On the Singularities at Infinity of Plane Algebraic Curves

Sagraloff

2015

Journal of Complexity

In this paper, we give improved bounds for the computational complexity of computing with planar algebraic curves. More specifically, for arbitrary coprime polynomials f , g ∈ Z[x, y] and an arbitrary polynomial h ∈ Z[x, y], each of total degree less than n and with integer coefficients of absolute value less than 2 τ , we show that each of the following problems can be solved in a deterministic way with a number of bit operations bounded byÕ(n 6 + n 5 τ), where we ignore polylogarithmic factors in n and τ:• The computation of isolating regions in C 2 for all complex solutions of the system f = g = 0,• the computation of a separating form for the solutions of f = g = 0,• the computation of the sign of h at all real valued solutions of f = g = 0, and• the computation of the topology of the planar algebraic curve C defined as the real valued vanishing set of the polynomial f .Our bound improves upon the best currently known bounds for the first three problems by a factor of n 2 or more and closes the gap to the state-of-the-art randomized complexity for the last problem.

Section: Introductionmentioning

confidence: 99%

On the complexity of computing with planar algebraic curves

Sagraloff

2015

Journal of Complexity

“…Computation of n + α . The following result due to Teissier [17,18] is crucial for our approach: Lemma 2 (Teissier). For an x-critical point p = (α, β) of C, it holds that…”

Section: Lift-nt-a Symbolic-numeric Approach For Fiber Computationmentioning

confidence: 99%

“…Namely, as in the algorithm Bisolve, we managed to reduce the amount of purely symbolic computations, that is, we only use resultants and gcds, where both computations are outsourced again to graphics hardware. Furthermore, based on a result from Teissier [17,18] which relates the intersection multiplicities of the curves f , f x and f y , and the multiplicity of a root of f α , we derive additional information about the number n α of distinct complex roots of f α . In fact, we compute an upper bound n + α which matches n α except in the case where the curve C is in a very special geometric location.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exact symbolic–numeric computation of planar algebraic curves

Berberich

Emeliyanenko

Theoretical Computer Science

et al. 2013

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic curve.Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of involved exact operations, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry.We have implemented our algorithms as prototypical contributions to the C++-project Cgal. They exploit graphics hardware to expedite the symbolic computations. We have also compared our implementation with the current reference implementations, that is, Lgp and Maple's Isolate for polynomial system solving, and Cgal's bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones. (Michael Sagraloff) 1 G C is isotopic to C if there exists a continuous mapping φ :Outline. The bivariate solver Bisolve is discussed in Section 2. In Section 3, we introduce GeoTop to analyze a single algebraic curve. The latter section particularly features two parts, that is, the presentation of a complete method Lift-BS in Section 3.2.1 that is based on Bisolve, and the presentation of the symbolic-numeric method Lift-NT in Section 3.2.2. Lift-NT uses a numerical solver whose details are given in Appendix A. Bisolve and GeoTop are finally utilized in Section 4 in order to enable the computation of arrangements of algebraic curves. The presented algorithms allow speedups, among other things, due to the use of graphics hardware for symbolic operations as described in Section 5. Our algorithms are prototypically implemented in the Cgal project. Section 6 gives necessary details and also features many experiments that show the performance of the new approach. We conclude in Section 7 and outline further directions of research.

“…The main reason for this approach is that we can outsource both computations to graphics hardware [20,21,22], removing a bottleneck of previous methods which was due to the high amount of symbolic operations. Secondly, for curve analysis, we use a result from Teissier [24,33] to obtain additional information for the number of distinct complex roots of f (α, y) along a critical fiber (actually, an upper bound which most likely matches the exact number). We combine this information with a new certified complex root solver [26] to isolate the roots of f (α, y).…”

Section: Introductionmentioning

confidence: 99%

Arrangement computation for planar algebraic curves

Berberich

Emeliyanenko

Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation

et al. 2012

We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly reduce the amount of exact operations, that is, our algorithms only uses resultant and gcd as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm which combines the usage of a certified numerical complex root solver and information derived from the resultant computation. Additionally, we never consider any coordinate transformation and the output is also given with respect to the initial coordinate system. We implemented our algorithm as a prototypical package of the C++-library Cgal. Our implementation exploits graphics hardware to expedite the resultant and gcd computation. We also compared our implementation with the current reference implementation, that is, Cgal's curve analysis and arrangement for algebraic curves. For various series of challenging instances, our experiments show that the new implementation outperforms the existing one.