An Upper Bound of the Bezout Number for Piecewise Algebraic Curves over a Rectangular Partition

Abstract

A piecewise algebraic curve is a curve defined by the zero set of a bivariate spline function. Given two bivariate spline spaces 𝑆^𝑟_𝑚 (Δ) and 𝑆^𝑡_𝑛 (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)∈𝑆^𝑟_𝑚 (Δ) and 𝑔(𝑥,𝑦)∈𝑆^𝑡_𝑛 (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 𝐏_𝑘 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,…,𝐷_𝑁 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]: 𝑆^𝜇_𝑘(Δ)∶=𝑠∈𝐶^𝜇(𝐷)∣𝑠_∣𝐷𝑖∈𝐏_𝑘,𝑖=1,…,𝑁. (1.1) For a spline 𝑠∈𝑆^𝜇_𝑘(Δ), the zero set 𝑍(𝑠)∶=𝑥,𝑦∈𝐷∣𝑠𝑥,𝑦=0 (1.2) 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 𝑆^𝑟_𝑚(Δ) and 𝑆^𝑡_𝑛(Δ), the following number [2] BN(𝑚,𝑟;𝑛,𝑡;Δ)∶=max_𝑓,𝑔#𝑓,𝑔<∞∣𝑓𝑥,𝑦∈𝑆^𝑟_𝑚(Δ),𝑔𝑥,𝑦∈𝑆^𝑡_𝑛(Δ) (1.3) 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 𝑆^𝑟_𝑚(Δ) and 𝑆^𝑡_𝑛(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 BN(𝑚,𝑟;𝑛,𝑡;Δ)≤𝐵𝑁(𝑚,𝑟;𝑛,𝑟;Δ)≤𝑁𝑚𝑛. (1.4) 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 𝑆^𝑟_𝑚(Δ) and 𝑆^𝑡_𝑛(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 Δ_𝑘 the partition containing only 𝑘 parallel lines (Figure 1), we have the followeing theorem.

Figure 1 

Δ𝑘.

Theorem 2.1 (see [10]). One hasBN𝑚,𝑟;𝑛,𝑡;Δ_𝑘≤(𝑘+1)𝑚𝑛−min{𝑟,𝑡}𝑘. (2.1)

For a triangulation Δ, the Bezout number for two continuous piecewise algebraic curves satisfying the following theorem.

Theorem 2.2 (see [6]). Consider ()𝑉BN𝑚,0;𝑛,0;Δ≤𝑁𝑚𝑛−^odd+23, (2.2) where 𝑁 and 𝑉_odd 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).

Figure 2 

An interior star.

Figure 3 

A boundary star.

Theorem 2.3 (see [9]). If 𝑣 is an interior vertex, then BN(𝑚,1;𝑛,1;st(𝑣))≤𝑑(𝑣)𝑚𝑛−(𝑑(𝑣)−1), (2.3) and if 𝑣 is a boundary vertex, then BN(𝑚,1;𝑛,1;st(𝑣))≤(𝑑(𝑣)−1)𝑚𝑛−(𝑑(𝑣)−2). (2.4)

Based on Theorem 2.3, by a combinatorial optimization method, a result for a general nonobtuse triangulation is obtained.

Theorem 2.4 (see [7]). For any given nonobtuse triangulation Δ, BN(𝑚,1;𝑛,1;Δ)≤𝑁𝑚𝑛−2𝐸_𝐼−𝑉_𝐼−𝑉_𝐵3, (2.5) where 𝑁, 𝐸_𝐼, 𝑉_𝐼, and 𝑉_𝐵 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 [7]). For any nonobtuse star 𝑠𝑡(𝑣), if 𝑣 is an interior vertex, then ((𝑣(𝑣)𝑑(𝑣)𝑑(𝑣)BN𝑚,𝑟;𝑛,𝑟;st))≤𝑑𝑚𝑛−max−4,+12𝑟, (2.6) and if 𝑣 is a boundary vertex, then ((𝑣≤(𝑑(𝑣))𝑑(𝑣)𝑑(𝑣)BN𝑚,𝑟;𝑛,𝑟;st))−1𝑚𝑛−max−2,−12𝑟. (2.7)

