A piecewise algebraic curve is a curve defined by the zero set of a bivariate
spline function. Given two bivariate spline spaces Smr
(Δ) and Snt
(Δ) over
a domain D with a partition Δ, the Bezout number BN(m,r;n,t;Δ) is defined
as the maximum finite number of the common intersection points of two arbitrary
piecewise algebraic curves f(x, y)=0 and g(x, y)=0, where f(x, y)∈Smr
(Δ)
and g(x, y)∈Snt
(Δ). In this paper, an upper bound of the Bezout number for
piecewise algebraic curves over a rectangular partition is obtained.
1. Introduction
Let D⊂R2 be a bounded domain, and let Pk be the collection of real bivariate polynomials with total degree not greater than k. Divide D by using finite number of irreducible algebraic curves we get a partition denoted by Δ. The subdomains D1,D2,…,DN are called the cells. The line segments that form the boundary of each cell are called the edges. Intersection points of the edges are called the vertices. The vertices in the inner of the domain are called interior vertices, otherwise are called boundary vertices. For a vertex v, its so-called star st(v) means the union of all cells in Δ sharing v as a common vertex, and its degree d(v) is defined as the number of the edges sharing v as a common endpoint. If d(v) is odd, we call v an odd vertex. For integers k and μ with k>μ≥0, the bivariate spline space with degree k and smoothness μ over D with respect to Δ is defined as follows [1, 2]:
(1.1)Skμ(Δ):={s∈Cμ(D)∣s∣Di∈Pk,i=1,…,N}.
For a spline s∈Skμ(Δ), the zero set
(1.2)Z(s):={(x,y)∈D∣s(x,y)=0}
is called a piecewise algebraic curve [1, 2]. Obviously, the piecewise algebraic curve is a generalization of the usual algebraic curve [3, 4].
In fact, the definition of piecewise algebraic curve is originally introduced by Wang in the study of bivariate spline interpolation. He pointed out that the given interpolation knots are properly posed if and only if they are not lying on a same nonzero piecewise algebraic curve [1, 2]. Hence, to solve a bivariate spline interpolation problem, it is necessary to deal with the properties of piecewise algebraic curve. Moreover, piecewise algebraic curve is also helpful for us to study the usual algebraic curve. Besides, piecewise algebraic curve also relates to the remarkable Four-Color conjecture [5–7]. In fact, the Four-Color conjecture holds if and only if, for any triangulation, there exist three linear piecewise algebraic curves such that the union of them equals the union of all central lines of all triangles in the triangulation. We know that any triangulation is 2-vertex signed [6, 7], which means the vertices of the triangulation can be marked by -1 or 1 such that the vertices of every triangle in the triangulation are marked by different numbers. So, we remark that the Four-Color conjecture holds if and only if there exist three nonequivalent mark methods. Based on these observations, in a word, piecewise algebraic curve is a new and important topic of computational geometry and algebraic geometry. It is of important theoretical and practical significance in many fields such as bivariate spline interpolation and computer-aided geometric design (CAGD). Hence, it is necessary to continue to study it.
In this paper, we mainly focus on the Bezout number for piecewise algebraic curves. It is well known that the Bezout theorem for usual algebraic curves is very important in algebraic geometry [3, 4]. Its weak form says that two algebraic curves with degree m and n, respectively, will have infinite number of common intersection points if they have more intersection points (including multiple points) than the product mn of their degrees, that is, the so-called Bezout number. Similarly, for two given bivariate spline spaces Smr(Δ) and Snt(Δ), the following number [2]
(1.3)BN(m,r;n,t;Δ):=maxf,g{#(f,g)<∞∣f(x,y)∈Smr(Δ),g(x,y)∈Snt(Δ)}
is called the Bezout number, where #(f,g) denotes the number of the common intersection points (including multiple points) of f(x,y)=0 and g(x,y)=0. It also implies that two piecewise algebraic curves determined by two splines in Smr(Δ) and Snt(Δ) respectively will have infinite number of common intersection points if they have more intersection points than BN(m,r;n,t;Δ). Needless to say, it is crucial for us to obtain BN(m,r;n,t;Δ). Obviously, if r≤t, then we have
(1.4)BN(m,r;n,t;Δ)≤BN(m,r;n,r;Δ)≤Nmn.
