Recursive Polynomial Remainder Sequence and its Subresultants

We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated "recursively" for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of initial polynomials. We give three different constructions of subresultant matrices for recursive subresultants; while the first one is built-up just with previously defined matrices thus the size of the matrix increases fast as the recursion deepens, the last one reduces the size of the matrix drastically by the Gaussian elimination on the second one which has a "nested" expression, i.e. a Sylvester matrix whose elements are themselves determinants.Comment: 30 pages. Preliminary versions of this paper have been presented at CASC 2003 (arXiv:0806.0478 [math.AC]) and CASC 2005 (arXiv:0806.0488 [math.AC]

Recursive Polynomial Remainder Sequence and the Nested Subresultants

We give two new expressions of subresultants, nested subresultant and reduced nested subresultant, for the recursive polynomial remainder sequence (PRS) which has been introduced by the author. The reduced nested subresultant reduces the size of the subresultant matrix drastically compared with the recursive subresultant proposed by the authors before, hence it is much more useful for investigation of the recursive PRS. Finally, we discuss usage of the reduced nested subresultant in approximate algebraic computation, which motivates the present work.Comment: 12 pages. Presented at CASC 2005 (Kalamata, Greece, Septermber 12-16, 2005

On the complexity of real solving bivariate systems

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.

Geometric algorithms for algebraic curves and surfaces

This work presents novel geometric algorithms dealing with algebraic curves and surfaces of arbitrary degree. These algorithms are exact and complete — they return the mathematically true result for all input instances. Efficiency is achieved by cutting back expensive symbolic computation and favoring combinatorial and adaptive numerical methods instead, without spoiling exactness in the overall result. We present an algorithm for computing planar arrangements induced by real algebraic curves. We show its efficiency both in theory by a complexity analysis, as well as in practice by experimental comparison with related methods. For the latter, our solution has been implemented in the context of the Cgal library. The results show that it constitutes the best current exact implementation available for arrangements as well as for the related problem of computing the topology of one algebraic curve. The algorithm is also applied to related problems, such as arrangements of rotated curves, and arrangments embedded on a parameterized surface. In R3, we propose a new method to compute an isotopic triangulation of an algebraic surface. This triangulation is based on a stratification of the surface, which reveals topological and geometric information. Our implementation is the first for this problem that makes consequent use of numerical methods, and still yields the exact topology of the surface.Diese Arbeit stellt neue Algorithmen für algebraische Kurven und Flächen von beliebigem Grad vor. Diese Algorithmen liefern für alle Eingaben das mathematisch korrekte Ergebnis. Wir erreichen Effizienz, indem wir aufwendige symbolische Berechnungen weitesgehend vermeiden, und stattdessen kombinatorische und adaptive numerische Methoden einsetzen, ohne die Exaktheit des Resultats zu zerstören. Der Hauptbeitrag ist ein Algorithmus zur Berechnung von planaren Arrangements, die durch reelle algebraische Kurven induziert sind. Wir weisen die Effizienz des Verfahrens sowohl theoretisch durch eine Komplexitätsanalyse, als auch praktisch durch experimentelle Vergleiche nach. Dazu haben wir unser Verfahren im Rahmen der Softwarebibliothek Cgal implementiert. Die Resultate belegen, dass wir die zur Zeit beste verfügbare exakte Software bereitstellen. Der Algorithmus wird zur Arrangementberechnung rotierter Kurven, oder für Arrangements auf parametrisierten Oberflächen eingesetzt. Im R3 geben wir ein neues Verfahren zur Berechnung einer isotopen Triangulierung einer algebraischen Oberfläche an. Diese Triangulierung basiert auf einer Stratifizierung der Oberfläche, die topologische und geometrische Informationen berechnet. Unsere Implementierung ist die erste für dieses Problem, welche numerische Methoden konsequent einsetzt, und dennoch die exakte Topologie der Oberfläche liefert

New structure theorem for subresultants

We give a new structure theorem for subresultants precising their gap structure and derive from it a new algorithm for computing them. If d is a bound on the degrees and τ a bound on the bitsize of the minors extracted from Sylvester matrix, our algorithm has O(d²) arithmetic operations and size of intermediate computations 2τ. The key idea is to precise the relations between the successive Sylvester matrix of A and B in one hand and of A and XB on the other hand, using the notion of G-remainder we introduce. We also compare our new algorithm with another algorithm with the same characteristics already appeared in [4]