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- Eric Berberich, Pavel Emeliyanenko, Alexander Kobel, Michael Sagraloff
- Theor. Comput. Sci.
- 2013

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main… (More)

- Alexander Kobel, Michael Sagraloff
- ArXiv
- 2013

It is well known that, using fast algorithms for polynomial multiplication and division, evaluation of a polynomial F ∈ C[x] of degree n at n complex-valued points can be done with˜O(n) exact field operations in C, where˜O(·) means that we omit polylogarithmic factors. We complement this result by an analysis of approximate multipoint evaluation of F to a… (More)

We present a new <i>certified</i> and <i>complete</i> algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we… (More)

- Alexander Kobel, Michael Sagraloff
- J. Complexity
- 2015

In this paper, we give improved bounds for the computational complexity of computing with planar algebraic curves. More specifically, for arbitrary coprime polynomials f , g ∈ Z[x, y] and an arbitrary polynomial h ∈ Z[x, y], each of total degree less than n and with integer coefficients of absolute value less than 2 τ , we show that each of the following… (More)

- Alexander Kobel
- 2008

Non-plagiarism statement Hereby I confirm that this thesis is my own work and that I have documented all sources used. Statement of Consistence Hereby I confirm that this thesis is identical to the digital version submitted at the same time. Abstract In 1876, Alfred Bray Kempe stated a preliminary version of what nowadays is known as Kempe's Universality… (More)

- Alexander Kobel, Fabrice Rouillier, Michael Sagraloff
- ISSAC
- 2016

Very recent work introduces an asymptotically fast subdivision algorithm, denoted ANewDsc, for isolating the real roots of a univariate real polynomial. The method combines Descartes? Rule of Signs to test intervals for the existence of roots, Newton iteration to speed up convergence against clusters of roots, and approximate computation to decrease the… (More)

- Alexander Kobel, Michael Sagraloff
- ArXiv
- 2014

In this paper, we give improved bounds for the computational complexity of computing with planar algebraic curves. More specifically, for arbitrary coprime polynomials f , g ∈ Z[x, y] and an arbitrary polynomial h ∈ Z[x, y], each of total degree less than n and with integer coefficients of absolute value less than 2 τ , we show that each of the following… (More)

Hereby I confirm that this thesis is my own work and that I have documented all sources used. Statement of Consistency Hereby I confirm that this thesis is identical to the digital version submitted at the same time. Abstract Root isolation of univariate polynomials is one of the fundamental problems in computational algebra. It aims to find disjoint… (More)

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