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The maximum number and its distribution of singular points for parametric piecewise algebraic curves
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The maximum number and its distribution of singular points for parametric piecewise algebraic curves✩ Jinming Wu a,∗ , Chungang Zhu b a
Department of Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, China
b
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
article
info
Article history: Received 5 September 2016 Received in revised form 30 November 2016 Keywords: Parametric piecewise algebraic curve Singular points Semi-algebraic set
abstract The piecewise algebraic curve, as the zero set of a bivariate spline function, is a generalization of the classical algebraic curve. Based on the previous method presented by Lai et al. (2009), we show that computing singular points of parametric piecewise algebraic curves amounts to solving parametric piecewise polynomial systems. In this article, we give a method to compute the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve. This method also produces necessary and sufficient conditions of its parameters must be satisfied. An illustrated example shows that our method is flexible. © 2017 Elsevier B.V. All rights reserved.
1. Introduction We denote by R[x, y] the polynomial ring in variables x and y over real number field R and Pk [x, y] the set of bivariate polynomials in R[x, y] with real coefficients and total degree 6 k. A polynomial f (x, y) ∈ R[x, y] is called an irreducible polynomial if the polynomial f cannot be divided by any other polynomial except a constant or itself. Algebraic curve
Z(f ) := {(x, y) ∈ R2 | f (x, y) = 0, f (x, y) ∈ Pk [x, y]} is called an irreducible algebraic curve if f (x, y) is an irreducible polynomial. Let Ω be a connected and bounded region in R2 . Using a finite number of irreducible algebraic curves in R2 , we divide the region Ω into several simply connected regions, which are called the partition cells. Denote by ∆ the partition of the T region Ω , let δ1 , δ2 , . . . , δT be a given ordering of the cells in ∆, and let Ω = i=1 δi . Also, we write ∆ = {δ1 , δ2 , . . . , δT } and the interior of each cell δi can be described as
δi = {(x, y) ∈ R2 | g1[i] (x, y) > 0, g2[i] (x, y) > 0, . . . , gN[ii] (x, y) > 0}, where g1 (x, y), . . . , gNi (x, y) ∈ R[x, y] are irreducible polynomials. Denoted by C µ (∆) the set of C µ functions s(x, y) on Ω . Assume the restriction s(x, y)|δi is a polynomial which belongs to R[x, y] and let [i]
[i]
Pk (∆) = {s(x, y)|s(x, y)|δi ∈ Pk [x, y], i = 1, 2, . . . , T } be the set of piecewise polynomials defined on ∆ with total degree 6 k. ✩ This work was supported by the National Natural Science Foundation of China (Nos.11401526, 11101366, 11671068 and 11271328), the Zhejiang Provincial Natural Science Foundation of China (No.LY14A010001). ∗ Corresponding author. E-mail address: [email protected] (J. Wu).
http://dx.doi.org/10.1016/j.cam.2017.01.023 0377-0427/© 2017 Elsevier B.V. All rights reserved.
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The bivariate spline space with smoothness µ and degree k over Ω with respect to ∆ is defined as follows: µ Sk (∆) = {s(x, y) | s(x, y) ∈ C µ (∆) ∩ Pk (∆)}.
The zero set µ
Z(s) := {(x, y) ∈ Ω | s(x, y) = 0, s(x, y) ∈ Sk (∆)} is called a piecewise algebraic curve. Obviously, the piecewise algebraic curve is a generalization of the classic algebraic curve. However, it is difficult to study piecewise algebraic curve not only because of the complexity of the partition but also because of the possibility of {(x, y) | s(x, y) = 0} ∩ δi = ∅. Piecewise algebraic curve is originally introduced by Wang in the study of multivariate spline interpolation. He pointed out that the given interpolation knots are properly posed if and only if they are not lie in a non-zero piecewise algebraic curve [1]. In recent years, Wang and his researcher group have done a lot of significant work on piecewise algebraic curves (see [1–19]). For example, the Bezout theorem [3,4,13,16], Nöther-type theorem [7,9,10], Cayley–Bacharach theorem [8,14] and Riemann–Roch type theorems [9] of piecewise algebraic curves were established. Besides, piecewise algebraic curve also relates to the remarkable Four-Color conjecture [4]. Moreover, Lai et al. [11,12] discussed the real zeros of the zerodimensional parametric piecewise algebraic variety and produced the corresponding necessary and sufficient conditions of coefficients. Also, the Viro method for construction of Bernstein–Bézier algebraic hypersurface piece and piecewise algebraic curve are discussed [17,18]. In sum, the piecewise algebraic curve is a new and important topic in Computer Aided Geometry Design and Computational Geometry and has many applications in various fields. It is necessary and significant to study the related problems on piecewise algebraic curves. The singular points of a given real algebraic curve Z(f ) (SPT (f ) for short) are those points on the curve where both partial derivatives vanish, i.e.,
SPT (f ) = (x, y) | f (x, y) = fx (x, y) = fy (x, y) = 0 . The singular points and its classification for a given algebraic curve is a basic and important topic in algebraic geometry [20,21]. The detection of singular points helps to determine the geometrical shape and topology of a real algebraic curve. In 1990, Sakkalisa et al. [21] described an algorithm for determining whether an irreducible algebraic curve f (x, y) = 0 with total degree >3 is singular, and if so, isolating its singular points and computing their multiplicities. For a given plane rational curve in parametric form, there are several efficient methods to detect and compute the singular points [22]. Hence, it is of theoretical and practical significance to study singular points of a given parametric piecewise algebraic curve, especially the maximum number and its distribution of its singular points. Certainly, it helps us to determine the topological structure of piecewise algebraic curves better. Now, we firstly give the definition of parametric piecewise algebraic curves. Let V = (v1 , v2 , . . . , vr ) be the parameters and denoted by R[V ][x, y] the set of all polynomials in variables x and y with coefficients in R[V ]. Obviously, p ∈ R[V ][x, y] can be viewed as p = p(V , x, y). µ
Definition 1.1. Let s = s(V , x, y) ∈ Sk (∆) and s|δi ∈ R[V ][x, y] for each cell δi in ∆, where V = (v1 , v2 , . . . , vr ) and (x, y) are viewed as parameters and variables, respectively. Then Z(s) is called a parametric piecewise algebraic curve. In other words, piecewise algebraic curve is ‘‘parametric’’ means it contains symbolic coefficients and also allows to have some certain constant coefficients. From the theory of smoothing cofactor of bivariate spline [1], we can easily have µ
Lemma 1.1. Let s = s(V , x, y) ∈ Sk (∆) and s|δi ∈ R[V ][x, y], i = 1, 2, . . . , T . If the point p ∈ Z(s) and p ∈ δi ∩ δj , i, j ∈ {1, 2, . . . , T }, then p is the singular (regular) point of algebraic curve Z(s|δi ) = {(x, y) ∈ δi | s|δi (V , x, y) = 0} if and only if p is the singular (regular) point of algebraic curve Z(s|δj ) = {(x, y) ∈ δj | s|δj (V , x, y) = 0}. Proof. By using smoothing cofactor of bivariate spline [1], we have s|δi (V , x, y) − s|δj (V , x, y) = lij (x, y)µ+1 qij (V , x, y), where Z(lij ) is the common algebraic curve of δi ∩ δj , and qij (V , x, y) ∈ Pd [x, y], d = k − (µ + 1) deg(lij ). From simple computation, the result is proved. Therefore, singular points of a given parametric piecewise algebraic curve are defined as follows: µ
Definition 1.2. Let s = s(V , x, y) ∈ Sk (∆). If p ∈ Z(s) and there exists i ∈ {1, 2, . . . , T } such that p ∈ δi (the closure of δi ) and p is the singular point of algebraic curve Z(s|δi ), then p is called a singular point of parametric piecewise algebraic curve Z(s). Otherwise, p is called a regular point of Z(s).
