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 $\sOB(N^{14})$ for the purely projection-based method, and $\sOB(N^{12})$ for two sub\-result\-ant-based methods: we ignore polylogarithmic factors, and $N$ bounds the degree and the bitsize of the polynomials. The previous record bound was $\sOB(N^{14})$. Our main tool is signed subresultant sequences, 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, % We thus derive new bounds for the sign evaluation of bi- and multi-variate polynomials 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 $\sOB( N^{16})$. All algorithms have been implemented in \maple, in conjunction with numeric filtering. We compare them against \gbrs and system solvers from \synaps; we also consider \maple libraries \func{insulate} and \func{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. }
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[Research Report] RR-6116, INRIA. 2007
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Dimitrios Diochnos, Ioannis Z. Emiris, Elias Tsigaridas. On the complexity of real solving bivariate systems. [Research Report] RR-6116, INRIA. 2007. 〈inria-00129309v5〉

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