However, we remark that it is very hard to obtain exact BN(m,r;n,t;Δ). On the one hand, piecewise algebraic curve itself is difficult; on the other hand, the Bezout number BN(m,r;n,t;Δ) is also complicated; we know it not only relies on the degrees m, n and the smoothness orders r, t, but also the dimensions of Smr(Δ) and Snt(Δ) and the geometric characteristics of Δ as well [6, 7]. Hence, we can only get an upper bound for BN(m,r;n,t;Δ) sometimes.
Throughout the relative literatures, we find that there have been some results on the Bezout number for piecewise algebraic curves over triangulations [6–9], while this paper, as a continue paper to fill a gap, is mainly devoted to an upper bound of the Bezout number for two piecewise algebraic curves over a rectangular partition. Our method is different from the methods in [6–9] and is new and effective. The remainder is organized as follows. In Section 2, for the sake of integrity, some prevenient results on the Bezout number for piecewise algebraic curves over parallel lines partition and triangulation are introduced; Section 3 is the main section, in this section; assume Δ to be a rectangular partition, an upper bound of BN(m,r;n,r;Δ) is well estimated; finally, this paper is concluded in Section 4 with a conjecture.
2. Preliminary
Denoted by Δk the partition containing only k parallel lines (Figure 1), we have the followeing theorem.
Δk.
Theorem 2.1 (see [<xref ref-type="bibr" rid="B4">10</xref>]).
One has(2.1)BN(m,r;n,t;Δk)≤(k+1)mn-min{r,t}k.
For a triangulation Δ, the Bezout number for two continuous piecewise algebraic curves satisfying the following theorem.
Theorem 2.2 (see [<xref ref-type="bibr" rid="B5">6</xref>]).
Consider
(2.2)BN(m,0;n,0;Δ)≤Nmn-[Vodd+23],
where N and Vodd is the number of the triangles and the odd interior vertices in Δ, respectively, and [x] means the maximum integer not greater than x.
By using resultants and polar coordinates, Zhao studied the Bezout number for two C1 piecewise algebraic curves over a non-obtuse-angled star st(v) (Figures 2 and 3).
An interior star.
A boundary star.
Theorem 2.3 (see [<xref ref-type="bibr" rid="B14">9</xref>]).
If v is an interior vertex, then
(2.3)BN(m,1;n,1;st(v))≤d(v)mn-(d(v)-1),
and if v is a boundary vertex, then
(2.4)BN(m,1;n,1;st(v))≤(d(v)-1)mn-(d(v)-2).
Based on Theorem 2.3, by a combinatorial optimization method, a result for a general nonobtuse triangulation is obtained.
Theorem 2.4 (see [<xref ref-type="bibr" rid="B9">7</xref>]).
For any given nonobtuse triangulation Δ,
(2.5)BN(m,1;n,1;Δ)≤Nmn-2EI-VI-VB3,
where N, EI, VI, and VB is the number of the triangles, the interior edges, the interior vertices, and the boundary vertices in Δ.
We know that the Bezout number for piecewise algebraic curves over stars is the key issue for the Bezout number for piecewise algebraic curves over a general partition. In order to get BN(m,r;n,r;Δ), Wang and Xu generalized the smoothness order of Theorem 2.3 from 1 to r.
Theorem 2.5 (see [<xref ref-type="bibr" rid="B9">7</xref>]).
For any nonobtuse star st(v), if v is an interior vertex, then
(2.6)BN(m,r;n,r;st(v))≤d(v)mn-max{d(v)-4,[d(v)+12]}r,
and if v is a boundary vertex, then
(2.7)BN(m,r;n,r;st(v))≤(d(v)-1)mn-max{d(v)-2,[d(v)-12]}r.