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The singular points of a given parametric piecewise algebraic curve Z(s) is equivalent to solving the following parametric piecewise polynomial system
s(V , x, y) = 0, ∂ s(V , x, y) = 0, sx (V , x, y) = Z(s, ∇ s) : ∂x sy (V , x, y) = ∂ s(V , x, y) = 0 ∂y
(1)
µ
where s ∈ Sk (∆) and s|δi ∈ R[V ][x, y], i = 1, 2, . . . , T . µ
µ−1
Since s ∈ Sk (∆), we have sx , sy ∈ Sk−1 (∆) and Z(s, ∇ s) is indeed a piecewise algebraic variety contained in Z(s). Obviously, the singular points of Z(s) is equal to the real zeros of Z(s, ∇ s). We assume that all the singular points of Z(s) is torsion-free if it lies in the interior of a cell δi in ∆ if it is not specified. This assumption is convenient to count the number of singular points of a given parametric piecewise algebraic curve. In order to guarantee the number of singular points of a given piecewise algebraic curve is finite, we introduce the following concept. Definition 1.3. A bivariate spline s is said to be square-free if and only if for each i ∈ {1, 2, . . . , T } at least one of the following conditions holds:
• s|δi = s |δi , where
s |δi = s|δi / gcd s|δi ,
∂ s|δi ∂ s|δi , ∂x ∂y
.
• Z(fki ) ∩ δi = ∅, where, fki are the factors of s|δi with multiplicity greater than one. In our discussion, s is assumed to be square-free for almost all the parametric values V . That is to say, Z(s, ∇ s) is assumed to be zero-dimensional if and only if for almost all the parametric values V and for each i ∈ {1, 2, . . . , T } the set
Z (s, ∇ s) : [i]
s|δi (V , x, y) = 0, sx |δi (V , x, y) = 0, sy |δi (V , x, y) = 0
(2)
is generally zero-dimensional. It means that Z(s) has a finite number of singular points. The purpose of our paper is to present an algorithm to give the maximum number of singular points of a given parametric piecewise algebraic curve Z(s), and to get its distribution of the number of singular points in every cell, which is primarily based on our early proposed method in paper [11]. It can be viewed as a good and successful application in studying the related problems of piecewise algebraic curves. The rest of this paper is organized as follows. In Section 2, we give the basic framework of computing singular points of a given parametric piecewise algebraic curve. In Section 3, we recall several basic definitions on triangular parametric semialgebraic set. In Section 4, we present an algorithm to determine the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve. Meanwhile, the algorithm also gives the necessary and sufficient conditions on parameters must be satisfied. In Section 5, an illustrated example is provided to show that the algorithm is flexible. Finally, we conclude our paper in Section 6. 2. Singular points of parametric piecewise algebraic curve In this section, we firstly show that the coefficients of a given C µ parametric piecewise polynomial satisfy a system of linear equations. Theorem 2.1 ([11,12]). A piecewise polynomial function s is in C µ (∆) if and only if in for any adjacent pair δi , δj in ∆, which µ+1
intersects along detij = δi ∩ δj contained in the algebraic curve Z(lij ), all coefficients of the polynomial rem(s|δi − s|δj , lij , xij ) are null, where rem is either the usual Euclidean remainder or the pseudo-remainder, and lij is an irreducible polynomial with degree >1 with respect to some variables xij ∈ {x, y} appearing in lij . These coefficients are homogeneous linear functions of the coefficients of s|δi and s|δj , and these nullity can be written into a homogeneous linear system. µ
For a given s ∈ Sk (∆), s|δi can be written in the form s|δi =
λ1 +λ2 6k
where v
[i]
vλ[i1] λ2 xλ1 yλ2 = s|δi (v [i] , x, y),
is a row vector whose components are formed by coefficients vλ1 λ2 of s|δi in a fixed order. [i]
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Let Q = (v [1] , v [2] , . . . , v [T ] )′ . By Theorem 2.1, the system of homogeneous linear equations with respect to variables Q corresponding to C µ continuity condition of parametric piecewise polynomial s is BQ = 0, µ+1
where the elements in the matrix B are determined by the coefficients of all polynomials rem(s|δi − s|δj , lij Theorem 2.1. From Theorem 2.1, we have the following result
, xij ) in
µ
Theorem 2.2. Let s ∈ Sk (∆) and s|δi ∈ R[V ][x, y], i = 1, . . . , T . Then the following system of parametric piecewise polynomials has the same real zeros as Z(s, ∇ s) for any given values of parameters V
s(V , x, y) = 0, sx (V , x, y) = 0, ZV (s, ∇ s) : sy (V , x, y) = 0, BQ = 0,
(3)
where V = (v1 , v2 , . . . , vr ) is viewed as the parameters and B, Q are defined as above. It tells us that computing singular points of parametric piecewise algebraic curves amounts to solving parametric piecewise polynomial system ZV (s, ∇ s). Let ΘBQ = {(u1 , u2 , . . . , ur )| BQ = 0} be the set of solutions of the system BQ and dim(ΘBQ ) = d. Without loss of generality, U = (u1 , u2 , . . . , ud ) is viewed as free parameters and (ud+1 , ud+2 , . . . , ur ) can be uniquely determined by U. Define a mapping Π : Rr −→ Rd by
Π (u1 , . . . , ud , ud+1 , . . . , ur ) := (u1 , u2 , . . . , ud ). The method for computing Π (ΘBQ ) can be found in any textbook on linear algebra. Using Gaussian elimination method to eliminate parameters (ud+1 , ud+2 , . . . , ur ), the ZV (s, ∇ s) can be transformed into a system of parametric piecewise polynomials
s(U , x, y) = 0, sx (U , x, y) = 0, ZU (s, ∇ s) : sy (U , x, y) = 0,
(4)
where U = (u1 , u2 , . . . , ud ) ∈ Rd ∩ Π (ΘBQ ) are viewed as actual parameters. We have the following conclusion immediately from Theorem 2.2 and the above discussion. µ
Theorem 2.3. Let s ∈ Sk (∆) and s|δi ∈ R[V ][x, y], i = 1, . . . , T . We define:
ZU (s, ∇ s) : [i]
s|δi (U , x, y) = 0, sx |δi (U , x, y) = 0, sy |δi (U , x, y) = 0,
(5)
g1 (x, y) > 0, . . . , gNi (x, y) > 0. [i]
[i]
Let NSP (s) be the number of singular points of a given parametric piecewise algebraic curve Z(s) and Rzero(·) be the set of real zeros of a given system of parametric (piecewise) polynomials. Then for any given U ∈ Rd ∩ Π (ΘBQ ), we have NSP (s) = Rzero(Z(s, ∇ s)), Rzero(Z(s, ∇ s)) = Rzero(ZV (s, ∇ s)) = Rzero(ZU (s, ∇ s)) =
T
Rzero(ZU (s, ∇ s)) [i]
i=1
and Rzero(Z[i] (s, ∇ s)) = Rzero(ZU (s, ∇ s)), [i]
i = 1, . . . , T .
