y2?1?k xk = y + x. The quotient is then 4x + 7x2 ? 4y ? 7y2 = 4 1 + 7 (y + x) x?y as expected.
4 The Transformation Matrix From Sylvester to Bezout If we perform the division in the Cayley expression (2) using the method of truncated formal power series and delay expanding f(t) and g(t) as sums, we obtain a relationship between the Sylvester matrix Sn and the Bezout matrix Bn . Recall that L = [ f(t) g(t) ] and let Rj = [ bj ?aj ]T . Then n(t; ) = (f(t) = = =
n X j =0 n X
n X j =0
bj j ? g(t)
n X j =0
aj j )=( ? t)
LRj j =( ? t) LRj
j ?1 X
v=0 j =0 nX ?1 X n v=0 j =v+1
tj ?1?v v
LRj tj ?1?v v :
(7)
Hence there is a matrix Fn such that
2 1 3 n(t; ) = L tL tn?1L Fn 64 ... 75 n?1 2 1 3 = 1 t2n?1 Sn Fn 64 ... 75 n?1
Comparing Equation (8) with Equations (3) and (4), we see that Sn Fn = 0Bnnn : 4
(8)
There are n rows of zeros appended below Bn because, unlike Sn , Bn does not involve the monomials t , n 2n ? 1. To nd the entry of Fn indexed by (t L; v ), let = j ? 1 ? v. Then j = +v +1. Hence by Equation (7) the entry of Fn indexed by (t L; v ) is simply R +v+1 ; (9) for + v + 1 n and zero otherwise. From this entry formula, we easily see that: 2 R R R R 3 66 R12 R23 Rnn?1 n 77 77 ; .. (10) Fn = 666 ... . ... 75 4 Rn?1 Rn Rn since the entries are constant along the diagonals + v = constant, and zero below the diagonal + v = n ? 1. Though the dimension of Fn is 2n n, Fn can be treated as a square matrix of dimension n n if its entries are viewed as column vectors Rj . From this perspective, we see that Fn is striped, symmetric, and upper triangular.
5 Fast Computation of the Entries of the Bezout Resultant An explicit entry formula for the Bezout matrix is provided in [Goldman et al 1984]. In this section, we derive this formula from the transformation matrix of Section 4. We also present a new, more ecient method for computing the entries of Bezout matrices and sums of Bezout matrices. From Section 2 and Section 4, we know that 3 2 L0 2 R1 R n 3 . 66 .. . . . 77 6 .. . . . . 77 7 6 Fn = 664 ... . . . Sn = 666 LLn?1 LL0 777 ; 75 ; . .. 64 n . . ..1 75 . . Rn Ln and B Sn Fn = 0 n : nn Hence 2 3 2 L0 3 6 R.1 Rn 7 . . 77 : 5 66 ... . . . . . Bn = 4 ... . . . (11) .. . . 4 5 Ln?1 L0 Rn Write 2 Z 3 0;0 Z0;n?1 .. 75 : Bn = 64 ... (12) . Zn?1;0 Zn?1;n?1 Then by Equation (11) and Equation (12), Zi;j =
X
min(i;n?1?j )
l=0
Li?l Rj +1+l ; 5
0 i; j n ? 1:
(13)
Since Li Rj + Lj Ri = 0, it is easy to verify that
X
max(0;i?j ?1)
l=0
Li?l Rj +1+l = 0:
Therefore, we can rewrite Equation (13) as Zi;j =
X
min(i;n?1?j )
l=max(0;i?j )
Li?l Rj +1+l ;
0 i; j n ? 1:
(14)
Equation (14) is equivalent to the explicit entry formula for the Bezout matrix given in [Goldman et al 1984]. By comparing the entry formulas in Equation (14), it follows that Zi;j = Zj;i . Thus we see again that Bn is a symmetric matrix. From Equation (14), it is easy to recognize recursion along the diagonals i+j = k, for 0 k 2n ? 2. In fact, Zi;j = Zi?1;j +1 + Li Rj +1: (15) Equation (15) can be applied to compute the entries of Bn very eciently using the following three-step algorithm: 1. Initialization:
2 L0 R 1 L 0 R n 3 .. ... 5; (Bn )init = 4 . Ln?1 Rn
that is, (Zi;j )init = Li Rj +1 = aibj +1 ? aj +1 bi ;
0 i j n ? 1:
2. Recursion: for i = 1 to n ? 2 for j = n ? 2 to i (step ?1) Zi;j Zi;j + Zi?1;j +1: That is, we add along the diagonals from upper right to lower left, so schematically,
2Z 64 0;0
2 L0 R1 L0 R2 L0 R3 . . 6 Z0;n?1 3 666 . .. 75 = 6 ... . 66 Zn?1;n?1 64
L0 Rn?3 L0 Rn?2 . . . . ...