Here, we note that if v is a boundary vertex, then d(v)≥2, so d(v)-2≥[(d(v)-1)/2] holds unconditionally; hence we have
(2.8)BN(m,r;n,r;st(v))≤(d(v)-1)mn-(d(v)-2)r.
Recently, by using the theory of resultants, polar coordinates and periodic trigonometric spline, Theorem 2.5 is improved greatly when v is an interior vertex.
Theorem 2.6 (see [<xref ref-type="bibr" rid="B1">5</xref>, <xref ref-type="bibr" rid="B10">8</xref>]).
For any nonobtuse star st(v), if v is an interior vertex, then
(2.9)BN(m,r;n,r;st(v))≤2[d(v)(mn-r)2],
and if v is a boundary vertex, then
(2.10)BN(m,r;n,r;st(v))≤(d(v)-1)(mn-r)+r.
Here, we also remark that if v is a boundary vertex, then the second formula is likewise equivalent to
(2.11)BN(m,r;n,r;st(v))≤(d(v)-1)mn-(d(v)-2)r.
By Theorem 2.6, we get a better upper bound of BN(m,r;n,r;Δ) as follows.
Theorem 2.7.
For any nonobtuse triangulation Δ, one has
(2.12)BN(m,r;n,r;Δ)≤{Nmn-23EIr,ifmn-riseven,Nmn-23EIr-13Vodd,ifmn-risodd,
where Vodd stands for the number of the odd interior vertices in Δ.
Proof.
Suppose that the number of the common intersection points of two piecewise algebraic curves f(x,y)=0 and g(x,y)=0 is finite and equals BN(m,r;n,r;Δ), where f∈Smr(Δ) and g∈Snr(Δ). For a vertex v in Δ, let k(v) be the number of the common intersection points of f(x,y)=0 and g(x,y)=0 in st(v). Summing k(v) for each vertex, so every common intersection point is counted triply. Hence, we get
(2.13)∑vk(v)=3BN(m,r;n,r;Δ).
By Theorem 2.6, we have
(2.14)k(v)≤BN(m,r;n,r;st(v))≤δ(v),
where
(2.15)δ(v)={{d(v)(mn-r),ifd(v)(mn-r)isevend(v)(mn-r)-1,ifd(v)(mn-r)isoddifvisaninteriorvertex(d(v)-1)mn-(d(v)-2)r,ifvisaboundaryvertex.
By the Euler formulae EI=VB+3VI-3,N=VB+2VI-2 and the equations ∑vd(v)=2(EI+EB),EB=VB, if mn-r is even, we get
(2.16)BN(m,r;n,r;Δ)=13∑vk(v)≤13∑vδ(v)=13{∑interiorvd(v)(mn-r)+∑boundaryv((d(v)-1)mn-(d(v)-2)r)}=13(∑vd(v)-VB)mn-13(∑vd(v)-2VB)r=Nmn-23EIr.
Similarly, if mn-r is odd, we have
(2.17)BN(m,r;n,r;Δ)≤Nmn-23EIr-13Vodd.
In the next section, we will apply Theorems 2.1 and 2.6 to a rectangular partition. A good upper bound of the Bezout number for piecewise algebraic curves over a rectangular partition is derived, which fills a gap in the study of the Bezout number for piecewise algebraic curves over any partition.
3. Results
Without loss of generality, let integers a≥b≥1. For a rectangular domain D, subdivide it into N:=ab subrectangular cells by (a-1) vertical lines and (b-1) horizontal lines, and denote by Δa×b the partition; see Figure 4 for an example. Let V0, V10, and V11 be the collection of the interior vertices (d(v)=4), the boundary vertices that lying in the interior of the boundary edges (d(v)=3), and the else four corner vertices (d(v)=2), respectively. The cardinality of V0, V10, and V11 is (a-1)(b-1), 2(a+b-2), and 4, respectively. If a cell lies in the ith column (from left to right) and in the jth row (from bottom to top), we denote it by D(i,j), i=1,2,…,a; j=1,2,…,b. For an interior vertex v∈V0, if it is the intersection point of the pth vertical interior line and the qth horizontal interior line, then we denote it by v(p,q),p=1,2,…,a-1; q=1,2,…,b-1. The else boundary vertices can also be denoted similarly.