Therefore, computing singular points of a given parametric piecewise algebraic curve amounts to solving these [1] [T ] parametric semi-algebraic systems ZU (s, ∇ s), . . . , ZU (s, ∇ s) with U = (u1 , . . . , ud ) as actual parameters. It is well known that we can use the Ritt–Wu method [23], Gröbner basis method or subresultant method [24] to [i] transform the parametric ZU (s, ∇ s) into one or more systems in the triangular form (TSA for short). Then we check whether it is a regular TSA and, if not, transform it into regular TSA by WR algorithm [25] or Wang’s algorithm [26]. Let the regular [i] TSA for ZU (s, ∇ s) be
[i] S (U ) = 0 0[,ik] ϕ ( U , x) = 0 1,k [i] TSAk (s, ∇ s) : [i] ϕ 2,k (U , x, y) = 0, [i] [i] g1 (x, y) > 0, . . . , gNi (x, y) > 0 for some k = 1, 2, . . . , βi , i = 1, 2, . . . , T .
(6)
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We found that we need not care about S0,k (U ) = 0 when we solve the real zeros of TSAk (s, ∇ s). In fact, S0,k (U ) = 0 is one of the conditions we wanted for determining the maximum number and its distribution of the number of singular [i] [i] points for a given parametric piecewise algebraic curve Z(s). Denoted by TSAk the semi-algebraic system TSAk (s, ∇ s) after [i]
[i]
[i]
deleting the condition S0,k (U ) = 0, i.e., [i]
[i] ϕ1,k (U , x) = 0 [i] [i] TSAk : ϕ2,k (U , x, y) = 0, [i] [i] g1 (x, y) > 0, . . . , gNi (x, y) > 0.
(7)
Thus, TSAk (s, ∇ s) = TSAk [i]
[i]
[i] {S0,k (U ) = 0}.
3. Triangular semi-algebraic set [i]
In order to solve TSAk , we ought to introduce some basic notations on triangular semi-algebraic set. A semi-algebraic set is a system consisting of polynomials equations, polynomial inequations and polynomial inequalities, where all polynomials are of integer coefficients. In this article, it requires to study the real zeros of bivariate polynomials, we slightly simplify the notations compared to our previous paper [11]. Given the following special triangular parametric semi-algebraic set f1 (u, x) = 0, f2 (u, x, y) = 0, g1 (x, y) > 0, g2 (x, y) > 0, . . . , gt (x, y) > 0,
TSA :
(8)
where u = (u1 , u2 , . . . , ud ) and (x, y) are viewed as parameters and variables, respectively; f1 , f2 ∈ Z(u)[x, y], gj ∈ Z[x, y], 1 6 j 6 t. Definition 3.1. Given a polynomial g (x), let res(g , gx′ , x) be the Sylvester resultant of g and gx′ with respect to x. We call it the discriminant of g with respect to x and denote it by Discrim(g , x). Definition 3.2. Given a triangular set {f1 , f2 }, I1 denotes the leading coefficient of f1 in x, and I2 denotes the leading coefficient of f2 in y. Triangular set {f1 , f2 } is called a normal ascending chain if I1 ̸= 0,
res(I2 , f1 ) ̸= 0,
where res(I2 , f1 ) denotes the resultant of I2 and f1 with respect to x. Definition 3.3. Given a parametric normal ascending chain TSA. For every fi , let R1 = Discrim(f1 , x), Rgi = res(gi , f2 , f1 ),
R2 = res(Discrim(f2 , y), f1 ), i = 1, . . . , t .
We define BPs := {Ri |1 6 i 6 2} ∪ {Rgi |1 6 i 6 t } and call it the set of critical polynomials of system TSA. Definition 3.4. Given a parametric TSA as (8) and {f1 , f2 } is a normal ascending chain. Then TSA is regular if 0 ̸∈ BPs . Definition 3.5. Given two polynomials f (x) and g (x), let Discr(f , g ) be the generalized discrimination matrix of f (x) with respect to g (x), and D0 = 1. Denote by
{D1 (f , g ), D2 (f , g ), . . . , Dn (f , g )} the even order principal minors of Discr(f , g ). We call
[D0 , D1 (f , g ), D2 (f , g ), . . . , Dn (f , g )] the generalized discriminant sequence of f (x) with respect to g (x), and denote it by GDL(f , g ).
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Let ps = {pi |1 6 i 6 w} be a nonempty and finite set of polynomials. We define mset (ps) = {1} ∪ {pi1 pi2 · · · pik |1 6 k 6 w, 1 6 i1 < i2 < · · · < ik 6 w}. For example, if ps = {p1 , p2 , p3 }, then mset (ps) = {1, p1 , p2 , p3 , p1 p2 , p1 p3 , p2 p3 , p1 p2 p3 }. Definition 3.6. Given a parametric TSA (8), we define P3 = {g1 , g2 , . . . , gt },
U2 =
GDL(f2 , q),
q∈mset (P3 )
P2 = {h(u, x1 )| h ∈ U2 },
U1 =
GDL(f1 , q),
q∈mset (P2 )
TSAP1 (g1 , g2 , . . . , gt ) = {h(u)| h ∈ U1 }, where Ui means the set consisting of all the polynomials in each GDL(fi , q) where q belongs to mset (Pi+1 ). Similarly, we can define TSAP1 (g1 , g2 , . . . , gi ). Theorem 3.1 ([27]). The necessary and sufficient condition for system TSA (8) to have a given number of distinct real solutions can be expressed in terms of the signs of the polynomials in TSAP1 (g1 , g2 , . . . , gt ). 4. Maximum number and its distribution of singular points By using WR algorithm [25] or Wang’s algorithm [26], for every ZU (s, ∇ s), it can be decomposed into regular triangular [i]
[i] TSA1 (s, ∇ s), . . . , TSAβi (s, ∇ s), [i]
systems We define
BPs (ZU (s, ∇ s)) :=
where TSAk (s, ∇ s) = TSAk
[i]
[i]
[i] {S0,k (U ) = 0}.
(BPs )k[i] ,
(9)
16i6T 16k6βi
where (BPs )k is the set of critical polynomials of semi-algebraic system TSAk . Set [i]
[i]
P1 (ZU (s, ∇ s)) :=
TSAk P1 (g1 , . . . , gNi ), [i]
16i6T 16k6βi
[i]
[i]
(10)
where TSAk P1 (g1 , . . . , gj ) is defined as Definition 3.6. Let [i]
[i]
Q (ZU (s, ∇ s)) :=
[i]
16i6T 16k6βi
{S0[i,]k (U ) = 0}.
(11)
From Theorems 2.3, 3.1 and using similar techniques in paper [11], we have the following important conclusions. Theorem 4.1. The necessary and sufficient condition for a given parametric piecewise algebraic curve Z(s) to have a given number of singular points in every cell in ∆ can be expressed in terms of the signs of the polynomials in P1 (ZU (s, ∇ s)) and the elements in Q (ZU (s, ∇ s)). Now, we present the algorithm to deal with the maximum number and its distribution of distinct singular points of a given piecewise algebraic curve Z(s). Algorithm 1 (Max-Distribution-SingularPoints-PAC). Input: A given parametric piecewise algebraic curve Z(s), or a given parametric piecewise algebraic variety Z(s, ∇ s). Output: MaxNSP denotes by the maximum number of distinct singular points of Z(s); DistSP denotes by its distribution of distinct singular points in cells δ1 , . . . , δT when the number of singular points reach MaxNSP; The necessary and sufficient condition Φ (NS[1] , . . . , NS[T ] ) on the parameters for the Z(s) to have exactly (NS[1] , . . . , NS[T ] ) distinct singular points in cells δ1 , . . . , δT . Step 1. Obtain ZV (s, ∇ s) with Theorem 2.1, and transform it into ZU (s, ∇ s) by using elimination method.
