Computer Aided Geometric Design 29 (2012) 296–314
Contents lists available at SciVerse ScienceDirect
Computer Aided Geometric Design
www.elsevier.com/locate/cagd
Using Smith normal forms and
of rational planar curves ✩
μ-bases to compute all the singularities
Xiaohong Jia a,∗ , Ron Goldman b
a
b
KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences Beijing, China
Department of Computer Science, Rice University, Houston, TX, USA
a r t i c l e
i n f o
a b s t r a c t
Article history:
Received 13 July 2011
Received in revised form 18 November 2011
Accepted 10 February 2012
Available online 14 February 2012
Keywords:
Rational planar curve
Singularities
Infinitely near points
Blow-up
Intersection multiplicity
μ-Basis
We prove a result similar to the conjecture of Chen et al. (2008) concerning how to calculate the parameter values corresponding to all the singularities, including the infinitely
near singularities, of rational planar curves from the Smith normal forms of certain Bezout
resultant matrices derived from μ-bases. A great deal of mathematical lore is hidden behind their conjecture, involving not only the classical blow-up theory of singularities from
algebraic geometry, but also the intrinsic relationship between μ-bases and the singularities of rational planar curves. Here we explore these mathematical foundations in order
to reveal the true nature of this conjecture. We then provide a novel approach to proving
a related conjecture, which in addition to these mathematical underpinnings requires only
an elementary knowledge of classical resultants.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
The nature and number of the singularities of planar algebraic curves contain a great deal of information about the
geometry and topology of these curves. In Computer Graphics knowledge of these singularities is particularly important
for developing robust algorithms for rendering these curves (Alberti and Mourrain, 2007a; Alberti and Mourrain, 2007b).
Many classical texts in algebraic geometry such as (Hilton, 1920; Coolidge, 1959; Walker, 1950) concentrate on the theory
of singularities for planar algebraic curves. More recent research focuses on procedures for calculating the singularities of
planar algebraic curves.
For algebraic planar curves, Sakkalis and Farouki (1990) provide an algorithm to compute the singularities and their
multiplicities which requires only rational arithmetic operations on the coefficients of the implicit equation of the curve. For
polynomially parametrized planar curves, Abhyankar (1990) develops a Taylor resultant for computing the singularities of
the curve. Later, Peŕez-Diáz (2006) generalizes the result of Abhyankar (1990) to rational parametric curves — they find the
singularities as well as the infinitely near singularities and their multiplicities by analyzing the degree of a function directly
derived from the parametrization. Their method has the advantage that their computations are free of algebraic numbers.
A very resent investigation by Cox et al. (2011) studies the singularities of rational projective curves via a Hilbert–Burch
matrix constructed from the parametrization.
A few years ago, Chen, Wang, and Liu presented a conjecture concerning how to use the Smith normal forms of
certain Bezout resultant matrices derived from μ-bases to calculate the parameter values corresponding to all the singularities, including the infinitely near singularities, of rational planar curves (Chen et al., 2008). The algorithm presented by
✩
*
This paper has been recommended for acceptance by Thomas Sederberg.
Corresponding author.
E-mail address: [email protected] (X. Jia).
0167-8396/$ – see front matter
doi:10.1016/j.cagd.2012.02.001
© 2012
Elsevier B.V. All rights reserved.
X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
297
Chen et al. (2008) succeeds in directly extracting the singularity trees of rational planar curves, and because of the symbolic
nature of the method eliminates numerical problems. The goal of this paper is to reveal the underlying mathematical methods behind their conjecture and also to inspire a possible novel approach to the proof of their conjecture, which requires in
addition to these essential mathematical methods only an elementary knowledge of classical resultants.
Simultaneously with this paper, Buse and D’Andrea (2009) also studied the conjecture of Chen, Wang and Liu. Buse
and D’Andrea focus on factoring D-resultants, whereas this paper is a sequel to our paper on μ-Bases and Singularities of
Rational Planar Curves (Jia and Goldman, 2009). In (Jia and Goldman, 2009), we show how the parameters corresponding to
each singularity, including the infinitely near singularities, of rational planar curves are related to the intersections of two
algebraic curves derived from μ-bases. This result provides the initial step towards a proof of the conjecture of Chen, Wang,
and Liu. However, there is still some way to go to complete the proof of the original conjecture because in the conjecture of
Chen, Wang, and Liu, one can directly read the parameter values and the multiplicities from the entries of the Smith form
matrices constructed from a μ-basis, while all the results in (Jia and Goldman, 2009) are represented by the determinants of
those Smith form matrices; hence much of the detailed information about the parameters and their multiplicities for each
singularity is hidden. Here we shall further excavate the information hidden in (Jia and Goldman, 2009) using an elaborate
analysis on the matrix level. This analysis will allow us to read the singularity tree directly from the Smith normal forms
of the Bezout matrix of the two algebraic curves, and hence totally reveal what lies behind the conjecture of Chen, Wang,
and Liu.
We proceed in the following fashion. In Section 2 we review the notion of μ-bases for rational planar curves, and in
Section 3 we summarize previous results on singularities of rational planar curves. We state the conjecture of Chen, Wang
and Liu in Section 4. Section 5 is devoted to a proof of our main result. Here we focus on a single singularity. First we
compute the Smith normal forms of two hybrid Bezout matrices, one of which provides all the parameters of the infinitely
near singularities while the other provides all the parameters of the original singularity. We then combine these two Smith
normal forms together by invoking companion matrices to factor the k-th determinant factors of the Bezout matrix that
appears in our main theorem and thereby complete our proof. We close in Section 6 with a discussion of the relationship
between our main theorem and the conjecture of Chen, Wang and Liu. We also provide two appendices. Appendix A is
devoted to the definitions and properties of Smith normal forms, while Appendix B introduces companion matrices in order
to factor hybrid Bezout matrices.
2. Preliminaries on μ-bases
Let R[s, u ] be the set of homogeneous polynomials in the homogeneous parameter s : u with real coefficients.
A parametrization for a degree n rational planar curve C is usually written in homogeneous form as
P(s, u ) = a(s, u ), b(s, u ), c (s, u ) ,
(1)
where a(s, u ), b(s, u ), c (s, u ) are degree n homogeneous polynomials in R[s, u ]. To avoid the degenerate case where P(s, u )
parameterizes a line, we shall assume that the three homogeneous polynomials a(s, u ), b(s, u ), c (s, u ) are relatively prime
and linearly independent. Moreover, throughout this paper we will assume that the parametrization P(s, u ) is proper. For
non-proper parameterizations one can reparameterize the curve — see (Peŕez-Diáz, 2006).
A polynomial vector L(s, u ) = ( A (s, u ), B (s, u ), C (s, u )) is a syzygy of the parametrization (1) if
1
L(s, u ) · P(s, u ) = A (s, u )a(s, u ) + B (s, u )b(s, u ) + C (s, u )c (s, u ) ≡ 0.
(2)
The set M p of all syzygies of a parameterization P(s, u ) is a module over the ring R[s, u ], called the syzygy module. The
syzygy module M p is known to be a free module with two generators (Chen and Wang, 2003).
Definition 2.1. Two syzygies p(s, u ) and q(s, u ) are called a μ-basis for a parameterization P(s, u ) of the rational planar
curve C if p and q form a basis for M p , i.e., every syzygy L(s, u ) ∈ M p can be written as
L(s, u ) = α (s, u )p(s, u ) + β(s, u )q(s, u ),
where
(3)
α (s, u ), β(s, u ) ∈ R[s, u ].
Note that since we are using homogeneous polynomials, Definition 2.1 implicitly implies the following degree constraint
on the elements of a μ-basis (Cox et al., 1998):
deg(p) + deg(q) = deg(P).
We usually set deg(p) = μ, and therefore deg(q) = deg(P) − μ.
1
Since even a real singular point may correspond to a pair of complex conjugate parameter values, throughout this paper we shall perform our analysis
of parameters over the complex projective domain P1 (C). Nevertheless, even though our parameter values may be complex, we shall be concerned only
with points with real coordinates.
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Every rational planar curve has a μ-basis. Moreover, there is a fast algorithm for computing μ-bases based on Gaussian
elimination (Chen and Wang, 2003).