A rectangular partition Δa×b, a=5, b=4.
Assume that the number of the common intersection points of two piecewise algebraic curves f(x,y)=0 and g(x,y)=0 is finite and equals BN(m,r;n,r;Δa×b), where f∈Smr(Δa×b) and g∈Snr(Δa×b). If b=1, then Δa×1 is a partition Δa-1 containing only a-1 parallel lines essentially (Figure 1); by Theorem 2.1, we have
(3.1)BN(m,r;n,r;Δa×1)≤amn-(a-1)r.
In the rest of this section, assume a≥b≥2.
3.1. The First Method
For a vertex v in Δa×b, let k(v) be the number of the common intersection points of f(x,y)=0 and g(x,y)=0 in st(v). Summing k(v) for each vertex, so that every common intersection point is counted fourfold. Hence, we get
(3.2)∑vk(v)=4BN(m,r;n,r;Δa×b).
By Theorem 2.6, we have
(3.3)k(v)≤BN(m,r;n,r;st(v))≤δ(v),
where
(3.4)δ(v)={4mn-4r,ifv∈V0,d(v)=42mn-r,ifv∈V10,d(v)=3mn,ifv∈V11,d(v)=2.
We have
(3.5)BN(m,r;n,r;Δa×b)=14∑vk(v)≤14∑vδ(v)=14(∑v∈V0δ(v)+∑v∈V10δ(v)+∑v∈V11δ(v))=14{(4mn-4r)(4mn-4r)(a-1)(b-1)+(2mn-r)(2(a+b-2))+4mn}=Nmn-r(N-a+b2).
Let BN1=Nmn-r(N-((a+b)/2)).
3.2. The Second Method
Let k(i,j) be the number of the common intersection points of f(x,y)=0 and g(x,y)=0 in the cell D(i,j)(i=1,2,…,a;j=1,2,…,b) summing k(i,j) for all cells, we get
(3.6)∑i,jk(i,j)=BN(m,r;n,r;Δa×b).
For the cells in the jth row, by Theorem 2.1, we have
(3.7)∑ik(i,j)≤BN(m,r;n,r;Δa-1)≤amn-(a-1)r.
Hence,
(3.8)BN(m,r;n,r;Δa×b)=∑i,jk(i,j)=∑j(∑ik(i,j))≤(amn-(a-1)r)b=Nmn-r(N-b).
Similarly,
(3.9)BN(m,r;n,r;Δa×b)=∑i,jk(i,j)=∑i(∑jk(i,j))≤(bmn-(b-1)r)a=Nmn-r(N-a).
Let BN2=Nmn-r(N-b), BN3=Nmn-r(N-a). Considering a≥b, we have BN2≤BN1≤BN3. This tells us a better summation method results in a better upper bound.
3.3. The Third Method
In this subsection, we will derive a rather better upper bound of BN(m,r;n,r;Δa×b) than BN1, BN2, and BN3 by using a mixed method based on Theorems 2.1 and 2.6. We prefer this summation method.
If a and b are both even, then Δa×b=∪p=1a/2∪q=1b/2st(v(2p-1,2q-1)); so
(3.10)BN(m,r;n,r;Δa×b)=∑p=1a/2∑q=1b/2k(v(2p-1,2q-1))≤∑p=1a/2∑q=1b/2δ(v(2p-1,2q-1))=(4mn-4r)×a2×b2=Nmn-Nr.
If a is odd and b is even, then Δa×b=∪p=1((a-1)/2)∪q=1b/2st(v(2p-1,2q-1))∪j=1bD(a,j); so we have
(3.11)BN(m,r;n,r;Δa×b)=∑p=1((a-1)/2)∑q=1b/2k(v(2p-1,2q-1))+∑j=1bk(a,j)≤∑p=1((a-1)/2)∑q=1b/2δ(v(2p-1,2q-1))+BN(m,r;n,r;Δb-1)=(4mn-4r)×a-12×b2+(bmn-(b-1)r)=Nmn-(N-1)r.