.. .
.
.. .
. .
L0 Rn?1 3 L1 Rn?1 77 L2 Rn?1 77 .. 77 : . Ln?3 Rn?1 775 Ln?2 Rn?1 Ln?1 Rn?1
3. Symmetry: Zi;j = Zj;i, i > j. That is, the entries below the diagonal i = j are obtained via symmetry from the entries above the diagonal. Notice how much more ecient this recursive algorithm is compared to the explicit formula (Equation (14)). Using the explicit formula to nd all the entries of Bn , we perform O(n3 ) additions and we 6
recompute each product Li Rj many times. In the recursive algorithm we perform only O(n2) additions and we calculate each product Li Rj only once, thus removing redundant computations. This method can also be applied to compute sums of Bezout matrices of the same size, say n n, eciently. Instead of computing each Bezout matrix separately, and then adding them together, we can save time and space by adopting the following strategy of initialization and marching: 1. initialize the entries in this n n matrix with indices 0 i j n ? 1 using the sums of the initializations for each Bezout matrix|the previous initialize{and{march method assigns an initialization for each Bezout matrix in the sum; 2. march to the southwest along the diagonals 1 i + j 2n ? 3. 3. obtain the entries with indices i > j via symmetry. In this way, we save not only the space for storing the individual Bezout matrices, but also we march along the diagonals just once; hence this approach greatly reduces the computational complexity. This fast computation of sums of Bezout matrices has applications in the ecient computation of resultants for bivariate polynomials [Chionh et al 1998].
6 Hybrid Resultants A collection of n ? 1 hybrids of the Sylvester resultant and the Bezout resultant for f and g are constructed in [Weyman and Zelevinsky 1994; Sederberg et al 1997]. These hybrid resultants are composed of some columns from the Sylvester resultant and some columns from the Bezout resultant. In this section, we will generate these hybrids based on the transformation in Section 4 from the Sylvester to the Bezout resultant. Let Hj , j = 0; ; n, be the j ?th hybrid resultant matrix 2 L0 Z0;0 Z0;j ?1 3 .. .. .. 77 66 ... . . . . . . 7 66 66 Ln?j?1 . . . L0 Zn?j?1;0 Zn?j?1;j?1 777 77 66 ... . . . ... .. .. .. . . . 7: 66 66 Ln?1 . . . Lj Zn?1;0 Zn?1;j?1 777 77 66 . 77 66 Ln . . Lj.+1 . . . .. 5 4 Ln That is, Hj contains the rst 2(n ? j) truncated columns of Sn , and the rst j columns of Bn . Note that Hj is a square matrix of order 2n ? j; moreover, H0 = Sn and Hn = Bn . Below we will show that jHj j = jHj +1j, j = 0; ; n ? 1. Since Sn and Bn are known to be resultants for f and g, it follows that Hj is a resultant of f and g for j = 0; ; n. To proceed with our proof, let 2 Ri 3 Rij = 4 ... 5 ; Rj 0 J = 1 , and consider the (2n ? j) (2n ? j) matrix 3 2I j +1 2(n?j ?1) 0 Rn?1 0 0 Rn J 5 : Tj = 4 0 0 Ij 0 0 7
Multiplying Hj by Tj , we get 2 L0 Z0;0 Z0;j ?1 3 .. .. .. 77 66 ... . . . . . . 7 66 66 Ln?j?1 . . . L0 Zn?j?1;0 Zn?j+1;j?1 777 2 j +1 0 3 7 66 ... . . . ... .. .. .. I 0 R 2( n ? j ? 1) n?1 7 . . . 7 4 0 0 R Hj Tj = 66 n J5 7 66 Ln?1 . . . Lj Zn?1;0 Zn?1;j?1 77 0 Ij 0 0 77 66 . 77 66 Ln . . Lj.+1 . . . .. 5 4 Ln 3 2 L0 Z0;0 Z0;j .. .. .. 77 66 ... . . . . . . 77 66 77 66 Ln?j?2 . . . L0 Zn?j?2;0 Zn?j?2;j 7 66 ... . . . ... .. .. .. . . . b0 77 6 . 7: . (16) = 66 L 66 n?1 . . . Lj Zn?1;0 Zn?1;j ... 777 .. 77 66 Ln . . Lj+1 66 .. 77 . . . ... . 7 64 Ln bn?1 5 0 bn Notice that the top-left (2n ? j ? 1) (2n ? j ? 1) submatrix is exactly Hj +1, so Equation (16) can be rewritten as (17) Hj Tj = Hj0+1 b : n Therefore, jHj j jTj j = jHj +1j bn But by construction, jTj j = j [Rn J ] j = ?ban 01 n = bn ; so jHj j = jHj +1j: (18) Thus we generate a sequence of resultants that are hybrids of the Sylvester and Bezout resultants. Since H0 = Sn and Hn = Bn , we obtain in this fashion a direct proof that jSn j = jBnj (19) without appealing to any speci c properties of resultants.