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Step 2. For every i ∈ {1, 2, . . . , T }, perform WR algorithm or Wang’s algorithm on every ZU (s, ∇ s) and obtain regular [i]
triangular systems
[i] TSA1 (s, ∇ s), . . . , TSAβi (s, ∇ s), [i]
[i] where TSAk (s, ∇ s) = TSAk {S0,k (U ) = 0}, k = 1, . . . , βi . [i]
[i]
Step 3. Compute (BPs )k (the set of critical polynomials of system TSAk ) for each i ∈ {1, . . . , T } and k ∈ {1, . . . , βi } and obtain BPs (ZU (s, ∇ s)). Step 4. Set Polyset := BPs (ZU (s, ∇ s)) and l := 1. Step 5. Compute a PolySet-invariant cylindrical algebraic decomposition with the algorithm of PCAD [28]. Denote by D′ the set of cad-cell δ in D with δ ∩ Π (ΘBQ ) ̸= 0 and its cylindrical algebraic sample S. Step 6. For each cad-cell δ in D′ and its sample points sδ ∈ S, compute the signs of polynomials in PolySet on this cad-cell after substituting sδ into them. The signs of polynomials in PolySet on this cad-cell δ form a first order formula, denoted by Φδ . If Q (Z(s, ∇ s)) ∩ Φδ ̸= ∅, then D′ := D′ ∪ {δ}. Otherwise, D′ := D′ . [i]
[i]
Step 7. For each cad-cell δ in D′ and its sample point sδ ∈ S, substitute sδ into TSAk and compute the number of distinct [i]
real solutions of every system TSAk (sδ ) (say mk (δ)). If all mk (δ) are computed, then set [i]
MaxNSP := max N |N =
[i]
βi T
[i]
mk (δ), δ ∈ D [i]
′
,
i=1 k=1
DistSP :=
(NS , . . . , NS )|NS = [1]
[T ]
[i]
βi
mk (δ), δ ∈ D [i]
′
.
k=1
Step 8. For each (NS[1] , . . . , NS[T ] ) ∈ DistSP, set
Θ (NS , . . . , NS ) := [1]
Φδ |
[T ]
βi
mk (δ) = NS , i = 1, . . . , m, δ ∈ D [i]
[i]
′
.
k=1
Step 9. Decide whether all the Θ (NS[1] , . . . , NS[T ] ) can form a necessary and sufficient conditions similar to our previous method in [11]. If so, go to Step 10. Otherwise, we set Polyset := Polyset ∪ P1,l (ZU (s, ∇ s)), where P1,l (ZU (s, ∇ s)) :=
TSAk P1 (g1 , . . . , gl ), [i]
[i]
[i]
1 6 l 6 max Ni ,
16i6T 16k6βi
16i6T
let l := l + 1 and go to Step 4. Step 10. Output MaxNSP and DistSP. If we let Θ (NS[1] , . . . , NS[T ] ) = {Φδ1 , . . . , Φδt } for a given (NS[1] , . . . , NS[T ] ) ∈ DistSP, then
Φ NS[1] , . . . , NS[T ] = Φδ1 ··· , Φδt Q ′ (Z(s, ∇ s)), where Q ′ (Z(s, ∇ s)) =
k,i S0,k (U ) = 0 | mk (δ) ̸= 0 .
[i]
[i]
We found that the termination of this algorithm is guaranteed by Theorem 4.1. In fact, we use the tofind command in DISCOVERER package developed by B.C. Xia [29] to help to determine whether Q (Z(s, ∇ s))∩ Φδ ̸= ∅ or not. It is pointed out that the equality constraints are a significant improvement compared to our previous method [11]. We find the results are interesting and incredible which can be easily seen from the later illustrated example. It is viewed as a successful application and extension. 5. An illustrated example A simple example is provided to illustrate the proposed algorithm is flexible. Let ∆ = {δ1 , δ2 } be a partition of a given triangle V1 V2 V3 in R2 , where δ1 = [V1 V2 V0 ], δ2 = [V2 V3 V0 ], V1 = (1, 0), V2 = (0, 1), V3 = (−1, 0), and V0 = (0, 0). It is obvious that the partition straight line is L : x = 0. Let s(x, y) ∈ S31 (∆), where s(x, y) =
s(x, y)|δ1 = x3 + ax2 + by2 + kxy + cy + d s(x, y)|δ2 = ex3 + hxy2 + fx2 + my2 + ny + l.
From Theorem 2.1, these coefficients of s(x, y) satisfy the following conditions m = b,
n = c,
l = d,
k = h = 0.
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Fig. 1. Singular points of a piecewise algebraic curve Z(s∗ ).
Substituting these conditions into s(x, y), we have s(x, y) =
s(x, y)|δ1 = x3 + ax2 + by2 + cy + d s(x, y)|δ2 = ex3 + fx2 + by2 + cy + d.
If we perform the algorithm Max-Distribution-SingularPoints-PAC, then we have the following facts (1) The maximum number of the singular points for the given piecewise algebraic curve Z(s) is two, i.e., MaxNSP = 2. (2) Its distribution in δ1 and δ2 is {1, 1} when the number of singular points reaches maximum, i.e., DistSP = {1, 1}. (3) The piecewise algebraic curve Z(s) has one singular point in every cell if and only if Case A : [a > 0 ∧ b > 0 ∧ c < 0 ∧ e < 0 ∧ f > 0 ∧ ∆1 < 0 ∧ ∆2 < 0 ∧ S1 = 0 ∧ S2 = 0] or Case B : [a > 0 ∧ b < 0 ∧ c > 0 ∧ e < 0 ∧ f > 0 ∧ ∆1 > 0 ∧ ∆2 > 0 ∧ S1 = 0 ∧ S2 = 0] where
∆1 = −6b − 3c + 4ba, ∆2 = 6eb + 3ec + 4bf , S1 = 108bd + 16ba3 − 27c 2 , S2 = (108e2 d + 16f 3 )b − 27e2 c 2 . Take a =
1 2
, b = 1; c = −1, d =
25 108
, e = −8, f = 2 for example. We find that Case A is satisfied and s∗ (x, y) becomes
1 25 s∗1 (x, y) = s∗ (x, y)|δ1 = + x2 + x3 − y + y2 , 108
2
s∗ (x, y) = s∗ (x, y)|δ = 25 + 2x2 − 8x3 − y + y2 . 2 2 108
The graph of Z(s∗ ) is shown in Fig. 1 and we found that the total number of singular points of Z(s∗ ) is two and its distribution in every cell is one. This experiment presents an interesting result with respect to singular points of parametric piecewise algebraic curves till now. It indicates that our proposed method is flexible. 6. Conclusion It is of theoretical and practical significance to study the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve. In this paper, we extend our previous method [11] to solve this important issue. Numerical experiments show that our proposed method is flexible and our results are considerable interesting. It is pointed out that it is interesting to study the classification of singular points for parametric piecewise algebraic curves in our future work. In fact, we have successfully discussed the real root classification of parametric spline functions [30]. Thus, we have reasons to believe that this problem is solvable. Acknowledgments We appreciate the reviewers and editors for their careful reading, valuable comments, timely review and reply.