μ-Bases have many advantageous properties. For example, we can recover the parametrization P(s, u ) of the rational
planar curve C from the outer product of a μ-basis:
p(s, u ) × q(s, u ) = kP(s, u ),
(4)
where k is a nonzero constant. We can also retrieve the implicit equation f (x, y , w ) = 0 of the rational planar curve C by
taking the resultant of a μ-basis:
f (x, y , w ) = Ress,u p(s, u ) · X, q(s, u ) · X ,
(5)
where X = (x, y , w ) (Chen and Wang, 2003).
Example 2.1. Given a rational planar curve C with parameterization
P(s, u ) = (s − u )2 (s − 2u )2 (s − 3u )u 2 , (s − u )2 (s − 2u )4 (s − 3u ), u 7 ,
μ-basis for P(s, u ):
p(s, u ) = −u 5 , 0, s5 − 9s4 u + 31s3 u 2 − 51s2 u 3 + 40su 4 − 12u 5 ,
q(s, u ) = s2 − 4su + 4u 2 , −u 2 , 0 .
by the algorithm in (Chen and Wang, 2003), we compute a
One can check that p(s, u ) × q(s, u ) = P(s, u ). Moreover, Ress,u (p(s, u ) · X, q(s, u ) · X) = x7 + 2x5 y w − 2x4 y 2 w + x3 y 2 w 2 −
3x2 y 3 w 2 + 3xy 4 w 2 − y 5 w 2 = 0 gives the implicit equation of the curve C . Hence μ-bases are a bridge connecting the
parametric and implicit equations. 2
3. Preliminaries on singularities
We next review some basic concepts concerning singularities. Readers are referred to (Chen et al., 2008) and (Jia and
Goldman, 2009) for further details.
Definition 3.1. A point Q = (x0 , y 0 , w 0 ) is a singularity of multiplicity r on the algebraic curve f (x, y , w ) = 0 if and only if
∂ r −1 f
∂ xi ∂ y j ∂ w k
(x0 , y 0 , w 0 ) = 0,
i + j + k = r − 1,
(6)
and at least one r-th partial derivative of f at Q does not vanish.
Definition 3.1 is a classical definition for the multiplicity of singularities of algebraic curves. For rational planar curves,
we have the following alternative characterization of the multiplicity of a singularity in terms of the number of parameter
values corresponding to the singular point, see (Jia and Goldman, 2009).
Proposition 3.1. A point Q is of multiplicity r on the rational planar curve C with parameterization P(s, u ) if and only if there are
exactly r parameters (si , u i ) counting multiplicity such that P(si , u i ) = λi Q for some nonzero constants λi . When r > 1, the point Q
is called a singular point.
Definition 3.2. For a singular point Q as in Proposition 3.1, we call a polynomial h(s, u ) an inversion formula for Q if
r
h(s, u ) = λ i =1 (u i s − si u ) for some nonzero constant λ. Hence an inversion formula h(s, u ) for a point Q is unique up to a
nonzero constant multiple.
The inversion formula for a point Q can be computed by using the following proposition.
Proposition 3.2. (See Chen and Wang (2003).) Let Q be a singularity on a rational planar curve C parameterized by P(s, u ) with a
μ-basis p(s, u ), q(s, u ). The inversion formula for Q is given by
h(s, u ) = gcd p(s, u ) · Q , q(s, u ) · Q .
Example 3.1. Consider the rational planar curve given in Example 2.1. The inversion formula for the point Q = (0, 0, 1) is
given by
h(s, u ) = gcd p(s, u ) · Q , q(s, u ) · Q = (s − u )2 (s − 2u )2 (s − 3u ).
Hence the point Q is a singularity of multiplicity deg(h) = 5.
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299
Proposition 3.2 tells whether a point Q is a singular point, and deg(h) is just the multiplicity of the point Q . However,
this multiplicity does not totally reveal the nature of a singularity. The infinitely near singularities that are in the neighborhood of the point Q form a singularity tree for Q (see Jia and Goldman (2009) for details). Next we shall briefly explain how
to blow up the parameterization P(s, u ) at the point Q to get all the infinitely near singularities of Q .
Definition 3.3. To blow up the parameterization P(s, u ) of the curve C at the singular point Q , we first perform a coordinate
transformation so that the point Q is moved to the origin. Then the parametrization P(s, u ) becomes
P(s, u ) = a(s, u )h(s, u ), b(s, u )h(s, u ), c (s, u ) ,
where h(s, u ) is the inversion formula for the point Q , and gcd(a, b) = gcd(h, c ) = 1. The first blow-up curve of P(s, u ) is
, b )).
defined by P1 (s, u ) = (a2 h, bc , ac ) (the homogenization of the curve ( X , Y ) = ( ah
c a
Let Q1 be a point of multiplicity r1 on the curve P1 (s, u ), whose inversion formula is h1 (s, u ). If h1 (s, u )|h(s, u ), then we
say that the point Q1 is an infinitely near point of multiplicity r1 in the first neighborhood of Q .
If we continue to blow up the parametric curve P1 (s, u ) at the point Q1 using the above approach, we shall get a second
blow-up curve P2 (s, u ) of the original curve P(s, u ). The points on P2 (s, u ) whose inversion formulas divide h(s, u ) are
called infinitely near points in the second neighborhood of Q . Continuing with this approach, we get all the infinitely near
singularities in all the neighborhoods of Q .
Example 3.2. We continue to use the rational planar curve C parameterized by P(s, u ) in Example 2.1. By Example 3.1 we
already know that Q = (0, 0, 1) is a point of multiplicity 5 on the curve C . Now we blow up the parameterization P(s, u ) at
Q to get a new curve P1 (s, u ) = ((s − 3u )(s − u )2 (s − 2u )2 , (s − 2u )2 u 3 , u 5 ), which contains a double point Q∗1 = (0, 0, 1) with
inversion formula (on P1 ) h1 (s, u ) = (s − 2u )2 , and a double point Q∗2 = (0, 1, 1) with inversion formula (on P1 ) h2 (s, u ) =
(s − u )(s − 3u ). Since h1 |h and h2 |h, both Q∗1 and Q∗2 are infinitely near singularities of Q . We can continue blowing up
P1 (s, u ) at Q∗1 and Q∗2 to see that there are no additional infinitely near singularities. 2
The above blow-up approach aims more at explaining what infinitely near singularities are than exactly computing these
infinitely near singularities. The blow-up technique is not practical for computation, since we first need to know where the
singularities of the original curve are; and even if we already know the locations of the basic singularities, for each blow-up
we need to translate the coordinate system to move the singularity to the origin, which will lead to numerical problems.
There is, however, an efficient symbolic approach to computing all the singularities including infinitely near singularities on
a rational planar curve presented by Chen et al. (2008), which they left as an unproved conjecture. Next we shall review
this conjecture.
4. The conjecture for computing singularities on rational planar curves
Let P(t , v ) be the parameterization of a rational planar curve C of degree n with a
B (t , v ) := HBezs,u
p(s, u ) · P(t , v ), q(s, u ) · P(t , v )
μ-basis p(t , v ), q(t , v ), and let
be the hybrid Bezout matrix of the two polynomials with respect to the parameter (s, u ). Then B (t , v ) is a square matrix of
size deg(q) = n − μ, and the Smith normal form2 of B (t , v ) has the form3
S ( B ) = diag dn−μ (t , v ), dn−μ (t , v )dn−μ−1 (t , v ), . . . , dn−μ (t , v ) · · · d2 (t , v ), 0 ,
where di (t , v ) are polynomials, i = 2, . . . , n − μ.
Chen et al. (2008) state the following conjecture.
Conjecture.
dr (t , v ) = hr (t , v )
ψri (t , v ),
(7)
i r
where hr (t , v ) is the product of the inversion formulas of all the order r singularities on the curve parameterized by P(t , v ), and ψri (t , v )
is the inversion formula for all the order r infinitely near singularities in the neighborhood of order i r singular points on P(t , v ).
Example 4.1. We continue with our analysis of the rational planar curve P(s, u ) in Example 2.1. Computing the Smith normal
form S ( B ) (see Appendix A, Definition A.4 for computing Smith normal forms of matrices with homogeneous polynomial
entries) of B (t , v ) = HBezs,u (p(s, u ) · P(t , v ), q(s, u ) · P(t , v )), we get
2
The definitions and properties of Smith normal forms are provided in Appendix A.
The last element is zero because Ress,u (p(s, u ) · P(t , v ), q(s, u ) · P(t , v )) ≡ 0, since Ress,u (p(s, u ) · X, q(s, u ) · X) gives the implicit equation of the curve
P(t , v ).
3
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S ( B ) = diag 12(t − 3v )(t − v )2 (t − 2v )2 , 12(t − 3v )(t − v )2 (t − 2v )2 ,
12(t − 3v )(t − v )2 (t − 2v )2 , v 6 (t − 3v )2 (t − v )3 (t − 2v )4 , 0 .
Hence d2 = v 6 (t − 3v )(t − v )(t − 2v )2 , d3 = d4 = 1, d5 = (t − 3v )(t − v )2 (t − 2v )2 . By the conjecture, d2 (t , v ) is the product
of the inversion formulas for all the double points including the infinitely near ones on P(t , v ). Since deg(d2 ) = 10, there are
five double points, two of which are infinitely near double points of Q1 = (0 : 0 : 1), given by (t − 3v )(t − v ) and (t − 2v )2 ;
the other three are the double point Q2 = (1 : 1 : 0) together with its two infinitely near double points, given by v 6 . In
addition, d5 (t , v ) is the inversion formula for the singularity Q1 = (0 : 0 : 1) of multiplicity 5 on P(s, u ).
The conjecture in (Chen et al., 2008) expands the inversion formulas for all the singularities including the infinitely
near singularities on a rational planar curve symbolically by computing the Smith normal form of the hybrid Bezout matrix
constructed from a μ-basis of the curve. Chen et al. (2008) also point out that taking the Smith normal form of the following
Bezout matrix provides an equivalent result:
B (s, u ) := Bezt , v p(s, u ) · P(t , v ), q(s, u ) · P(t , v ) ,
where Bezt , v is the symmetric Bezout matrix4 with respect to the parameter (t , v ). Actually since degt , v (p(s, u ) · P(t , v )) =
degt , v (q(s, u ) · P(t , v )) = n, the Smith normal form S ( B (s, u )) is
diag 1, . . . , 1, dn−μ (s, u ), dn−μ (s, u )dn−μ−1 (s, u ), . . . , dn−μ (s, u ) · · · d2 (s, u ), 0 .
μ
Therefore, by Corollary A.1 in Appendix A, the conjecture is also equivalent to
Theorem 4.1. Let
B (s, u ) := Bezt , v
p(s, u ) · P(t , v ) q(s, u ) · P(t , v )
,
sv − tu
sv − tu
.
Then the Smith normal form S ( B (s, u )) is
diag 1, . . . , 1, dn−μ (s, u ), dn−μ (s, u )dn−μ−1 (s, u ), . . . , dn−μ (s, u ) · · · d2 (s, u ) ,
μ
where dr (s, u ), r = 2, . . . , n − μ is the product of the inversion formulas of all the singularities including the infinitely near singularities
of multiplicity r of the curve C parameterized by P(s, u ).
In Section 5 we shall prove Theorem 4.1.
5. Proof of the conjecture
We are going to prove Theorem 4.1 by applying an approach that is a matrix version parallel to the analysis in (Jia
and Goldman, 2009). We begin in Section 5.1 by explaining the equivalence of different pairs of syzygies for representing
the singularities of the curve by the Smith normal forms of their Bezout matrices, so that Theorem 4.1 is transformed
to computing the Smith normal form constructed from another obvious pair of syzygies for the curve. In Section 5.2, we
decompose this Smith normal form into two Smith normal forms, one of which provides all the parameters of the infinitely
near singularities while the other provides all the parameters of the original singularity. Then in Section 5.3, we combine
the two Smith normal forms together to complete the proof.
5.1. Equivalent Smith normal forms from different pairs of syzygies
Suppose Q is a singular point on the curve C with parameterization P(t , v ). Let p(t , v ), q(t , v ) be a μ-basis for P(t , v ),
and let p̃(t , v ), q̃(t , v ) be an arbitrary pair of syzygies for P(t , v ) that are always independent for any parameter (t , v )
corresponding to the point Q . Set
F (s, u ; t , v ) :=
F̃ (s, u ; t , v ) :=
p(s, u ) · P(t , v )
sv − tu
p̃(s, u ) · P(t , v )
sv − tu
,
G (s, u ; t , v ) :=
,
G̃ (s, u ; t , v ) :=
q(s, u ) · P(t , v )
sv − tu
q̃(s, u ) · P(t , v )
sv − tu
,
.
(8)
4
Throughout this paper, for any two bivariate polynomials F (s, t ) and G (s, t ), we shall use HBezt ( F , G ) and Bezt ( F , G ) to denote the hybrid Bezout
matrix and symmetric Bezout matrix of F and G with respect to the parameter t. When degt ( F ) = degt (G ), HBezt ( F , G ) = Bezt ( F , G ), and in this case we
shall prefer the expression Bezt ( F , G ) (see Fuhrmann (1996) for further details).
X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
301
We shall show that S (Bezt , v ( F , G )) is equivalent to S (Bezt , v ( F̃ , G̃ )) for representing the singularity tree of Q . But before we
continue, a word about our notation.
Remark 5.1. For a polynomial d(s, u ), we use dQ (s, u ) to denote all the factors of d(s, u ) whose roots are parameters corresponding to the point Q . For example, if the inversion formula for the point Q is s2 (s + u ), and d(s, u ) = s(s + u )2 (s − u ),
then dQ (s, u ) = s(s + u )2 . Consequently, if A is a polynomial matrix and
S ( A ) = diag( gn , gn−1 , . . . , g 1 ),
then we shall write
Q
Q
Q
S Q ( A ) := diag gn , gn−1 , . . . , g 1
to denote the Smith normal form of A restricted to Q .
Remark 5.2. Since we do not care about the constant coefficients of the polynomials in the Smith normal forms, from now
on we shall write S ( A ) = S ( B ) if the corresponding elements in the Smith normal forms of the matrices A and B are equal
polynomials up to a nonzero constant.
Theorem 5.3.
S Q Bezt , v ( F , G ) = S Q Bezt , v ( F̃ , G̃ ) .
Proof. Suppose
S Bezt , v ( F , G ) := diag gn−1 (s, u ), . . . , g 1 (s, u ) ,
S Bezt , v ( F̃ , G̃ ) := diag g̃n−1 (s, u ), . . . , g̃ 1 (s, u ) .
It suffices to prove that
Q
Q
g i = λ g̃ i
for some constant λ = 0. Since p̃(s, u ), q̃(s, u ) are a pair of syzygies and p(s, u ), q(s, u ) are a
α (s, u ), β(s, u ), γ (s, u ), δ(s, u ) such that
μ-basis, there are polynomials
p̃(s, u ) = α (s, u )p(s, u ) + β(s, u )q(s, u ),
q̃(s, u ) = γ (s, u )p(s, u ) + δ(s, u )q(s, u ).
Therefore
F̃ (s, u ; t , v ) = α (s, u ) F (s, u ; t , v ) + β(s, u )G (s, u ; t , v ),
G̃ (s, u ; t , v ) = γ (s, u ) F (s, u ; t , v ) + δ(s, u )G (s, u ; t , v ).
Now by definition Bezt , v ( F̃ , G̃ ) is the coefficient matrix of the following Bezoutian:
α (s, u ) F (s, u ; t , v ) + β(s, u )G (s, u ; t , v ) γ (s, u ) F (s, u ; t , v ) + δ(s, u )G (s, u ; t , v ) α (s, u ) F (s, u ; t̄ , v̄ ) + β(s, u )G (s, u ; t̄ , v̄ ) γ (s, u ) F (s, u ; t̄ , v̄ ) + δ(s, u )G (s, u ; t̄ , v̄ ) t v̄ − t̄ v
α (s, u ) γ (s, u ) F (s, u ; t , v ) G (s, u ; t , v ) β(s, u ) δ(s, u ) F (s, u ; t̄ , v̄ ) G (s, u ; t̄ , v̄ ) =
t v̄ − t̄ v
F (s, u ; t , v ) G (s, u ; t , v ) F (s, u ; t̄ , v̄ ) G (s, u ; t̄ , v̄ ) = κ (s, u )
t v̄ − t̄ v
where
κ (s, u ) = α (s, u )δ(s, u ) − β(s, u )γ (s, u ).