If a is even and b is odd, then Δa×b=∪p=1a/2∪q=1((b-1)/2)st(v(2p-1,2q-1))∪i=1aD(i,b); so
(3.12)BN(m,r;n,r;Δa×b)=∑p=1a/2∑q=1((b-1)/2)k(v(2p-1,2q-1))+∑i=1ak(i,b)≤∑p=1a/2∑q=1((b-1)/2)δ(v(2p-1,2q-1))+BN(m,r;n,r;Δa-1)=(4mn-4r)×a2×b-12+(amn-(a-1)r)=Nmn-(N-1)r.
If a and b are both odd, then Δa×b=∪p=1((a-1)/2)∪q=1((b-1)/2)st(v(2p-1,2q-1))∪j=1bD(a,j)∪i=1a-1D(i,b), hence
(3.13)BN(m,r;n,r;Δa×b)=∑p=1((a-1)/2)∑q=1((b-1)/2)k(v(2p-1,2q-1))+∑j=1bk(a,j)+∑i=1a-1k(i,b)≤∑p=1((a-1)/2)∑q=1((b-1)/2)δ(v(2p-1,2q-1))+BN(m,r;n,r;Δb-1)+BN(m,r;n,r;Δa-2)=(4mn-4r)×a-12×b-12+(bmn-(b-1)r)+((a-1)mn-(a-2)r)=Nmn-(N-2)r.
Theorem 3.1.
Let Δa×b be a rectangular partition, then
(3.14)BN(m,r;n,r;Δa×b)≤{Nmn-Nr,ifaandbarebothevenNmn-(N-1)r,if(a-b)isoddNmn-(N-2)r,ifaandbarebothodd.
Let BN denotes the upper bound given in Theorem 3.1. We remark that BN is better than BN1, BN2, and BN3. Since a≥b≥2 and BN2≤BN1≤BN3, we only give the comparisons between BN and BN2=Nmn-r(N-b). For fixed m, n, and r, we have the following results; see Table 1. In a word, we have BN≤BN2.
Comparisons of BN and BN2 (where a≥b≥2 and N=ab).
BN
BN2
BN2-BN
a and b are both even
Nmn-Nr
Nmn-r(N-b)
br≥0
(a-b) is odd
Nmn-(N-1)r
Nmn-r(N-b)
(b-1)r≥0
a and b are both odd
Nmn-(N-2)r
Nmn-r(N-b)
(b-2)r≥0
4. Conclusions
In this paper, we mainly derive an upper bound for the Bezout number BN(m,r;n,r;Δa×b). The results of Theorem 3.1 are excellent than BN1, BN2, and BN3. It is very useful in the fields of CAGD. For example, we are frequently needed to get the common intersection points of two piecewise algebraic curves [11, 12]. By Theorem 3.1, a prior estimation of the number of the common intersection points of two piecewise algebraic curves over a rectangular partition will be obtained. In future, we will try best to improve Theorem 3.1 to get the exact number or a lower upper bound for BN(m,r;n,r;Δa×b). In order to attract readers’ interest, here, we also give a conjecture on the Bezout number BN(m,r;n,r;Δa×b).
Conjecture 4.1.
Consider
(4.1)BN(m,r;n,r;Δa×b)=mn+[(a-1)+(b-1)](mn-r)+(a-1)(b-1)(mn-2r)=Nmn-(2N-a-b)r.
If this conjecture holds, it can be applied into the study of the Nöther-type theorem [13, 14] and the Cayley-Bacharach theorem [15] for piecewise algebraic curves over rectangular partition.
Acknowledgments
This work was supported by the Fundamental Research Funds for the Central Universities (no. 201113037). The author appreciate the reviewers and editors for their careful reading, valuable suggestions, timely review, and reply.
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