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Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
The maximum number and its distribution of singular points for parametric piecewise algebraic curves✩ Jinming Wu a,∗ , Chungang Zhu b a
Department of Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, China
b
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
article
info
Article history: Received 5 September 2016 Received in revised form 30 November 2016 Keywords: Parametric piecewise algebraic curve Singular points Semi-algebraic set
abstract The piecewise algebraic curve, as the zero set of a bivariate spline function, is a generalization of the classical algebraic curve. Based on the previous method presented by Lai et al. (2009), we show that computing singular points of parametric piecewise algebraic curves amounts to solving parametric piecewise polynomial systems. In this article, we give a method to compute the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve. This method also produces necessary and sufficient conditions of its parameters must be satisfied. An illustrated example shows that our method is flexible. © 2017 Elsevier B.V. All rights reserved.
1. Introduction We denote by R[x, y] the polynomial ring in variables x and y over real number field R and Pk [x, y] the set of bivariate polynomials in R[x, y] with real coefficients and total degree 6 k. A polynomial f (x, y) ∈ R[x, y] is called an irreducible polynomial if the polynomial f cannot be divided by any other polynomial except a constant or itself. Algebraic curve
Z(f ) := {(x, y) ∈ R2 | f (x, y) = 0, f (x, y) ∈ Pk [x, y]} is called an irreducible algebraic curve if f (x, y) is an irreducible polynomial. Let Ω be a connected and bounded region in R2 . Using a finite number of irreducible algebraic curves in R2 , we divide the region Ω into several simply connected regions, which are called the partition cells. Denote by ∆ the partition of the T region Ω , let δ1 , δ2 , . . . , δT be a given ordering of the cells in ∆, and let Ω = i=1 δi . Also, we write ∆ = {δ1 , δ2 , . . . , δT } and the interior of each cell δi can be described as
δi = {(x, y) ∈ R2 | g1[i] (x, y) > 0, g2[i] (x, y) > 0, . . . , gN[ii] (x, y) > 0}, where g1 (x, y), . . . , gNi (x, y) ∈ R[x, y] are irreducible polynomials. Denoted by C µ (∆) the set of C µ functions s(x, y) on Ω . Assume the restriction s(x, y)|δi is a polynomial which belongs to R[x, y] and let [i]
[i]
Pk (∆) = {s(x, y)|s(x, y)|δi ∈ Pk [x, y], i = 1, 2, . . . , T } be the set of piecewise polynomials defined on ∆ with total degree 6 k. ✩ This work was supported by the National Natural Science Foundation of China (Nos.11401526, 11101366, 11671068 and 11271328), the Zhejiang Provincial Natural Science Foundation of China (No.LY14A010001). ∗ Corresponding author. E-mail address: [email protected] (J. Wu).
http://dx.doi.org/10.1016/j.cam.2017.01.023 0377-0427/© 2017 Elsevier B.V. All rights reserved.
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The bivariate spline space with smoothness µ and degree k over Ω with respect to ∆ is defined as follows: µ Sk (∆) = {s(x, y) | s(x, y) ∈ C µ (∆) ∩ Pk (∆)}.
The zero set µ
Z(s) := {(x, y) ∈ Ω | s(x, y) = 0, s(x, y) ∈ Sk (∆)} is called a piecewise algebraic curve. Obviously, the piecewise algebraic curve is a generalization of the classic algebraic curve. However, it is difficult to study piecewise algebraic curve not only because of the complexity of the partition but also because of the possibility of {(x, y) | s(x, y) = 0} ∩ δi = ∅. Piecewise algebraic curve is originally introduced by Wang in the study of multivariate spline interpolation. He pointed out that the given interpolation knots are properly posed if and only if they are not lie in a non-zero piecewise algebraic curve [1]. In recent years, Wang and his researcher group have done a lot of significant work on piecewise algebraic curves (see [1–19]). For example, the Bezout theorem [3,4,13,16], Nöther-type theorem [7,9,10], Cayley–Bacharach theorem [8,14] and Riemann–Roch type theorems [9] of piecewise algebraic curves were established. Besides, piecewise algebraic curve also relates to the remarkable Four-Color conjecture [4]. Moreover, Lai et al. [11,12] discussed the real zeros of the zerodimensional parametric piecewise algebraic variety and produced the corresponding necessary and sufficient conditions of coefficients. Also, the Viro method for construction of Bernstein–Bézier algebraic hypersurface piece and piecewise algebraic curve are discussed [17,18]. In sum, the piecewise algebraic curve is a new and important topic in Computer Aided Geometry Design and Computational Geometry and has many applications in various fields. It is necessary and significant to study the related problems on piecewise algebraic curves. The singular points of a given real algebraic curve Z(f ) (SPT (f ) for short) are those points on the curve where both partial derivatives vanish, i.e.,
SPT (f ) = (x, y) | f (x, y) = fx (x, y) = fy (x, y) = 0 . The singular points and its classification for a given algebraic curve is a basic and important topic in algebraic geometry [20,21]. The detection of singular points helps to determine the geometrical shape and topology of a real algebraic curve. In 1990, Sakkalisa et al. [21] described an algorithm for determining whether an irreducible algebraic curve f (x, y) = 0 with total degree >3 is singular, and if so, isolating its singular points and computing their multiplicities. For a given plane rational curve in parametric form, there are several efficient methods to detect and compute the singular points [22]. Hence, it is of theoretical and practical significance to study singular points of a given parametric piecewise algebraic curve, especially the maximum number and its distribution of its singular points. Certainly, it helps us to determine the topological structure of piecewise algebraic curves better. Now, we firstly give the definition of parametric piecewise algebraic curves. Let V = (v1 , v2 , . . . , vr ) be the parameters and denoted by R[V ][x, y] the set of all polynomials in variables x and y with coefficients in R[V ]. Obviously, p ∈ R[V ][x, y] can be viewed as p = p(V , x, y). µ
Definition 1.1. Let s = s(V , x, y) ∈ Sk (∆) and s|δi ∈ R[V ][x, y] for each cell δi in ∆, where V = (v1 , v2 , . . . , vr ) and (x, y) are viewed as parameters and variables, respectively. Then Z(s) is called a parametric piecewise algebraic curve. In other words, piecewise algebraic curve is ‘‘parametric’’ means it contains symbolic coefficients and also allows to have some certain constant coefficients. From the theory of smoothing cofactor of bivariate spline [1], we can easily have µ
Lemma 1.1. Let s = s(V , x, y) ∈ Sk (∆) and s|δi ∈ R[V ][x, y], i = 1, 2, . . . , T . If the point p ∈ Z(s) and p ∈ δi ∩ δj , i, j ∈ {1, 2, . . . , T }, then p is the singular (regular) point of algebraic curve Z(s|δi ) = {(x, y) ∈ δi | s|δi (V , x, y) = 0} if and only if p is the singular (regular) point of algebraic curve Z(s|δj ) = {(x, y) ∈ δj | s|δj (V , x, y) = 0}. Proof. By using smoothing cofactor of bivariate spline [1], we have s|δi (V , x, y) − s|δj (V , x, y) = lij (x, y)µ+1 qij (V , x, y), where Z(lij ) is the common algebraic curve of δi ∩ δj , and qij (V , x, y) ∈ Pd [x, y], d = k − (µ + 1) deg(lij ). From simple computation, the result is proved. Therefore, singular points of a given parametric piecewise algebraic curve are defined as follows: µ
Definition 1.2. Let s = s(V , x, y) ∈ Sk (∆). If p ∈ Z(s) and there exists i ∈ {1, 2, . . . , T } such that p ∈ δi (the closure of δi ) and p is the singular point of algebraic curve Z(s|δi ), then p is called a singular point of parametric piecewise algebraic curve Z(s). Otherwise, p is called a regular point of Z(s).