Hence
Bezt , v ( F̃ , G̃ ) = κ (s, u )Bezt , v ( F , G ),
(9)
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X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
so
g̃ i (s, u ) = κ (s, u ) g i (s, u ).
Therefore,
Q
Q
g̃ i (s, u ) = κ Q (s, u ) g i (s, u ).
Now κ (s, u ) = 0 if and only if the two syzygies p̃, q̃ are linearly dependent, which by our assumption at the beginning of
this subsection never occurs for parameters corresponding to the point Q , so κ Q (s, u ) = λ, where λ is a nonzero constant.
Therefore
Q
Q
g̃ i = λ g i .
The proof is now complete.
2
Next we shall focus only on one singularity Q on the rational planar curve C to see how information for the singularity
branch rooted at the point Q is transferred from a μ-basis to another obvious syzygy of the parametrization. At the end of
the proof we shall put all the singularities on the curve C together to prove Theorem 4.1.
Now suppose that Q = (0, 0, 1). Then
P(t , v ) = a(t , v )h(t , v ), b(t , v )h(t , v ), c (t , v ) ,
where a, b, c ∈ R[t , v ], h is an inversion formula for Q , and gcd(a, b) = gcd(h, c ) = 1. We can also rotate the coordinates
around Q so that gcd(a, c ) = 1. Note that M(s, u ) := (−b, a, 0), L(s, u ) := (c , 0, −ah) are a pair of syzygies for P(t , v ), which
are always linearly independent for any parameter (s, u ) corresponding to Q . Let
M (s, u ; t , v ) :=
M(s, u ) · P(t , v )
sv − tu
L (s, u ; t , v ) :=
,
L(s, u ) · P(t , v )
sv − tu
(10)
.
Then by Theorem 5.3 we have
Corollary 5.1.
S Q Bezt , v ( F , G ) = S Q Bezt , v ( M , L ) .
Example 5.1. We can verify Corollary 5.1 using Example 2.1 with the singularity Q = (0, 0, 1). Since
F (s, u ; t , v ) =
p(s, u ) · P(t , v )
sv − tu
= v 4 s4 − 9s3 uv 4 + s3 tuv 3 + 31s2 u 2 v 4 − 9s2 tu 2 v 3 + s2 t 2 u 2 v 2
− 51su 3 v 4 + 31stu 3 v 3 − 9st 2 u 3 v 2 + st 3 u 3 v + 40u 4 v 4 − 51tu 4 v 3
+ 31t 2 u 4 v 2 − 9t 3 u 4 v + t 4 u 4 v 2 ,
q(s, u ) · P(t , v )
G (s, u ; t , v ) =
sv − tu
= svt 5 − 9st 4 v 2 − 12sv 6 + 31st 3 v 3 − 51st 2 v 4 + 40st v 5 − 172v 5tu
+ 48v 6 u + 67v 2 ut 4 − 13vut 5 + 244v 4 ut 2 − 175v 3 ut 3 + t 6 u ,
we can compute
S Bezt , v ( F , G ) = diag 1, 1, 12(s − 3u )(s − u )2 (s − 2u )2 , 12(s − 3u )(s − u )2 (s − 2u )2 ,
12(s − 3u )(s − u )2 (s − 2u )2 , u 6 (s − 3u )2 (s − u )3 (s − 2u )4 .
Also since
M (s, u ; t , v ) = −(sv − 4uv + tu )(t − v )2 (t − 2v )2 (t − 3v ),
L (s, u ; t , v ) = − v 4 s4 − 9s3 uv 4 + s3 tuv 3 + 31s2 u 2 v 4 − 9s2 tu 2 v 3 + s2 u 2 t 2 v 2 − 51su 3 v 4 + 31stu 3 v 3
− 9st 2 u 3 v 2 + st 3 u 3 v + 40u 4 v 4 − 51tu 4 v 3 + 31t 2 u 4 v 2 − 9t 3 u 4 v + t 4 u 4 v 2 u 2 ,
we can compute
S Bezt , v ( M , L ) = diag 144u 2 , 144u 2 , 12(s − 3u )(s − u )2 (s − 2u )2 u 2 , 12(s − 3u )(s − u )2 (s − 2u )2 u 2 ,
12(s − 3u )(s − u )2 (s − 2u )2 u 2 , u 6 (s − 3u )2 (s − u )3 (s − 2u )4 .
X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
303
Since the inversion formula of the point Q = (0, 0, 1) is h(s, u ) = (s − 3u )(s − u )2 (s − 2u )2 , we can see that
S Q Bezt , v ( F , G ) = S Q Bezt , v ( M , L )
= diag 1, 1, 12(s − 3u )(s − u )2 (s − 2u )2 , 12(s − 3u )(s − u )2 (s − 2u )2 ,
12(s − 3u )(s − u )2 (s − 2u )2 , (s − 3u )2 (s − u )3 (s − 2u )4 .
From the above observations, we need only examine S (Bezt , v ( M , L )) instead of S (Bezt , v ( F , G )). Before moving on, we
review some observations by Jia and Goldman (2009) on the intersections of F and G as well as M and L.
Q
Q
Proposition 5.1. (See Jia and Goldman (2009).) Suppose that S Q (Bezt , v ( F , G )) := diag( gn−1 , . . . , g 1 ). The intersection multiplicity
of F (s, u ; t , v ) = 0 and G (s, u ; t , v ) = 0 related to the point Q can be computed by
I Q ( F , G ) :=
n −1
Q
deg g i .
i =1
Remark 5.4. Remark 4.1 and Corollary 4.4 in (Jia and Goldman, 2009) together imply that we can compute the intersection
multiplicity this way. Also note that the Bezout matrix Bezt , v can be replaced by any resultant matrix.
Proposition 5.1 together with Corollary 5.1 yield the following result given by Jia and Goldman (2009), which is somewhat
weaker than Corollary 5.1.
Proposition 5.2.
I Q ( F , G ) = I Q ( M , L ).
The following proposition reveals the total number of intersections of F and G related to the singularity Q:
Proposition 5.3. (See Jia and Goldman (2009).) Let νQ∗ denote the multiplicity of an infinitely near point Q∗ of a singularity Q . Then
IQ( F , G ) =
νQ∗ (νQ∗ − 1),
Q∗
where the sum is taken over all the infinitely near points Q∗ of the point Q including Q itself.
5.2. Factoring S (Bezt , v ( M , L ))
Recall that by definition
M (s, u ; t , v ) =
L (s, u ; t , v ) =
M(s, u ) · P(t , v )
sv − tu
L(s, u ) · P(t , v )
sv − tu
=
=
a(s, u )b(t , v ) − b(s, u )a(t , v )
h(t , v ),
sv − tu
c (s, u )a(t , v )h(t , v ) − c (t , v )a(s, u )h(s, u )
sv − tu
.
Let
M̄ (s, u ; t , v ) =
a(s, u )b(t , v ) − b(s, u )a(t , v )
sv − tu
.
Next we shall separately study S (HBezt , v ( M̄ , L )) and S (HBezt , v (h, L )).
First we shall assume that all the singularities on the curve P(t , v ) have infinitely near singularities only in their first
neighborhoods. The more general cases will be treated later by induction (see Remark 5.11 at the end of Section 5.3).
5.2.1. S (HBezt , v ( M̄ , L ))
To begin, we blow up the parameterization P(t , v ) of the original curve C at the point Q = (0, 0, 1) to get
P1 (t , v ) = a2 h, bc , ca .
Let p1 (t , v ), q1 (t , v ) be a
F 1 (s, u ; t , v ) :=
μ-basis for P1 (t , v ), and let
p1 (s, u ) · P1 (t , v )
sv − tu
,
G 1 (s, u ; t , v ) :=
q1 (s, u ) · P1 (t , v )
sv − tu
,
Suppose that the multiplicity of Q is r. Then degt , v ( F 1 ) = degt , v (G 1 ) = deg(P1 ) − 1 = 2 deg(P) − r − 1.
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Theorem 5.5.
S Q Bezt , v ( F 1 , G 1 ) = diag( 1, . . . , 1, ψr , ψr ψr −1 , . . . , ψr ψr −1 · · · , ψ2 )
(11)
2n−2r
where ψi (s, u ) is the product of the inversion formulas for all the order i infinitely near singularities of Q .
Proof. Suppose that
S Bezt,v ( F 1 , G 1 ) = diag( f 2n−r , f 2n−r f 2n−r −1 , . . . , f 2n−r · · · f 2 ).
Then by Corollary A.1 and Corollary 4 in (Chen et al., 2008),
i = 2, . . . , r ,
ψi (s, u )| f i (s, u ),
(12)
Q
and since ψi (s, u ) = ψi (s, u ),
Q
i = 2, . . . , r .
ψi (s, u )| f i (s, u ),
(13)
Suppose that there are mi infinitely near singularities of order i related to the point Q on the blow-up curve P1 (s, u ). Then
deg(ψi ) = mi × i ,
i = 2, . . . , r .
(14)
Note that the multiplicity of an infinitely near singularity cannot exceed that of the point Q , so
deg(ψi ) = 0,
i > r.
By Proposition 5.1 and Eq. (13),
IQ( F 1, G 1) =
2n
−r
Q
deg f i
× ( i − 1) r
i =2
r
Q
deg f i
× ( i − 1)
i =2
deg(ψi ) × (i − 1).
(15)
i =2
But by Proposition 5.3 and (14),
IQ( F 1, G 1) =
r
m i × i × ( i − 1) =
i =2
r
deg(ψi ) × (i − 1).
(16)
i =2
Hence (15) and (16) together with (13) yield
Q
deg f i
=
deg(ψi ),
0,
i = 2, . . . , r ;
i > r.
Therefore by Eq. (13)
Q
f i (s, u ) =
ψ i , i = 2, . . . , r ;
i > r,
1,
up to a constant multiple. The proof is now complete.
2
Next we transfer the information on the singularity Q from the Smith normal form of Bezt , v ( F 1 , G 1 ) to the Smith normal
form of HBezt , v ( M̄ , L ).
Theorem 5.6.
S Q HBezt , v ( M̄ , L ) = diag( 1, . . . , 1, ψr , ψr ψr −1 , . . . , ψr ψr −1 · · · , ψ2 ).
n−r
Proof. For the blow-up curve
P1 (s, u ) = a2 h, bc , ca ,
we have a pair of syzygies
S1 (s, u ) (0, a, −b),
T1 (s, u ) (c , 0, −ah).
(17)
X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
305
Construct two polynomials from S1 (s, u ) and T1 (s, u ):
S 1 (s, u ; t , v ) :=
=
S1 (s, u ) · P1 (t , v )
sv − tu
a(s, u )b(t , v ) − b(s, u )a(t , v )
sv − tu
c (t , v )
= M̄ (s, u ; t , v )c (t , v ),
T1 (s, u ) · P1 (t , v )
T 1 (s, u ; t , v ) :=
sv − tu
c (s, u )a(t , v )h(t , v ) − c (t , v )a(s, u )h(s, u )
=
a(t , v )
sv − tu
= L (s, u ; t , v )a(t , v ).
(18)
Recalling that gcd(h, c ) = gcd(a, h) = gcd(a, c ) = 1, we have
S Q HBezt , v ( M̄ , L ) = S Q Bezt , v ( S 1 , T 1 ) (n−1)×(n−1) ,
(19)
where the subscript (n − 1) × (n − 1) means the (n − 1) × (n − 1) submatrix in the lower right corner. Since S1 , T1 are a pair
of syzygies for the curve P1 (s, u ), by Theorem 5.3
S Q Bezt , v ( S 1 , T 1 ) = S Q Bezt , v ( F 1 , G 1 ) .
Then Theorem 5.5 and Eq. (19) together yield
S Q HBezt , v ( M̄ , L ) = diag( 1, . . . , 1, ψr , ψr ψr −1 , . . . , ψr ψr −1 · · · , ψ2 ).
2
n−r
Example 5.2. We continue with Example 5.1. Since M̄ = −(sv − 4uv + tu ), we compute
S HBezt , v ( M̄ , L ) = diag 1, 1, 1, 1, 1, u 6 (s − u )(s − 3u )(s − 2u )2 .
Since the inversion formula of Q = (0, 0, 1) is h(s, u ) = (s − u )2 (s − 3u )(s − 2u )2 , we have
S Q HBezt , v ( M̄ , L ) = diag 1, 1, 1, 1, 1, (s − u )(s − 3u )(s − 2u )2 .
Therefore, by Theorem 5.6, there are two double infinitely near singularities in the neighborhood of the point Q on P(s, u ),
whose inversion formulas are ψ2 = (s − u )(s − 3u )(s − 2u )2 . One can check that on the first blow-up curve of P(s, u ),
h1 = (s − u )(s − 3u ) corresponds to a double infinitely near singularity of Q , and h2 = (s − 2u )2 corresponds to the other
double infinitely near singularity of Q .
5.2.2. S (HBezt , v (h, L ))
From the previous theorems we know that S (HBezt , v ( M̄ , L )) provides the parameters for all the infinitely near singularities of the singular point Q not including Q itself. Next we shall show that S (HBezt , v (h, L )) provides all the parameters for
the singularity Q itself.
Lemma 5.7. (See Jia and Goldman (2009).) Let r be the multiplicity of the singularity Q . Then
I Q (h, L ) = r (r − 1).
Theorem 5.8.
S Q HBezt , v h(t , v ), L (s, u ; t , v )
= diag 1, . . . , 1, h(s, u ), . . . , h(s, u ) .
n−r
Proof. Denote by
r −1
B := HBezt , v (sv − tu )h(t , v ), (sv − tu ) L .
By Corollary A.1, we need only prove that
S Q ( B ) = diag 1, . . . , 1, h(s, u ), . . . , h(s, u ), 0 .
n−r
r −1
Since
(sv − tu ) L (s, u ; t , v ) = c (s, u )a(t , v )h(t , v ) − c (t , v )a(s, u )h(s, u ),
(20)
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X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
and the polynomials c (s, u )a(t , v )h(t , v ) and c (t , v )a(s, u )h(s, u ) are both of degree n in (t , v ), by the construction of the
hybrid Bezout matrix we have
B ≈ B1 + B2,
where
B 1 = c (s, u )HBezt , v h(t , v )(sv − tu ), a(t , v )h(t , v ) ,
B 2 = a(s, u )h(s, u )HBezt , v h(t , v )(sv − tu ), c (t , v ) ,
and the notation ≈ means that the first r + 1 rows of the matrices B and B 1 + B 2 are the same, while the entries in
the last n − r − 1 rows of the matrix B 1 + B 2 are equal to twice the corresponding entries in the matrix B. We can
examine S ( B 1 + B 2 ) instead of S ( B ) because we are interested only in the polynomials in the Smith normal form, so the
multiplication by a constant in some rows does not matter.
Denote by d(s, u ) the (n − r + 1)-st determinant factor, which is the GCD of all the (n − r + 1) × (n − r + 1) minors of the
matrix B 1 + B 2 . We claim that h(s, u )|d(s, u ).
Let H be an order n − r + 1 submatrix of the matrix B 1 + B 2 . Then H = H 1 + H 2 , where H 1 and H 2 are order n − r + 1
submatrices of B 1 and B 2 , so by Xu et al. (1993)
det( H ) = det( H 1 ) +
n−r
Γni det H 1 / H 2i + det( H 2 ),
(21)
i =1
where Γni det( H 1 / H 2i ) is the sum of the combination of determinants in which i rows of H 1 are replaced by the corresponding rows of the matrix H 2 . Note that
deg gcd h(t , v )(sv − tu ), a(t , v )h(t , v )
= deg h(t , v ) = r ,
hence
rank( B 1 ) = n − r .
Therefore,
det( H 1 ) = 0.
(22)
Also note that the matrix B 2 has a polynomial scaler factor a(s, u )h(s, u ), so
n−r
h(s, u )
Γni det H 1 / H 2i + det( H 2 ).
