On the complexity of real solving bivariate systems

Abstract

This paper is concerned with exact real solving of well-constrained, bivariate algebraic 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 O B (N 14 ) for the purely projectionbased method, and O B (N 12 ) for two subresultant-based methods: we ignore polylogarithmic factors, and N bounds the degree and the bitsize of the polynomials. The previous record bound was O B (N 16 ). Our main tool is signed subresultant sequences, extended to several variables by the technique of binary segmentation. We exploit recent advances on the complexity of univariate root isolation, and extend them to multipoint evaluation, 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 O B (N 12 ), whereas the previous bound was O B (N 16 ). All algorithms have been implemented in maple, in conjunction with numeric filtering. We compare them against fgb/rs and system solvers from synaps; we also consider 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.

Matrix Representations by Means of Interpolation

“…If these polynomials have total degree δ in s,t and bitsize τ, then each system is solved in bit complexity O * (δ 12 + δ 10 τ 2 ), see e.g. [DET09], and this is repeated for each of the O(D) roots ρ j . It turns out that this is the bottleneck in practice, which can be explained by the analysis since D is only a few times larger than d. For example, the bicubic surface has D = 18 and δ = 6.…”

On the Complexity of Reliable Root Approximation

“…Also, they are relatively easy to implement, and have shown good practical performance. Real root solving is a cornerstone, for instance, for the computation of Cylindrical Algebraic Decomposition [4], for related problems such as topology computation [11,8] and arrangement computation [10], and many more.…”

“…It is used both in the COCOA library 1 [1] and the (experimental) algebraic kernel of the CGAL library 2 (used, for instance, in [11,10]). Its application is also attested in [8]. In this work, however, we focus on the complexity analysis, and do not address its practical performance.…”

On the Topology of Real Algebraic Plane Curves

“…Previous Work There have been many papers addressing the problem of computing the topology of algebraic plane curves (or closely related problems) defined by a bivariate polynomial with rational coefficients [1,3,4,8,10,18,19,22,26,30,32,33,40,41,46,48]. Most of the algorithms assume generic position for the input curve.…”

“…As for the complexity of the problem, the best known bound so far is O(N 12 ) [19], where N is essentially the maximum of the degree of f and of the maximum number of bits needed for representing the input coefficients and the notation O denotes that the poly-logarithmic factors are omitted. This assumes that the real algebraic numbers are represented by isolating intervals.…”