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The singular points of a given parametric piecewise algebraic curve Z(s) is equivalent to solving the following parametric piecewise polynomial system
s(V , x, y) = 0, ∂ s(V , x, y) = 0, sx (V , x, y) = Z(s, ∇ s) : ∂x sy (V , x, y) = ∂ s(V , x, y) = 0 ∂y
(1)
µ
where s ∈ Sk (∆) and s|δi ∈ R[V ][x, y], i = 1, 2, . . . , T . µ
µ−1
Since s ∈ Sk (∆), we have sx , sy ∈ Sk−1 (∆) and Z(s, ∇ s) is indeed a piecewise algebraic variety contained in Z(s). Obviously, the singular points of Z(s) is equal to the real zeros of Z(s, ∇ s). We assume that all the singular points of Z(s) is torsion-free if it lies in the interior of a cell δi in ∆ if it is not specified. This assumption is convenient to count the number of singular points of a given parametric piecewise algebraic curve. In order to guarantee the number of singular points of a given piecewise algebraic curve is finite, we introduce the following concept. Definition 1.3. A bivariate spline s is said to be square-free if and only if for each i ∈ {1, 2, . . . , T } at least one of the following conditions holds:
• s|δi = s |δi , where
s |δi = s|δi / gcd s|δi ,
∂ s|δi ∂ s|δi , ∂x ∂y
.
• Z(fki ) ∩ δi = ∅, where, fki are the factors of s|δi with multiplicity greater than one. In our discussion, s is assumed to be square-free for almost all the parametric values V . That is to say, Z(s, ∇ s) is assumed to be zero-dimensional if and only if for almost all the parametric values V and for each i ∈ {1, 2, . . . , T } the set
Z (s, ∇ s) : [i]
s|δi (V , x, y) = 0, sx |δi (V , x, y) = 0, sy |δi (V , x, y) = 0
(2)
is generally zero-dimensional. It means that Z(s) has a finite number of singular points. The purpose of our paper is to present an algorithm to give the maximum number of singular points of a given parametric piecewise algebraic curve Z(s), and to get its distribution of the number of singular points in every cell, which is primarily based on our early proposed method in paper [11]. It can be viewed as a good and successful application in studying the related problems of piecewise algebraic curves. The rest of this paper is organized as follows. In Section 2, we give the basic framework of computing singular points of a given parametric piecewise algebraic curve. In Section 3, we recall several basic definitions on triangular parametric semialgebraic set. In Section 4, we present an algorithm to determine the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve. Meanwhile, the algorithm also gives the necessary and sufficient conditions on parameters must be satisfied. In Section 5, an illustrated example is provided to show that the algorithm is flexible. Finally, we conclude our paper in Section 6. 2. Singular points of parametric piecewise algebraic curve In this section, we firstly show that the coefficients of a given C µ parametric piecewise polynomial satisfy a system of linear equations. Theorem 2.1 ([11,12]). A piecewise polynomial function s is in C µ (∆) if and only if in for any adjacent pair δi , δj in ∆, which µ+1
intersects along detij = δi ∩ δj contained in the algebraic curve Z(lij ), all coefficients of the polynomial rem(s|δi − s|δj , lij , xij ) are null, where rem is either the usual Euclidean remainder or the pseudo-remainder, and lij is an irreducible polynomial with degree >1 with respect to some variables xij ∈ {x, y} appearing in lij . These coefficients are homogeneous linear functions of the coefficients of s|δi and s|δj , and these nullity can be written into a homogeneous linear system. µ
For a given s ∈ Sk (∆), s|δi can be written in the form s|δi =
λ1 +λ2 6k
where v
[i]
vλ[i1] λ2 xλ1 yλ2 = s|δi (v [i] , x, y),
is a row vector whose components are formed by coefficients vλ1 λ2 of s|δi in a fixed order. [i]
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Let Q = (v [1] , v [2] , . . . , v [T ] )′ . By Theorem 2.1, the system of homogeneous linear equations with respect to variables Q corresponding to C µ continuity condition of parametric piecewise polynomial s is BQ = 0, µ+1
where the elements in the matrix B are determined by the coefficients of all polynomials rem(s|δi − s|δj , lij Theorem 2.1. From Theorem 2.1, we have the following result
, xij ) in
µ
Theorem 2.2. Let s ∈ Sk (∆) and s|δi ∈ R[V ][x, y], i = 1, . . . , T . Then the following system of parametric piecewise polynomials has the same real zeros as Z(s, ∇ s) for any given values of parameters V
s(V , x, y) = 0, sx (V , x, y) = 0, ZV (s, ∇ s) : sy (V , x, y) = 0, BQ = 0,
(3)
where V = (v1 , v2 , . . . , vr ) is viewed as the parameters and B, Q are defined as above. It tells us that computing singular points of parametric piecewise algebraic curves amounts to solving parametric piecewise polynomial system ZV (s, ∇ s). Let ΘBQ = {(u1 , u2 , . . . , ur )| BQ = 0} be the set of solutions of the system BQ and dim(ΘBQ ) = d. Without loss of generality, U = (u1 , u2 , . . . , ud ) is viewed as free parameters and (ud+1 , ud+2 , . . . , ur ) can be uniquely determined by U. Define a mapping Π : Rr −→ Rd by
Π (u1 , . . . , ud , ud+1 , . . . , ur ) := (u1 , u2 , . . . , ud ). The method for computing Π (ΘBQ ) can be found in any textbook on linear algebra. Using Gaussian elimination method to eliminate parameters (ud+1 , ud+2 , . . . , ur ), the ZV (s, ∇ s) can be transformed into a system of parametric piecewise polynomials
s(U , x, y) = 0, sx (U , x, y) = 0, ZU (s, ∇ s) : sy (U , x, y) = 0,
(4)
where U = (u1 , u2 , . . . , ud ) ∈ Rd ∩ Π (ΘBQ ) are viewed as actual parameters. We have the following conclusion immediately from Theorem 2.2 and the above discussion. µ
Theorem 2.3. Let s ∈ Sk (∆) and s|δi ∈ R[V ][x, y], i = 1, . . . , T . We define:
ZU (s, ∇ s) : [i]
s|δi (U , x, y) = 0, sx |δi (U , x, y) = 0, sy |δi (U , x, y) = 0,
(5)
g1 (x, y) > 0, . . . , gNi (x, y) > 0. [i]
[i]
Let NSP (s) be the number of singular points of a given parametric piecewise algebraic curve Z(s) and Rzero(·) be the set of real zeros of a given system of parametric (piecewise) polynomials. Then for any given U ∈ Rd ∩ Π (ΘBQ ), we have NSP (s) = Rzero(Z(s, ∇ s)), Rzero(Z(s, ∇ s)) = Rzero(ZV (s, ∇ s)) = Rzero(ZU (s, ∇ s)) =
T
Rzero(ZU (s, ∇ s)) [i]
i=1
and Rzero(Z[i] (s, ∇ s)) = Rzero(ZU (s, ∇ s)), [i]
i = 1, . . . , T .