(23)
i =1
Then Eqs. (21), (22) and (23) together yield
h(s, u )| det( H ).
Hence
h(s, u )|d(s, u ).
(24)
Suppose
S ( B ) = diag( f n , f n−1 , . . . , f 2 , f 1 ).
Since
Q
f i ( s , u ) = 1,
i > r.
(25)
Eq. (24) and Eq. (25) imply that
Q
h(s, u )| f r ,
(26)
and therefore
Q
h(s, u )| f i ,
i = 1, . . . , r .
(27)
Note that det( B ) ≡ 0 since gcd((sv − tu )h(t , v ), (sv − tu ) L ) = 1; hence
f 1 ≡ 0.
(28)
Moreover by Lemma 5.7 the intersection multiplicity
n
i =1
Q
deg f i
= I Q h(t , v ), L (s, u ; t , v ) = deg(h) × (r − 1),
(29)
X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
so Eqs. (25), (27), (28) and (29) together yield
307
S Q ( B ) = diag 1, . . . , 1, h(s, u ), . . . , h(s, u ), 0 .
n−r
This completes the proof.
r −1
2
Example 5.3. We continue with Example 5.1 and Example 5.2. By direct computation
S HBezt , v (h, L ) = diag u 2 , u 2 , u 2 h(s, u ), u 2 h(s, u ), u 2 h(s, u ), u 2 h(s, u ) .
Hence
S Q HBezt , v (h, L ) = diag 1, 1, h(s, u ), h(s, u ), h(s, u ), h(s, u ) ,
which agrees with Theorem 5.8.
5.3. Joining S (HBezt , v ( M̄ , L )) and S (HBezt , v (h, L ))
Now we are ready to join S (HBezt , v ( M̄ , L )) and S (HBezt , v (h, L )) together to compute S (Bezt , v ( M , L )). Note that although
det HBezt , v ( M , L ) = det HBezt , v ( M̄ , L ) det HBezt , v (h, L ) ,
unfortunately generally
HBezt , v ( M , L ) = HBezt , v ( M̄ , L )HBezt , v (h, L ),
so
S HBezt , v ( M , L ) = S HBezt , v ( M̄ , L ) S HBezt , v (h, L ) .
Therefore readers are referred to Appendix B for some additional preparation before we combine S (HBezt , v ( M̄ , L )) and
S (HBezt , v (h, L )) to get S (HBezt , v ( M , L )). In particular, we shall make use of the following result.
Theorem 5.9. Let D be an integral domain and let f , g , l be polynomials in D[t ] with deg( f ) = m, deg( g ) = n and deg(l) m + n .
Denote by αk , βk , γk the k-th invariant factors of the matrices HBez( f , l), HBez( g , l) and HBez( f g , l). Then
αi1 αi2 · · · αik β j1 β j2 · · · β jk |lk00 γi1 + j1 −1 γi2 + j2 −2 · · · γik + jk −k ,
where l0 is the leading coefficient of the polynomial l(t ), and k0 is some non-negative integer.
Proof. See the proof of Theorem B.1 in Appendix B.
2
Theorem 5.10.
S
Q
r
Bezt , v ( M , L ) = diag 1, . . . , 1, hψr , hψr ψr −1 , . . . , h(s, u )
ψi ,
n−r
i =2
where h(s, u ) is the inversion formula for the point Q , and ψi (s, u ) is the product of the inversion formulas for all the order i infinitely
near singularities of Q .
Proof. Suppose
S HBezt , v ( M̄ , L ) = diag( ḡn , ḡn−1 , . . . , ḡ 2 ),
S HBezt , v (h, L ) = diag( g̃n , g̃n−1 , . . . , g̃ 2 ),
S Bezt , v ( M , L ) = diag( gn , gn−1 , . . . , g 2 ).
By Theorem 5.5 and Theorem 5.6 we know that
Q
ḡ i
=
r
1,
k =i
ψk (s, u ), for 2 i r ,
for r < i n.
(30)
Also by Theorem 5.8 we know that
Q
g̃ i =
h(s, u ),
1,
for 2 i r ,
for i > r .
(31)
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X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
Since the multiplicity of Q is r, we also have
Q
g i = 1,
for i > r .
(32)
By Theorem 5.9, for i = 1, . . . , r − 1 we have
ḡn ḡn−1 · · · ḡr +1 ḡr +1−i g̃n g̃n−1 · · · g̃r +1 g̃r |lk0 gn gn−1 · · · gr +1 gr +1−i ,
(33)
for some non-negative integer k, where l0 (s, u ) is the leading coefficient of the polynomial L (s, u ; t , v ) in (t , v ). But
l0 (s, u ) = lc (ah)c (s, u ) − lc (c )a(s, u )h(s, u ),
where lc means the leading coefficient of the polynomial. Thus
gcd l0 (s, u ), h(s, u ) = 1.
Therefore, when restricted to the point Q , we have
( ḡn ḡn−1 · · · ḡr +1 ḡr +1−i )Q ( g̃n g̃n−1 · · · g̃r +1 g̃r )Q |lk0 ( gn gn−1 · · · gr +1 gr +1−i )Q ,
(34)
which by (30), (31) and (32) yield
Q
ψr +1−i · · · ψr h| gr +1−i ,
i = 1, . . . , r − 1,
(35)
i.e.,
Q Q
Q
ḡ i g̃ i | g i ,
i = 2, . . . , r .
(36)
Since det(HBezt , v ( M̄ , L )) det(HBezt , v (h, L )) = det(HBezt , v ( M , L )),
n
Q Q
ḡ i g̃ i =
i =2
n
Q
gi .
(37)
i =2
By Eqs. (36) and (37) and expressions (30) and (31) we immediately get
Q
gi
= h (s, u )
1,
r
k =i
ψk (s, u ), 2 i r ,
r <i n
up to a constant multiple. The proof is now complete.
(38)
2
By Theorem 5.10 and Corollary 5.1, we have now totally laid bare the inversion formulas of those infinitely near singularities in the neighborhood of (on the singularity branch rooted at) the singularity Q:
Corollary 5.2.
S
Q
r
Bezt , v ( F , G ) = diag 1, . . . , 1, hψr , hψr ψr −1 , . . . , h(s, u )
ψi .
n−r
i =2
Now we take all the singularities on the curve P(s, u ) into consideration and we conclude that
dk (s, u ) =
Q
dk = hk (s, u )
Q
ψri (s, u ).
i k
Theorem 4.1 is now proved.
Remark 5.11. At the beginning of Section 5.2, we assume that all the singularities on the original curve C parameterized
by P(s, u ) can be totally resolved after one blow-up of the curve. Actually our proof works in general where an arbitrary number of k blow-ups are needed to totally resolve all the singularities on the curve C . The proof is by induction.
Suppose that a singularity Q on the curve C has infinitely near singularities in at most the k-th neighborhood. We start
from the blow-up curves in the k-th neighborhood, whose singularities have no infinitely near singularities. Suppose that
Pk−1 (s, u ) is a blow-up curve in the (k − 1)-st neighborhood with a singularity Qk−1 . We blow up Pk−1 (s, u ) at Qk−1 to
get Pk (s, u ), which is in the k-th neighborhood. Let F k and G k be two bivariate homogeneous polynomials constructed
from a μ-basis for Pk (s, u ). Then S Q (Bezt , v ( F k , G k )) contains the infinitely near singularities of Q in the k-th neighborhood.
The singularity information is then transferred from S Q (Bezt , v ( F k , G k )) to S Q (HBezt , v )( M̄ k−1 , L k−1 ); on the other hand,
the information on the singularity in the (k − 1)-st neighborhood, i.e., Qk−1 is provided by S (HBez(hk−1 , L k−1 )), where
hk−1 is the inversion formula for the point Qk−1 . Then S Q (HBezt , v )( M̄ k−1 , L k−1 ) and S (HBez(hk−1 , L k−1 )) are combined to
S Q (Bezt , v ( F k−1 , G k−1 )), which gives the infinitely near singularities in the k-th and (k − 1)-st neighborhood, and will next
be replaced by S Q (HBezt , v )( M̄ k−2 , L k−2 ). We can continue with this method proceeding by induction until we reach the
bottom of the singularity tree. (See Fig. 1.)
