Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set $F \subset {\R}[y_1, ..., y_n]$ we apply comprehensive triangular decomposition in order to obtain an $F$-invariant cylindrical decomposition of the $n$-dimensional complex space, from which we extract an $F$-invariant cylindrical algebraic decomposition of the $n$-dimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.
Deep Dive into Computing Cylindrical Algebraic Decomposition via Triangular Decomposition.
Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set $F \subset {\R}[y_1, ..., y_n]$ we apply comprehensive triangular decomposition in order to obtain an $F$-invariant cylindrical decomposition of the $n$-dimensional complex space, from which we extract an $F$-invariant cylindrical algebraic decomposition of the $n$-dimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.
Cylindrical algebraic decomposition (CAD) is a fundamental and powerful tool in real algebraic geometry. The original algorithm introduced by Collins in 1973 [11] has been followed by many substantial ameliorations, including adjacency and clustering techniques [2], improved projection methods [25,18,27,6], partially built CADs [13,26,29], improved stack construction [14] and efficient projection orders [16].
The main application of CAD is quantifier elimination (QE) for which other approaches are also available. Some of them have more attractive complexity results [4] than CAD. However, as pointed out by Brown and Davenport in [8], “there is the issue of whether the asymptotic cross-over points between CAD and those other QE algorithms actually occur in the range of problems that are even close to accessible with current machines”. In addition, these authors observe that CAD can help solving certain QE problems [7,19] that other QE algorithms can not.
For a finite set Fn ⊂ R[y1, . . . , yn] the CAD algorithm [11] decomposes the real n-dimensional space into disjoint cells C1, . . . , Ce together with one sample point Si ∈ Ci, for all 1 ≤ i ≤ e, such that the sign of each f ∈ Fn does not change in Ci and can be determined at Si. Besides, this decomposition is cylindrical in the following sense: For all 1 ≤ j < n the projections on the first j coordinates (y1, . . . , yj) of any two cells are either disjoint or equal. We will make use of this notion of “cylindrical” decomposition in C n .
The algorithm of Collins is based on a projection and lifting procedure which computes from Fn a finite set Fn-1 ⊂ R[y1, . . . , yn-1] such that an Fn-invariant CAD of R n can be constructed from an Fn-1-invariant CAD of R n-1 . This construction and the base case n = 1 rely on real root isolation of univariate polynomials.
In this paper, we propose a different approach for computing CAD, which proceeds by successive transformation of an initial decomposition of the complex n-dimensional space. Our algorithm consists of three main steps:
Initial Partition: we decompose C n into disjoint constructible sets C1, . . . , Ce such that for all 1 ≤ i ≤ e, for each f ∈ Fn either f is identically zero in Ci or f vanishes at no points of Ci.
Make Cylindrical: we transform the initial partition and obtain another decomposition of C n into disjoint constructible sets such that this second decomposition is cylindrical in the above sense.
Make Semi-Algebraic: from the previous decomposition we produce an Fn-invariant CAD of R n .
Our first motivation is to understand the relation and possible interaction between cylindrical algebraic decompositions and triangular decompositions of polynomial systems. This latter kind of decompositions have been intensively studied since the work of Wu [32]. The papers [3,5,20] and book [30] contain surveys of the subject. The primary goal of triangular decompositions is to provide unmixed decompositions of algebraic varieties. However, the third and fourth authors have initiated the use of triangular decompositions in real algebraic geometry [35]. Moreover, real root isolation of zero-dimensional polynomial systems can be achieved via triangular decompositions [33,34,10].
A second motivation of this work is to investigate the possibility of improving the practical efficiency of CAD implementation by means of modular methods and fast polynomial arithmetic. Such techniques have been successfully introduced into triangular decomposition methods [15,24,22]. Each of the three main steps of the algorithm proposed in this paper relies on existing sub-algorithms for triangular decompositions taken from [28,9,34] and for which efficient implementation in the RegularChains library [21] is work in progress based on the highly optimized low-level routines of the Modpn library [23].
Our third motivation is to extend to real algebraic geometry the concept of Comprehensive Triangular Decomposition (CTD) introduced in [9]. The relation between CAD and parametric polynomial system solving is natural as pointed in [17] and the presentation therein of Weispfenning’s approach [31] for QE based on comprehensive Gröbner bases. This suggests that the algorithm proposed in this paper could support a similar QE method.
This paper is organized as follows. A summary of the theory of triangular decomposition is given in Section 2. Section 3 and Section 4 are dedicated to the first two main steps of our algorithm whereas Sections 5 presents the last one. In Section 6 we report on a preliminary experimentation of our new algorithm. No modular methods or fast polynomial arithmetic are being used yet and our code is just highlevel Maple interpreted code. However our code can already process well-known examples from the literature. We also analyze the performances of the different main steps and subroutines of our algorithm and implementation. This suggests that there is a large potential for improvement by means of modular m
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