Therefore, computing singular points of a given parametric piecewise algebraic curve amounts to solving these [1] [T ] parametric semi-algebraic systems ZU (s, ∇ s), . . . , ZU (s, ∇ s) with U = (u1 , . . . , ud ) as actual parameters. It is well known that we can use the Ritt–Wu method [23], Gröbner basis method or subresultant method [24] to [i] transform the parametric ZU (s, ∇ s) into one or more systems in the triangular form (TSA for short). Then we check whether it is a regular TSA and, if not, transform it into regular TSA by WR algorithm [25] or Wang’s algorithm [26]. Let the regular [i] TSA for ZU (s, ∇ s) be
[i] S (U ) = 0 0[,ik] ϕ ( U , x) = 0 1,k [i] TSAk (s, ∇ s) : [i] ϕ 2,k (U , x, y) = 0, [i] [i] g1 (x, y) > 0, . . . , gNi (x, y) > 0 for some k = 1, 2, . . . , βi , i = 1, 2, . . . , T .
(6)
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We found that we need not care about S0,k (U ) = 0 when we solve the real zeros of TSAk (s, ∇ s). In fact, S0,k (U ) = 0 is one of the conditions we wanted for determining the maximum number and its distribution of the number of singular [i] [i] points for a given parametric piecewise algebraic curve Z(s). Denoted by TSAk the semi-algebraic system TSAk (s, ∇ s) after [i]
[i]
[i]
deleting the condition S0,k (U ) = 0, i.e., [i]
[i] ϕ1,k (U , x) = 0 [i] [i] TSAk : ϕ2,k (U , x, y) = 0, [i] [i] g1 (x, y) > 0, . . . , gNi (x, y) > 0.
(7)
Thus, TSAk (s, ∇ s) = TSAk [i]
[i]
[i] {S0,k (U ) = 0}.
3. Triangular semi-algebraic set [i]
In order to solve TSAk , we ought to introduce some basic notations on triangular semi-algebraic set. A semi-algebraic set is a system consisting of polynomials equations, polynomial inequations and polynomial inequalities, where all polynomials are of integer coefficients. In this article, it requires to study the real zeros of bivariate polynomials, we slightly simplify the notations compared to our previous paper [11]. Given the following special triangular parametric semi-algebraic set f1 (u, x) = 0, f2 (u, x, y) = 0, g1 (x, y) > 0, g2 (x, y) > 0, . . . , gt (x, y) > 0,
TSA :
(8)
where u = (u1 , u2 , . . . , ud ) and (x, y) are viewed as parameters and variables, respectively; f1 , f2 ∈ Z(u)[x, y], gj ∈ Z[x, y], 1 6 j 6 t. Definition 3.1. Given a polynomial g (x), let res(g , gx′ , x) be the Sylvester resultant of g and gx′ with respect to x. We call it the discriminant of g with respect to x and denote it by Discrim(g , x). Definition 3.2. Given a triangular set {f1 , f2 }, I1 denotes the leading coefficient of f1 in x, and I2 denotes the leading coefficient of f2 in y. Triangular set {f1 , f2 } is called a normal ascending chain if I1 ̸= 0,
res(I2 , f1 ) ̸= 0,
where res(I2 , f1 ) denotes the resultant of I2 and f1 with respect to x. Definition 3.3. Given a parametric normal ascending chain TSA. For every fi , let R1 = Discrim(f1 , x), Rgi = res(gi , f2 , f1 ),
R2 = res(Discrim(f2 , y), f1 ), i = 1, . . . , t .
We define BPs := {Ri |1 6 i 6 2} ∪ {Rgi |1 6 i 6 t } and call it the set of critical polynomials of system TSA. Definition 3.4. Given a parametric TSA as (8) and {f1 , f2 } is a normal ascending chain. Then TSA is regular if 0 ̸∈ BPs . Definition 3.5. Given two polynomials f (x) and g (x), let Discr(f , g ) be the generalized discrimination matrix of f (x) with respect to g (x), and D0 = 1. Denote by
{D1 (f , g ), D2 (f , g ), . . . , Dn (f , g )} the even order principal minors of Discr(f , g ). We call
[D0 , D1 (f , g ), D2 (f , g ), . . . , Dn (f , g )] the generalized discriminant sequence of f (x) with respect to g (x), and denote it by GDL(f , g ).
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Let ps = {pi |1 6 i 6 w} be a nonempty and finite set of polynomials. We define mset (ps) = {1} ∪ {pi1 pi2 · · · pik |1 6 k 6 w, 1 6 i1 < i2 < · · · < ik 6 w}. For example, if ps = {p1 , p2 , p3 }, then mset (ps) = {1, p1 , p2 , p3 , p1 p2 , p1 p3 , p2 p3 , p1 p2 p3 }. Definition 3.6. Given a parametric TSA (8), we define P3 = {g1 , g2 , . . . , gt },
U2 =
GDL(f2 , q),
q∈mset (P3 )
P2 = {h(u, x1 )| h ∈ U2 },
U1 =
GDL(f1 , q),
q∈mset (P2 )
TSAP1 (g1 , g2 , . . . , gt ) = {h(u)| h ∈ U1 }, where Ui means the set consisting of all the polynomials in each GDL(fi , q) where q belongs to mset (Pi+1 ). Similarly, we can define TSAP1 (g1 , g2 , . . . , gi ). Theorem 3.1 ([27]). The necessary and sufficient condition for system TSA (8) to have a given number of distinct real solutions can be expressed in terms of the signs of the polynomials in TSAP1 (g1 , g2 , . . . , gt ). 4. Maximum number and its distribution of singular points By using WR algorithm [25] or Wang’s algorithm [26], for every ZU (s, ∇ s), it can be decomposed into regular triangular [i]
[i] TSA1 (s, ∇ s), . . . , TSAβi (s, ∇ s), [i]
systems We define
BPs (ZU (s, ∇ s)) :=
where TSAk (s, ∇ s) = TSAk
[i]
[i]
[i] {S0,k (U ) = 0}.
(BPs )k[i] ,
(9)
16i6T 16k6βi
where (BPs )k is the set of critical polynomials of semi-algebraic system TSAk . Set [i]
[i]
P1 (ZU (s, ∇ s)) :=
TSAk P1 (g1 , . . . , gNi ), [i]
16i6T 16k6βi
[i]
[i]
(10)
where TSAk P1 (g1 , . . . , gj ) is defined as Definition 3.6. Let [i]
[i]
Q (ZU (s, ∇ s)) :=
[i]
16i6T 16k6βi
{S0[i,]k (U ) = 0}.
(11)
From Theorems 2.3, 3.1 and using similar techniques in paper [11], we have the following important conclusions. Theorem 4.1. The necessary and sufficient condition for a given parametric piecewise algebraic curve Z(s) to have a given number of singular points in every cell in ∆ can be expressed in terms of the signs of the polynomials in P1 (ZU (s, ∇ s)) and the elements in Q (ZU (s, ∇ s)). Now, we present the algorithm to deal with the maximum number and its distribution of distinct singular points of a given piecewise algebraic curve Z(s). Algorithm 1 (Max-Distribution-SingularPoints-PAC). Input: A given parametric piecewise algebraic curve Z(s), or a given parametric piecewise algebraic variety Z(s, ∇ s). Output: MaxNSP denotes by the maximum number of distinct singular points of Z(s); DistSP denotes by its distribution of distinct singular points in cells δ1 , . . . , δT when the number of singular points reach MaxNSP; The necessary and sufficient condition Φ (NS[1] , . . . , NS[T ] ) on the parameters for the Z(s) to have exactly (NS[1] , . . . , NS[T ] ) distinct singular points in cells δ1 , . . . , δT . Step 1. Obtain ZV (s, ∇ s) with Theorem 2.1, and transform it into ZU (s, ∇ s) by using elimination method.
