A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems

We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly involves resultant computation and real root isolation of univariate polynomial equations. The roots of the system have a linear univariate representation. The complexity of the method is $\tilde{O}_B(N^{10})$ for the bivariate case, where $N=\max(d,\tau)$, $d$ resp., $\tau$ is an upper bound on the degree, resp., the maximal coefficient bitsize of the input polynomials. The algorithm is certified with probability 1 in the multivariate case. The implementation shows that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure

Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers

We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., $(x^n \in \mathbb{Q})$ and $(x^n \in \mathbb{Z})$. Our decision procedure combines computation over real algebraic cells with the rational root theorem and witness construction via algebraic number density arguments.Comment: To appear in CADE-25: 25th International Conference on Automated Deduction, 2015. Proceedings to be published by Springer-Verla

On Continued Fraction Expansion of Real Roots of Polynomial Systems, Complexity and Condition Numbers

International audienceWe elaborate on a correspondence between the coeffcients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions, that use only integer arithmetic (in contrast to Bernstein basis) and are feasible over unbounded regions. Then, we study an algorithm to split this representation and we obtain a subdivision scheme for the domain of multivariate polynomial functions. This implies a new algorithm for real root isolation, MCF, that generalizes the Continued Fraction (CF) algorithm of univariate polynomials. A partial extension of Vincent's Theorem for multivariate polynomials is presented, which allows us to prove the termination of the algorithm. Bounding functions, projection and preconditioning are employed to speed up the scheme. The resulting isolation boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. Finally, we present new complexity bounds for a simplified version of the algorithm in the bit complexity model, and also bounds in the real RAM model for a family of subdivision algorithms in terms of the real condition number of the system. Examples computed with our C++ implementation illustrate the practical aspects of our method

On the asymptotic and practical complexity of solving bivariate systems over the reals

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 \sOB(N^{14}) for the purely projection-based method, and \sOB(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 \sOB(N^{14}). Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over two algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in \sOB(N^{12}), whereas the previous bound was \sOB(N^{14}). All algorithms have been implemented in MAPLE, in conjunction with numeric filtering. We compare them against FGB/RS, system solvers from SYNAPS, and MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/C++ libraries. Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure