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

Abstract

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 OB(N 14 ) for the purely projection-based method, and 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 OB(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 OB(N 12 ), whereas the previous bound was OB(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.

Rational univariate representations of bivariate systems and applications

“…These equalities hold for infinitely many values of a, and ri(S), LR(S) and fi(S) are polynomials in S, thus ri(S) = LR(S)fi(S) and, by (8), R(T, S) = LR(S)fI (T, S).…”

“…We addressed in [4] the first phase of the above algorithm and proved that a separating linear form 1 The complexity of the isolation phase in [8,Theorem 19] is stated as e O B (d 12 +d 10 τ 2 ) but it trivially decreases to e O B (d 10 +d 9 τ ) with the recent result of Sagraloff [17] which improves the complexity of isolating the real roots of a univariate polynomial.…”

“…The idea of this algorithm comes originally from [12], where the Cauchy index of two polynomials is computed by means of sign variations of a particular remainder sequence called the Sylvester-Habicht sequence. In [8], this approach is slightly adapted to deduce the sign from the Cauchy index ( [21,Thm. 7.3]) and the bit complexity is given in terms of the two initial degrees and bitsizes.…”

Separating linear forms for bivariate systems

“…The computation of such parameterizations has been a focus of interest for a long time; see for example [1,9,13,8,3,6] and references therein. Most algorithms first shear the coordinate system, with a linear change of variables, so that the input algebraic system is in generic position, that is such that no two solutions are vertically aligned.…”

A Note on the Complexity of Real Algebraic Hypersurfaces

“…For the planar case, the complexity of the problem has been upper bounded by O(N 12 ) [8,10], where N is defined as the maximum of the degree of f and the bitsize of its coefficients. However, our question of how many segment/triangles are needed in principal to capture the topology of the object seems to be untreated in this context.…”