On the complexity of real solving bivariate systems

Journal of Symbolic Computation, 2009

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of e OB(N 14 ) for the purely projection-based method, and e OB(N 12 ) for two subresultant-based methods: this notation ignores polylogarithmic factors, where N bounds the degree, and the bitsize of the polynomials. The previous record bound was e OB(N 14 ).

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation - ISSAC '12, 2012

We study the complexity of computing the real solutions of a bivariate polynomial system using the recently proposed algorithm Bisolve [3]. Bisolve is a classical elimination method which first projects the solutions of a system onto the x-and y-axes and, then, selects the actual solutions from the so induced candidate set. However, unlike similar algorithms, Bisolve requires no genericity assumption on the input nor it needs any change of the coordinate system. Furthermore, extensive benchmarks from [3] confirm that the algorithm outperforms state of the art approaches by a large factor. In this work, we show that, for two polynomials f, g ∈ Z[x, y] of total degree at most n with integer coefficients bounded by 2 τ , Bisolve computes isolating boxes for all real solutions of the system f = g = 0 usingÕ(n 8 τ 2) bit operations 1 , thereby improving the previous record bound by a factor of at least n 2 .

Lecture Notes in Computer Science, 2005

We propose exact, complete and efficient methods for 2 problems: First, the real solving of systems of two bivariate rational polynomials of arbitrary degree. This means isolating all common real solutions in rational rectangles and calculating the respective multiplicities. Second, the computation of the sign of bivariate polynomials evaluated at two algebraic numbers of arbitrary degree. Our main motivation comes from nonlinear computational geometry and computer-aided design, where bivariate polynomials lie at the inner loop of many algorithms. The methods employed are based on Sturm-Habicht sequences, univariate resultants and rational univariate representation. We have implemented them very carefully, using advanced object-oriented programming techniques, so as to achieve high practical performance. The algorithms are integrated in the public-domain C++ software library synaps, and their efficiency is illustrated by 9 experiments against existing implementations. Our code is faster in most cases; sometimes it is even faster than numerical approaches.

Lecture Notes in Computer Science, 2008

We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ , using Sturm (-Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of e OB(d 4 τ 2 ). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non square-free polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities (SI) and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some experimentations on various data sets.

International Mathematical Forum, 2010

This paper presents a lecture on existing algorithms for solving polynomial systems with their complexity analysis from our experiments on the subject. It is based on our studies of the complexity of solving parametric polynomial systems. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. The paper provides different methods and techniques used for representing solutions of algebraic systems that include Rational Univariate Representations (RUR), Gröbner bases, etc.

2005

We present algorithmic and complexity results concerning computations with one and two real algebraic numbers, as well as real solving of univariate polynomials and bivariate polynomial systems with integer coefficients using Sturm-Habicht sequences.

Computing Research Repository, 2010

We present a new algorithm for solving the real roots of a bivariate polynomial system Σ = { f (x, y), g(x, y)} with a finite number of solutions by using a zero-matching method. The method is based on a lower bound for bivariate polynomial system when the system is non-zero. Moreover, the multiplicities of the roots of Σ = 0 can be obtained by a given neighborhood. From this approach, the parallelization of the method arises naturally. By using a multidimensional matching method this principle can be generalized to the multivariate equation systems.

Journal of Symbolic Computation, 1990

This paper is devoted to a precise algorithmical and complexity study of a new polynomial time method for formal computations with polynomial inequalities and real algebraic numbers.

Theoretical Computer Science, 2008

We present exact and complete algorithms based on precomputed Sturm-Habicht sequences, discriminants and invariants, that classify, isolate with rational points and compare the real roots of polynomials of degree up to 4. We have closed formulas for all isolating points. Moreover we combine these results with a simple version of rational univariate representation so as to isolate and compute the multiplicity of all common real roots of a bivariate system of integer polynomials of total degree ≤ 2. We present our implementation within synaps and we perform experimentation and comparison with all available software. Our package is 2-10 times faster, even when compared to inexact software or to sofware with intrinsic filtering.

2011 Proceedings of the Thirteenth Workshop on Algorithm Engineering and Experiments (ALENEX), 2011

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-of-the-art implementations, that is, Lgp and Maple's Isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.