X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
309
Fig. 1. Proof by induction on the height of the singularity tree.
6. A discussion of the conjecture of Chen, Wang and Liu
Let C be a rational planar curve parameterized by P(t , v ) = (a(t , v ), b(t , v ), c (t , v )) with a μ-basis p(s, u ), q(s, u ), and let
L1 (s, u ) = (c (s, u ), 0, −a(s, u )), L2 (s, u ) = (0, c (s, u ), −b(s, u )) be a pair of obvious syzygies of the parameterization P(t , v ).
Then to compute all the singularities on C , we can compute the Smith normal form for any one of the following four Bezout
resultant matrices.
1.
2.
3.
4.
HBezs,u (p(s, u ) · P(t , v ), q(s, u ) · P(t , v ));
Bezt , v (p(s, u ) · P(t , v ), q(s, u ) · P(t , v ));
Bezs,u (L1 (s, u ) · P(t , v ), L2 (s, u ) · P(t , v ));
Bezt , v (L1 (s, u ) · P(t , v ), L2 (s, u ) · P(t , v )).
Note that only matrix 1 which is the focus of the conjecture of Chen et al. is a hybrid Bezout matrix, while the other
three matrices are Bezout matrices. Theoretically, it is easier to study Bezout matrices than hybrid Bezout matrices — in
fact, one of the main obstructions to proving the conjecture of Chen, Wang and Liu is to prove an analogue of Theorem 5.3
for hybrid Bezout matrices; which is why we study the Bezout matrix in 2 rather than the hybrid Bezout matrix in the
conjecture.
The Smith normal forms of these four Bezout matrices are:
1.
2.
3.
4.
diag(dn−μ (t , v ), dn−μ (t , v )dn−μ−1 (t , v ), . . . , dn−μ (t , v ) · · · d2 (t , v ), 0);
diag(1, . . . , 1, dn−μ (s, u ), dn−μ (s, u )dn−μ−1 (s, u ), . . . , dn−μ (s, u ) · · · d2 (s, u ), 0);
c (s, u ) diag(1, . . . , 1, dn−μ (s, u ), dn−μ (s, u )dn−μ−1 (s, u ), . . . , dn−μ (s, u ) · · · d2 (s, u ), 0);
c (s, u ) diag(1, . . . , 1, dn−μ (s, u ), dn−μ (s, u )dn−μ−1 (s, u ), . . . , dn−μ (s, u ) · · · d2 (s, u ), 0).
Geometrically, since by (Jia and Goldman, 2009) the parameter pair (s, u ; t , v ) is an intersection point of the two algebraic
curves F (s, u ; t , v ) = 0 and G (s, u ; t , v ) = 0 if and only if the parameter pair (t , v ; s, u ) is also an intersection point of the
two algebraic curves F (s, u ; t , v ) = 0 and G (s, u ; t , v ) = 0, HBezs,u ( F , G ) or Bezt , v ( F , G ) should give the same Smith normal
form except that the latter matrix has larger size. However, currently we lack a rigorous algebraic proof for the equivalence
of these two Smith normal forms. Also note that although HBezs,u ( F , G ) in the conjecture of Chen et al. is smaller than
Bezt , v ( F , G ) in our main theorem — (n − μ) × (n − μ) vs. n × n — the entries in Bezt , v ( F , G ) are lower degree polynomials
than the polynomial entries in HBezs,u ( F , G ) of Chen et al. — degree n − μ vs. degree n.
We can also compute all the singularities of the curve C from the Smith normal forms of Bezs,u (L1 (s, u ) · P(t , v ),
L2 (s, u ) · P(t , v )) or Bezt , v (L1 (s, u ) · P(t , v ), L2 (s, u ) · P(t , v )) . Note that both L1 (s, u ) · P(t , v ) = c (s, u )a(t , v ) − a(s, u )c (t , v )
and L2 (s, u ) · P(t , v ) = c (s, u )b(t , v ) − b(s, u )c (t , v ) are antisymmetric with respect to the parameters (s, u ) and (t , v ). Therefore the Smith normal forms 3 and 4 are the same up to a sign. Here, however, we need to remove the extra factor c (s, u )
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X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
or c (t , v ) from the Smith normal forms 3 or 4 to get the true singularities of the curve C because for any root (s∗ , u ∗ ) of
the polynomial c (s, u ), gcd(L1 (s∗ , u ∗ ) · P(t , v ), L2 (s∗ , u ∗ ) · P(t , v )) = c (t , v ).
Acknowledgements
We would like to thank Laurent Buse for his help in pointing us to the companion matrices used in Appendix B. This
work was partially supported by a National Key Basic Research Project of China (2011CB302400) and by a grant from NSFC
(60821002).
Appendix A. Smith normal forms
The definition and main properties of Smith normal forms are reviewed below and summarized in Definitions A.1–A.3
and Propositions A.1–A.5; for further details and proofs, see (Lancaster and Tismenetsky, 1985).
Definition A.1. A polynomial matrix P ∈ M m×m (R[t ]) is said to be invertible if det( P ) = c ∈ R and c = 0.
Proposition A.1. The following elementary row matrices in M m×m (R[t ]) are invertible:
1. E i j : interchange rows i and j of the identity matrix I m ;
2. E i (λ): multiply row i of I m by λ ∈ R, λ = 0;
3. E i j ( f ): add f times row j of I m to row i, f ∈ R[t ].
Similarly the elementary column matrices F i j , F i (λ), F i j ( f ) in M m×m (R[t ]) are invertible.
Proposition A.2. Each invertible polynomial matrix P ∈ M m×m (R[t ]) is a product of elementary matrices.
Proposition A.3. For every nonzero polynomial matrix A ∈ M m×m (R[t ]) with r = rank( A ), there exist invertible polynomial matrices
P , Q ∈ M m×m (R[t ]) such that
⎛
⎜
⎜
⎜
⎜
⎜
P AQ = ⎜
⎜
⎜
⎜
⎝
⎞
fm
⎟
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎠
f m −1
..
.
f m−r +1
0
..
.
(A.1)
0
where f m , . . . , f m−r +1 ∈ R[t ] are polynomials with f k | f k−1 for m − r + 1 < k m.
Definition A.2. The matrix in (A.1) is called the Smith normal form of the polynomial
matrix A, and is denoted by S ( A ).
m
The polynomials f m−k+1 are called the k-th invariant factors of A, and D k := i =m−k+1 f i are called the k-th determinant
factors of the matrix A, k = 1, . . . , m.
Proposition A.4. Smith normal forms of polynomial matrices are unique up to constant multiples of the entries.
Definition A.3. Let A , B ∈ M m×m (R[t ]). Then A is said to be equivalent to B over R[t ] if and only if there are invertible
matrices P , Q ∈ M m×m (R[t ]) such that P A Q = B.
Proposition A.5. Equivalent matrices A , B ∈ M m×m (R[t ]) have the same Smith normal forms.
The following property of Smith normal forms is crucial to the main step in our proof (Theorem 5.10) of the conjecture.
Proposition A.6. (See Thompson (1982).) Let A , B ∈ M m×m (R[t ]) be nonsingular matrices, and denote by αk , βk , γk the k-th invariant
factor of A , B and A B, respectively. Then
αi1 αi2 · · · αik β j1 β j2 · · · β jk |γi1 + j1 −1 γi2 + j2 −2 · · · γik + jk −k ,
where the integer subscripts satisfy
1 i 1 < i 2 < · · · < ik ,
1 j 1 < j 2 < · · · < jk ,
i k + j k k + m.
X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
311
The following lemma allows us to switch between the Smith normal forms of two related pairs of bivariate polynomials.
Lemma A.1. Let F (s, t ), G (s, t ) be two bivariate polynomials. Then
S Bezt (s − t ) F , (s − t )G
=
S (Bezt ( F , G ))
0
0
0
.
Proof. Suppose that max(degt ( F ), degt (G )) = m. Then the symmetric Bezout matrix Bezt ((s − t ) F , (s − t )G ) is the coefficient
matrix of the following Bezoutian,
1, t , . . . , t m Bezt (s − t ) F , (s − t )G 1, α , . . . , αm
T
(s − t ) F (s, t ) (s − t ) G (s, t ) (s − α ) F (s, α ) (s − α ) G (s, α ) =
t−α
F (s, t ) G (s, t ) F (s, α ) G (s, α ) = (s − t )(s − α )
t−α
T
= (s − t )(s − α ) 1, t , . . . , t m−1 Bezt ( F , G ) 1, α , . . . , αm−1 .