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 BN(𝑚,𝑟;𝑛,𝑟;st(𝑣))≤(𝑑(𝑣)−1)𝑚𝑛−(𝑑(𝑣)−2)𝑟. (2.8)

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 [5, 8]). For any nonobtuse star 𝑠𝑡(𝑣), if 𝑣 is an interior vertex, then ((𝑣𝑑(𝑣)BN𝑚,𝑟;𝑛,𝑟;st))≤2)(𝑚𝑛−𝑟2, (2.9) and if 𝑣 is a boundary vertex, then BN(𝑚,𝑟;𝑛,𝑟;st(𝑣))≤(𝑑(𝑣)−1)(𝑚𝑛−𝑟)+𝑟. (2.10)

Here, we also remark that if 𝑣 is a boundary vertex, then the second formula is likewise equivalent to BN(𝑚,𝑟;𝑛,𝑟;st(𝑣))≤(𝑑(𝑣)−1)𝑚𝑛−(𝑑(𝑣)−2)𝑟. (2.11) 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 ()≤2BN𝑚,𝑟;𝑛,𝑟;Δ𝑁𝑚𝑛−3𝐸_𝐼2𝑟,𝑖𝑓𝑚𝑛−𝑟𝑖𝑠𝑒𝑣𝑒𝑛,𝑁𝑚𝑛−3𝐸_𝐼1𝑟−3𝑉_odd,𝑖𝑓𝑚𝑛−𝑟𝑖𝑠𝑜𝑑𝑑, (2.12) where 𝑉_odd 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∈𝑆^𝑟_𝑚(Δ) and 𝑔∈𝑆^𝑟_𝑛(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 _𝑣𝑘(𝑣)=3BN(𝑚,𝑟;𝑛,𝑟;Δ). (2.13) By Theorem 2.6, we have 𝑘(𝑣)≤BN(𝑚,𝑟;𝑛,𝑟;st(𝑣))≤𝛿(𝑣), (2.14) where 𝛿(𝑣)=𝑑(𝑣)(𝑣)𝑑(𝑣)(𝑣)(𝑑(𝑣))(𝑑(𝑣)))(𝑚𝑛−𝑟,if𝑑)(𝑚𝑛−𝑟iseven)(𝑚𝑛−𝑟−1,if𝑑)(𝑚𝑛−𝑟isoddif𝑣isaninteriorvertex−1𝑚𝑛−−2𝑟,if𝑣is𝑎boundaryvertex. (2.15) By the Euler formulae 𝐸_𝐼=𝑉_𝐵+3𝑉_𝐼−3,𝑁=𝑉_𝐵+2𝑉_𝐼−2 and the equations ∑_𝑣𝑑(𝑣)=2(𝐸_𝐼+𝐸_𝐵),𝐸_𝐵=𝑉_𝐵, if 𝑚𝑛−𝑟 is even, we get ()=1BN𝑚,𝑟;𝑛,𝑟;Δ3_𝑣𝑘(𝑣)≤13_𝑣𝛿(𝑣)=13_interior𝑣𝑑(𝑣)(𝑚𝑛−𝑟)+_boundary𝑣𝑑(𝑣))(𝑑(𝑣))𝑟)=1((−1𝑚𝑛−−23_𝑣𝑑(𝑣)−𝑉_𝐵1𝑚𝑛−3_𝑣𝑑(𝑣)−2𝑉_𝐵𝑟2=𝑁𝑚𝑛−3𝐸_𝐼𝑟. (2.16) Similarly, if 𝑚𝑛−𝑟 is odd, we have ()2BN𝑚,𝑟;𝑛,𝑟;Δ≤𝑁𝑚𝑛−3𝐸_𝐼1𝑟−3𝑉_odd. (2.17)

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 Δ_𝑎×𝑏 the partition; see Figure 4 for an example. Let 𝑉₀, 𝑉⁰₁, 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, 𝑉⁰₁, 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.

Figure 4 

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;Δ_𝑎×𝑏), where 𝑓∈𝑆^𝑟_𝑚(Δ_𝑎×𝑏) and 𝑔∈𝑆^𝑟_𝑛(Δ_𝑎×𝑏). If 𝑏=1, then Δ_𝑎×1 is a partition Δ_𝑎−1 containing only 𝑎−1 parallel lines essentially (Figure 1); by Theorem 2.1, we have BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×1≤𝑎𝑚𝑛−(𝑎−1)𝑟. (3.1) In the rest of this section, assume 𝑎≥𝑏≥2.

3.1. The First Method

For a vertex 𝑣 in Δ_𝑎×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 _𝑣𝑘(𝑣)=4BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏. (3.2) By Theorem 2.6, we have 𝑘(𝑣)≤BN(𝑚,𝑟;𝑛,𝑟;st(𝑣))≤𝛿(𝑣), (3.3) where 𝛿(𝑣)=4𝑚𝑛−4𝑟,𝑖𝑓𝑣∈𝑉₀,𝑑(𝑣)=42𝑚𝑛−𝑟,𝑖𝑓𝑣∈𝑉⁰₁,𝑑(𝑣)=3𝑚𝑛,𝑖𝑓𝑣∈𝑉¹₁,𝑑(𝑣)=2. (3.4) We have BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏=14_𝑣𝑘(𝑣)≤14_𝑣𝛿(𝑣)=14_𝑣∈𝑉0𝛿(𝑣)+_{𝑣∈𝑉01}𝛿(𝑣)+_{𝑣∈𝑉11}𝛿(𝑣=1)4{(4𝑚𝑛−4𝑟)(𝑎−1)(𝑏−1)+(2𝑚𝑛−𝑟)(2(𝑎+𝑏−2))+4𝑚𝑛}=𝑁𝑚𝑛−𝑟𝑁−𝑎+𝑏2. (3.5) 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 _𝑖,𝑗𝑘𝑖,𝑗=BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏. (3.6)

For the cells in the 𝑗th row, by Theorem 2.1, we have _𝑖𝑘𝑖,𝑗≤BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎−1≤𝑎𝑚𝑛−(𝑎−1)𝑟. (3.7) Hence, BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏=_𝑖,𝑗𝑘𝑖,𝑗=_𝑗𝑖𝑘≤(()𝑟)().𝑖,𝑗𝑎𝑚𝑛−𝑎−1𝑏=𝑁𝑚𝑛−𝑟𝑁−𝑏 (3.8) Similarly, BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏=_𝑖,𝑗𝑘𝑖,𝑗=_𝑖𝑗𝑘≤(()𝑟)().𝑖,𝑗𝑏𝑚𝑛−𝑏−1𝑎=𝑁𝑚𝑛−𝑟𝑁−𝑎 (3.9) 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;Δ_𝑎×𝑏) than BN₁, BN₂, and BN₃ by using a mixed method based on Theorems 2.1 and 2.6. We prefer this summation method.(1)If 𝑎 and 𝑏 are both even, then Δ_𝑎×𝑏=∪^𝑎/2_𝑝=1∪^𝑏/2_𝑞=1𝑠𝑡(𝑣(2𝑝−1,2𝑞−1)); so BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏=^𝑎/2_𝑝=1^𝑏/2_𝑞=1𝑘𝑣2𝑝−1,2𝑞−1≤^𝑎/2_𝑝=1^𝑏/2_𝑞=1𝛿𝑣=()×𝑎2𝑝−1,2𝑞−14𝑚𝑛−4𝑟2×𝑏2=𝑁𝑚𝑛−𝑁𝑟. (3.10)(2)If 𝑎 is odd and 𝑏 is even, then Δ_𝑎×𝑏=∪^{((𝑎−1)/2)}_𝑝=1∪^𝑏/2_𝑞=1𝑠𝑡(𝑣(2𝑝−1,2𝑞−1))∪^𝑏_𝑗=1𝐷(𝑎,𝑗); so we have BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏=^{((𝑎−1)/2)}_𝑝=1^𝑏/2_𝑞=1𝑘𝑣2𝑝−1,2𝑞−1+^𝑏_𝑗=1𝑘≤𝑎,𝑗_{((𝑎−1)/2)𝑝=1𝑏/2𝑞=1}𝛿𝑣2𝑝−1,2𝑞−1+𝐵𝑁𝑚,𝑟;𝑛,𝑟;Δ_𝑏−1=(4𝑚𝑛−4𝑟)×𝑎−12×𝑏2+(()𝑟)()𝑏𝑚𝑛−𝑏−1=𝑁𝑚𝑛−𝑁−1𝑟. (3.11)(3)If 𝑎 is even and 𝑏 is odd, then Δ_𝑎×𝑏=∪^𝑎/2_𝑝=1∪^{((𝑏−1)/2)}_𝑞=1𝑠𝑡(𝑣(2𝑝−1,2𝑞−1))∪^𝑎_𝑖=1𝐷(𝑖,𝑏); so BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏=^𝑎/2_𝑝=1^{((𝑏−1)/2)}_𝑞=1𝑘𝑣2𝑝−1,2𝑞−1+^𝑎_𝑖=1𝑘()≤𝑖,𝑏_{𝑎/2𝑝=1((𝑏−1)/2)𝑞=1}𝛿𝑣2𝑝−1,2𝑞−1+BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎−1=()×𝑎4𝑚𝑛−4𝑟2×𝑏−12+(()𝑟)()𝑎𝑚𝑛−𝑎−1=𝑁𝑚𝑛−𝑁−1𝑟. (3.12)(4)If 𝑎 and 𝑏 are both odd, then Δ_𝑎×𝑏=∪^{((𝑎−1)/2)}_𝑝=1∪^{((𝑏−1)/2)}_𝑞=1st(𝑣(2𝑝−1,2𝑞−1))∪^𝑏_𝑗=1𝐷(𝑎,𝑗)∪^𝑎−1_𝑖=1𝐷(𝑖,𝑏), hence BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏=^{((𝑎−1)/2)}_𝑝=1^{((𝑏−1)/2)}_𝑞=1𝑘𝑣2𝑝−1,2𝑞−1+^𝑏_𝑗=1𝑘𝑎,𝑗+^𝑎−1_𝑖=1𝑘()≤𝑖,𝑏_{((𝑎−1)/2)𝑝=1((𝑏−1)/2)𝑞=1}𝛿𝑣2𝑝−1,2𝑞−1+𝐵𝑁𝑚,𝑟;𝑛,𝑟;Δ_𝑏−1+BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎−2=(4𝑚𝑛−4𝑟)×𝑎−12×𝑏−12+(()𝑟)+)()𝑟)()𝑏𝑚𝑛−𝑏−1((𝑎−1𝑚𝑛−𝑎−2=𝑁𝑚𝑛−𝑁−2𝑟. (3.13)

Theorem 3.1. Let Δ_𝑎×𝑏 be a rectangular partition, then BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏≤()()()𝑁𝑚𝑛−𝑁𝑟,𝑖𝑓𝑎𝑎𝑛𝑑𝑏𝑎𝑟𝑒𝑏𝑜𝑡ℎ𝑒𝑣𝑒𝑛𝑁𝑚𝑛−𝑁−1𝑟,𝑖𝑓𝑎−𝑏𝑖𝑠𝑜𝑑𝑑𝑁𝑚𝑛−𝑁−2𝑟,𝑖𝑓𝑎𝑎𝑛𝑑𝑏𝑎𝑟𝑒𝑏𝑜𝑡ℎ𝑜𝑑𝑑. (3.14)

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.

Table 1 

Comparisons of 𝐵𝑁 and 𝐵𝑁₂ (where 𝑎≥𝑏≥2 and 𝑁=𝑎𝑏).

4. Conclusions

In this paper, we mainly derive an upper bound for the Bezout number BN(𝑚,𝑟;𝑛,𝑟;Δ_𝑎×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;Δ_𝑎×𝑏). In order to attract readers’ interest, here, we also give a conjecture on the Bezout number BN(𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏).

Conjecture 4.1. Consider BN𝑚,𝑟;𝑛,𝑟;Δ_𝑎×𝑏)+()+()()=𝑚𝑛+[(𝑎−1𝑏−1)](𝑚𝑛−𝑟𝑎−1)(𝑏−1)(𝑚𝑛−2𝑟=𝑁𝑚𝑛−2𝑁−𝑎−𝑏𝑟. (4.1) 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.