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Step 2. For every i ∈ {1, 2, . . . , T }, perform WR algorithm or Wang’s algorithm on every ZU (s, ∇ s) and obtain regular [i]
triangular systems
[i] TSA1 (s, ∇ s), . . . , TSAβi (s, ∇ s), [i]
[i] where TSAk (s, ∇ s) = TSAk {S0,k (U ) = 0}, k = 1, . . . , βi . [i]
[i]
Step 3. Compute (BPs )k (the set of critical polynomials of system TSAk ) for each i ∈ {1, . . . , T } and k ∈ {1, . . . , βi } and obtain BPs (ZU (s, ∇ s)). Step 4. Set Polyset := BPs (ZU (s, ∇ s)) and l := 1. Step 5. Compute a PolySet-invariant cylindrical algebraic decomposition with the algorithm of PCAD [28]. Denote by D′ the set of cad-cell δ in D with δ ∩ Π (ΘBQ ) ̸= 0 and its cylindrical algebraic sample S. Step 6. For each cad-cell δ in D′ and its sample points sδ ∈ S, compute the signs of polynomials in PolySet on this cad-cell after substituting sδ into them. The signs of polynomials in PolySet on this cad-cell δ form a first order formula, denoted by Φδ . If Q (Z(s, ∇ s)) ∩ Φδ ̸= ∅, then D′ := D′ ∪ {δ}. Otherwise, D′ := D′ . [i]
[i]
Step 7. For each cad-cell δ in D′ and its sample point sδ ∈ S, substitute sδ into TSAk and compute the number of distinct [i]
real solutions of every system TSAk (sδ ) (say mk (δ)). If all mk (δ) are computed, then set [i]
MaxNSP := max N |N =
[i]
βi T
[i]
mk (δ), δ ∈ D [i]
′
,
i=1 k=1
DistSP :=
(NS , . . . , NS )|NS = [1]
[T ]
[i]
βi
mk (δ), δ ∈ D [i]
′
.
k=1
Step 8. For each (NS[1] , . . . , NS[T ] ) ∈ DistSP, set
Θ (NS , . . . , NS ) := [1]
Φδ |
[T ]
βi
mk (δ) = NS , i = 1, . . . , m, δ ∈ D [i]
[i]
′
.
k=1
Step 9. Decide whether all the Θ (NS[1] , . . . , NS[T ] ) can form a necessary and sufficient conditions similar to our previous method in [11]. If so, go to Step 10. Otherwise, we set Polyset := Polyset ∪ P1,l (ZU (s, ∇ s)), where P1,l (ZU (s, ∇ s)) :=
TSAk P1 (g1 , . . . , gl ), [i]
[i]
[i]
1 6 l 6 max Ni ,
16i6T 16k6βi
16i6T
let l := l + 1 and go to Step 4. Step 10. Output MaxNSP and DistSP. If we let Θ (NS[1] , . . . , NS[T ] ) = {Φδ1 , . . . , Φδt } for a given (NS[1] , . . . , NS[T ] ) ∈ DistSP, then
Φ NS[1] , . . . , NS[T ] = Φδ1 ··· , Φδt Q ′ (Z(s, ∇ s)), where Q ′ (Z(s, ∇ s)) =
k,i S0,k (U ) = 0 | mk (δ) ̸= 0 .
[i]
[i]
We found that the termination of this algorithm is guaranteed by Theorem 4.1. In fact, we use the tofind command in DISCOVERER package developed by B.C. Xia [29] to help to determine whether Q (Z(s, ∇ s))∩ Φδ ̸= ∅ or not. It is pointed out that the equality constraints are a significant improvement compared to our previous method [11]. We find the results are interesting and incredible which can be easily seen from the later illustrated example. It is viewed as a successful application and extension. 5. An illustrated example A simple example is provided to illustrate the proposed algorithm is flexible. Let ∆ = {δ1 , δ2 } be a partition of a given triangle V1 V2 V3 in R2 , where δ1 = [V1 V2 V0 ], δ2 = [V2 V3 V0 ], V1 = (1, 0), V2 = (0, 1), V3 = (−1, 0), and V0 = (0, 0). It is obvious that the partition straight line is L : x = 0. Let s(x, y) ∈ S31 (∆), where s(x, y) =
s(x, y)|δ1 = x3 + ax2 + by2 + kxy + cy + d s(x, y)|δ2 = ex3 + hxy2 + fx2 + my2 + ny + l.
From Theorem 2.1, these coefficients of s(x, y) satisfy the following conditions m = b,
n = c,
l = d,
k = h = 0.
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Fig. 1. Singular points of a piecewise algebraic curve Z(s∗ ).
Substituting these conditions into s(x, y), we have s(x, y) =
s(x, y)|δ1 = x3 + ax2 + by2 + cy + d s(x, y)|δ2 = ex3 + fx2 + by2 + cy + d.
If we perform the algorithm Max-Distribution-SingularPoints-PAC, then we have the following facts (1) The maximum number of the singular points for the given piecewise algebraic curve Z(s) is two, i.e., MaxNSP = 2. (2) Its distribution in δ1 and δ2 is {1, 1} when the number of singular points reaches maximum, i.e., DistSP = {1, 1}. (3) The piecewise algebraic curve Z(s) has one singular point in every cell if and only if Case A : [a > 0 ∧ b > 0 ∧ c < 0 ∧ e < 0 ∧ f > 0 ∧ ∆1 < 0 ∧ ∆2 < 0 ∧ S1 = 0 ∧ S2 = 0] or Case B : [a > 0 ∧ b < 0 ∧ c > 0 ∧ e < 0 ∧ f > 0 ∧ ∆1 > 0 ∧ ∆2 > 0 ∧ S1 = 0 ∧ S2 = 0] where
∆1 = −6b − 3c + 4ba, ∆2 = 6eb + 3ec + 4bf , S1 = 108bd + 16ba3 − 27c 2 , S2 = (108e2 d + 16f 3 )b − 27e2 c 2 . Take a =
1 2
, b = 1; c = −1, d =
25 108
, e = −8, f = 2 for example. We find that Case A is satisfied and s∗ (x, y) becomes
1 25 s∗1 (x, y) = s∗ (x, y)|δ1 = + x2 + x3 − y + y2 , 108
2
s∗ (x, y) = s∗ (x, y)|δ = 25 + 2x2 − 8x3 − y + y2 . 2 2 108
The graph of Z(s∗ ) is shown in Fig. 1 and we found that the total number of singular points of Z(s∗ ) is two and its distribution in every cell is one. This experiment presents an interesting result with respect to singular points of parametric piecewise algebraic curves till now. It indicates that our proposed method is flexible. 6. Conclusion It is of theoretical and practical significance to study the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve. In this paper, we extend our previous method [11] to solve this important issue. Numerical experiments show that our proposed method is flexible and our results are considerable interesting. It is pointed out that it is interesting to study the classification of singular points for parametric piecewise algebraic curves in our future work. In fact, we have successfully discussed the real root classification of parametric spline functions [30]. Thus, we have reasons to believe that this problem is solvable. Acknowledgments We appreciate the reviewers and editors for their careful reading, valuable comments, timely review and reply.
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