A polynomial-time algorithm for the topological type of a real algebraic curve
Journal of Symbolic Computation
An improved upper complexity bound for the topology computation of a real algebraic plane curve
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An efficient method for analyzing the topology of plane real algebraic curves
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ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
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Fundamental problems of algorithmic algebra
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Efficient topology determination of implicitly defined algebraic plane curves
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On the exact computation of the topology of real algebraic curves
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On the asymptotic and practical complexity of solving bivariate systems over the reals
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On the topology of planar algebraic curves
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An efficient algorithm for the stratification and triangulation of an algebraic surface
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A generic algebraic kernel for non-linear geometric applications
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On the complexity of solving a bivariate polynomial system
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From approximate factorization to root isolation
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On the boolean complexity of real root refinement
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Journal of Symbolic Computation
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Computing the topology of an algebraic plane curve C means computing a combinatorial graph that is isotopic to C and thus represents its topology in R^2. We prove that, for a polynomial of degree n with integer coefficients bounded by 2^@r, the topology of the induced curve can be computed with O@?(n^8@r(n+@r)) bit operations (O@? indicates that we omit logarithmic factors). Our analysis improves the previous best known complexity bounds by a factor of n^2. The improvement is based on new techniques to compute and refine isolating intervals for the real roots of polynomials, and on the consequent amortized analysis of the critical fibers of the algebraic curve.