(A.2)
Let qi j be the entries in the matrix Bezt ((s − t ) F , (s − t )G ), i , j = 1, . . . , m + 1, and let p i j be the entries in the matrix
Bezt ( F , G ), i , j = 1, . . . , m. Also set p i ,m+1 = pm+1, j = 0 for any i , j = 1, . . . , m + 1. Then from (A.2), for i , j = 2, . . . , m + 1
q i j = s 2 p i j − s ( p i , j −1 + p i −1 , j ) + p i −1 , j −1 .
(A.3)
The right-hand side of Eq. (A.3) represents row and column operations on the matrix
0
0
0 Bezt ( F , G )
.
But by Proposition A.5 row and column operations do not alter the Smith normal form because by Proposition A.1 elementary matrices are invertible. Therefore,
S Bezt (s − t ) F , (s − t )G
=
S (Bezt ( F , G ))
0
0
0
.
2
The following definition from (Chen et al., 2008) allows for Smith normal forms of matrices whose entries are in the
homogeneous polynomial ring R[t , v ].
Definition A.4. (See Chen et al. (2008).) Let A (t , v ) ∈ M m×m (R[t , v ]). Suppose that
S A (t , 1) = diag d̄m (t ), d̄m (t )d̄m−1 (t ), . . . , d̄m (t ) · · · d̄1 (t ) ,
and that
S A (1, v ) = diag d̂m ( v ), d̂m ( v )d̂m−1 ( v ), . . . , d̂m ( v ) · · · d̂1 ( v ) .
Define
di (t , v ) LC M d̄i (t , v ), d̂i (t , v ) ,
(A.4)
where d̄i (t , v ) and d̂i (t , v ) are the homogenizations of the polynomials d̄i (t ) and d̂i ( v ). Then S ( A (t , v )) is defined by
diag dm (t , v ), dm (t , v )dm−1 (t , v ), . . . , dm (t , v ) · · · d1 (t , v ) .
We have the following property of Smith normal forms for homogeneous polynomial matrices similar to Lemma A.1
Corollary A.1. Let F (s, u ; t , v ), G (s, u ; t , v ) be polynomials in the homogeneous parameters s : u and t : v, and suppose that
max(degt , v ( F ), degt , v (G )) = m. Then
S Bezt , v (sv − tu ) F , (sv − tu )G =
S (Bezt , v ( F , G ))
0
0
0
.
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X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
Proof. Let
B 1 (s, u ) := Bezt , v (sv − tu ) F , (sv − tu )G ,
By Lemma A.1,
S B 1 ( s , 1) =
S ( B 2 (s, 1))
0
0
0
Therefore by Definition A.4
S B 1 (s, u ) =
S ( B 2 (s, u ))
0
0
0
B 2 (s, u ) := Bezt , v ( F , G ).
,
S B 1 (1, u ) =
.
2
S ( B 2 (1, u ))
0
0
0
.
Appendix B. Companion matrices
In this appendix we shall introduce companion matrices to factor hybrid Bezout matrices and prepare for the recombination of S (HBezt , v ( M̄ , L )) and S (HBezt , v (h, L )) in Section 5.3. We denote by D an integral domain of characteristic zero;
hence D can be the real number field R or the polynomial ring R[s].
Definition B.1. Let P (t ) be a degree n polynomial in D[t ]:
P (t ) = p 0 t n + p 1 t n−1 + · · · + pn ,
p 0 = 0.
The companion matrix of P (t ) is defined by:
⎛ 0
⎜ p0
⎜ 0
P := ⎜
⎜ .
⎝ .
.
0
..
.
···
···
···
..
.
0
· · · p0
0
0
p0
0
0
0
..
.
− pn ⎞
− p n −1 ⎟
− p n −2 ⎟
⎟.
.. ⎟
⎠
.
− p1
The following proposition states the well-known relationship between companion matrices and the resultant of two
univariate polynomials.
Proposition B.1. (See Diaz-Tocaa and Gonzalez-Vega (2002).) Let P , Q be two polynomials in D[t ] with m = deg( Q ) deg( P ) = n.
Then
Res( Q , P ) = pm
0 det Q ( P / p 0 ) ,
where Q ( P / p 0 ) refers to the evaluation of the polynomial Q at the matrix P / p 0 .
By Proposition B.1, Res( Q R , P ) = Res( Q , P ) Res( R , P ). Generally, however, HBez( Q R , P ) = HBez( Q , P )HBez( R , P ), but
the following proposition provides a resultant matrix which can be factored in this way.
Proposition B.2. (See Diaz-Tocaa and Gonzalez-Vega (2002).) Let P , Q , R be polynomials in D[t ] satisfying deg( Q ) + deg( R ) deg( P ). Let
H ( Q , P ) J n · Q tP / p 0 · J n ,
where
⎛
.
..
Jn = ⎝
1
1
(B.1)
⎞
⎠
,
n×n
and Q (tP / p 0 ) refers to the evaluation of the polynomial Q at the transpose of the matrix P / p 0 . Then
H ( Q R , P ) = H ( Q , P ) · H ( R , P ).
Proof.
H ( Q , P ) · H ( R , P ) = J n · Q tP / p 0 · J n · J n · R tP / p 0 · J n
= J n · Q tP / p 0 · R tP / p 0 · J n
= J n · Q R tP / p 0 · J n
= H ( Q R , P ).
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X. Jia, R. Goldman / Computer Aided Geometric Design 29 (2012) 296–314
313
The following factorization shows the relationship between hybrid Bezout resultant matrices and the companion resultant
matrices defined in Proposition B.2.
Proposition B.3. (See Diaz-Tocaa and Gonzalez-Vega (2002).) Let P , Q be two polynomials in D[t ] with m = deg( Q ) deg( P ) = n.
Let H ( Q , P ) be the matrix defined in Eq. (B.1). Then
HBez( Q , P ) = T m · H ( Q , P ),
where
⎛
p0
⎜
⎜
⎜
⎜
Tm = ⎜
⎜ 0
⎜ .
⎝ .
.
0
⎞
· · · p m −1 0 · · · 0
..
..
.. ⎟
..
.
.
.
.⎟
⎟
p0
0
0⎟
.
⎟
⎟
···
0
1
⎟
..
..
⎠
.
.
···
0
1 n×n
Theorem B.1. Let f , g , l be polynomials in D[t ] with deg( f ) = m, deg( g ) = n and deg(l) m + n . Denote by
invariant factors of matrices HBez( f , l), HBez( g , l) and HBez( f g , l). Then
αk , βk , γk the k-th
αi1 αi2 · · · αik β j1 β j2 · · · β jk |lk00 γi1 + j1 −1 γi2 + j2 −2 · · · γik + jk −k , ,
where l0 is the leading coefficient of the polynomial l(t ), and k0 is some non-negative integer.
Proof. By Proposition B.3,
HBez( f , l) = T m · H ( f , l),
HBez( g , l) = T n · H ( g , l),
and
HBez( f g , l) = T m+n · H ( f g , l).
Since by Proposition B.2,
H ( f g , l) = H ( f , l) · H ( g , l),
we have
−1
T m+n · T m
· HBez( f , l) · T n−1 · HBez( g , l) = HBez( f g , l).
(B.2)
1
−1
−1 have denominators lm and ln , so
Note that now this equality holds over D[t , l−
0 ]. The entries in the matrices T m and T n
0
0
m+n
multiplying both sides of Eq. (B.2) by l0
to clear these denominators yields
+n
−1
· HBez( f , l) · ln0 T n−1 · HBez( g , l) = lm
T m+n · lm
HBez( f g , l).
0 Tm
0
(B.3)
Now the left-hand side of Eq. (B.3) is a product of polynomial matrices. From Proposition A.6 we get directly that for some
integer k0
αi1 αi2 · · · αik β j1 β j2 · · · β jk |lk00 γi1 + j1 −1 γi2 + j2 −2 · · · γik + jk −k .
2
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