7 Non-Homogeneous Bezout Matrices Due to Unequal Degrees When the coecients ai , bi , 0 i n, of f and g are treated as formal symbols, the Bezout matrix Bn is homogeneous in the sense that each entry is quadratic in the ai's and bi 's. 8
If the degree of g is m and m < n, then we have bm+1 = = bn = 0. In this situation, jSn j is not exactly the resultant of f; g|it has an extraneous factor of (an )n?m . Rather the correct Sylvester matrix Sn;m , whose determinant is the exact resultant, is the coecient matrix of n + m polynomials:
L tL tm?1L tm g tn?1g = 1 tn+m?1 S : n;m
(20)
Note that Sn is of order 2n, while Sn;m is of order n + m, and
jSnj = (an )n?m jSn;m j:
(21)
Similarly, the Bezout matrix Bn obtained from the Cayley expression n(t; ) also contains the extraneous factor ann?m . It is pointed out in [Cox et al 1997] that in this case the correct Bezout resultant for f and g can be written as the determinant of a non-homogeneous matrix Bn;m of size n n. The matrix Bn;m has m columns of quadratic entries consisting of coecients of both f and g, and n ? m columns of linear entries consisting of coecients of g. Below we present a simple alternative proof of this fact. Recall from Section 4 that 2 R1 R n 3 B n 4. . 5 0 = Sn Fn = Sn .. . . 2 2 R1 Rn Rm 3 2 Rm+1 Rn 3 3 . . 66 66 .. 66 ... . . . ... 77 77 . . .. .. 777 6 6 = 66 Sn 66 Rn?m+1 . . Rn 77 Sn 666 Rn 0 777 777 : (22) . . . . 4 4 .. 4 .. 5 .. 5 5 .. Rn 0 0 Let us focus on the second matrix product on the right hand side of Equation (22). With bm+1 = = bn = 0, 2 L0 3 66 ... 77 ... 66 L 7 2 0 2 Rm+1 Rn 3 3 0 n?m?1 L0 7 6 7 66 ... . . . ... 77 66 .. .. .. 77 66 ?am+1 ?an 77 . . . 6 6 7 77 Sn 66 Rn 0 77 = 66 Ln?1 Lm 777 666 ... .. . 75 .. 5 66 Ln Lm+1 77 4 0 4 ... . 66 . . . ... 77 ?an 0 0 64 75 Ln 0m2 2 b0 3 66 ... . . . 77 66 b 7 66 n?.m?1 . b.0 777 2 ?a 3 .. .. 77 6 m+1 ?an 7 66 .. = 66 bn?1 bm 77 4 ... (23) 5: ... 66 bn bm+1 77 ?an 66 . . . ... 77 64 7 bn 5 0m1 Note that the term 0m2 in the rst factor on the right hand side of Equation (23) represents an m 2 9
matrix of zero entries, since Sn has 2n rows. Let 2 ?a 3 m+1 ?an 75 : Am;n = 64 ... ... ?an Then putting together Equations (22) and (23), and using the notation of Equation (12), we obtain
2 66 2 66 Z0.;0 66 . Bn = 66 664 .. 66 .. 64 Zn?1;0
Z0;m?1 3 .. . .. .
.. . .. .
Zn?1;m?1
77 75
2 b0 66 ... 66 66 bm 66 66 4
... ... ...
33 77 77 77 77 ... 77 77 Im . . . b 77 77 Am;n : 0 77 . . . ... 75 75
(24)
bm Let us denote the n n matrix in the rst factor on the right hand side of Equation (24) by Bn;m . Then we can rewrite Equation (24) as I m Bn = Bn;m : (25) A
Hence
m;n
jBn j = (?an )n?m jBn;m j:
(26)
jSn;m j = jBn;m j:
(27)
Now it follows from Equations (19), (21), (26) that
That is, the coecient matrix of the following n polynomials is the Bezout matrix for f and g:
D D D 0 1 m?1 g tn?m?1 g :
For example, we have
(28)
2 j0; 1j j0; 2j b 3 0 66 j0; 2j j0; 3j + j1; 2j b1 b0 77 B5;2 = 66 j0; 3j j0; 4j + j1; 3j b2 b1 b0 77 ; 4 j0; 4j j0; 5j + j1; 4j b2 b1 5
j0; 5j j1; 5j b2 2 j0; 1j j0; 2j j0; 3j 66 j0; 2j j0; 3j + j1; 2j j0; 4j + j1; 3j B5;3 = 66 j0; 3j j0; 4j + j1; 3j j0; 5j + j1; 4j + j2; 3j 4 j0; 4j j0; 5j + j1; 4j j1; 5j + j2; 4j j0; 5j j1; 5j j2; 5j where ji; j j = Li Rj = ai bj ? aj bi, 0 i; j n. In summary, there exist matrices Fn(m) and Fn;m such that Sn Fn(m) =
B
n;m
0
10
= Sn;m Fn;m:
3
b0 b1 b0 77 b2 b1 77 ; b 3 b2 5 b3
In fact, let J = 0 1 T ; then
2 R 1 66 R2 66 ... 6 Fn(m) = 66 Rn?m 66 Rn?m+1 64 .. . Rn
Moreover, when n ? m > m,
2 R1 66 ... 66 R 66 ?a m 66 m. +1 .. Fn;m = 66 66 ?an?m 66 ?an?m+1 64 .. . ?an
...
...
...
Rm J Rm+1 J .. ... . . Rn?1 . . Rn ...
3 77 77 ... 7 J 77 : 77 75
3 Rm J .. 77 ... . 77 R2m?1 J 77 ?a2m 1 7 .. . . . 77 ; . 7 ?an?1 1 77 77 ?an 75
and when n ? m m,
2 R1 66 ... 66 R 66 R n?m 66 n?.m+1 Fn;m = 66 .. 66 Rm 66 ?am+1 64 .. . ?an
...
Rn?m .. .
...
.. .
Rm J ...
.. .
R2n?2m?1 Rn?1 R2n?2m Rn ...
Rn?1 ?an
...
3 77 ... 7 J 77 77 77 : 77 77 77 5
References [1] J. Canny. The Complexity of Robot Motion Planning. Ph.D. Thesis, Department of Electrical Engineering and Computer Science, MIT, 1987. [2] J. Canny. Generalized Characteristic Polynomials. Proc. ISSAC'88, Springer Lect. Notes Comp. Sc., 1988. [3] E. W. Chionh, M. Zhang, R. N. Goldman. The Block Structure of Three Dixon Resultants and Their Accompanying Transformation Matrices, Technical Report, TR99-341, Computer Science Department, Rice University, 1999. [4] D. A. Cox, J. B. Little, D. O'Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Second Edition. Springer-Verlag New York, Inc., 1997. [5] D. A. Cox, T. W. Sederberg, F. Chen. The Moving Line Ideal Basis of Planar Rational Curves, to appear in Computer Aided Geometric Design. 11
[6] 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. [7] D. Manocha and J. F. Canny. Algorithms for Implicitizing Rational Parametric Surfaces. Computer Aided Geometric Design, 9(1):25{50, 1992. [8] Y. De Montaudouin, W. Tiller. The Cayley Method in Computer Aided Geometric Design. Computer Aided Geometric Design, 1:309|326, 1984. [9] T. W. Sederberg, R. N. Goldman and H. Du. Implicitizing Rational Curves by the Method of Moving Algebraic Curves. J. Symbolic Computation, 23:153|175, 1997. [10] T. W. Sederberg and S. R. Parry. Comparison of Three Curve Intersection Algorithms. Computer Aided Design, 18(1), 1986. [11] B.L. van der Waerden. Modern Algebra, Second Edition. New York, Frederick Ungar, 1950. [12] J. Weyman, A. Zelevinsky. Determinantal Formulas for Multigraded Resultants. J. Algebraic Geometry, 3: 569-597, 1994. [13] 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.
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