A worst-case bound for topology computation of algebraic curves

We consider the problem of approximating all real roots of a square-free polynomial f with real coefficients. Given isolating intervals for the real roots and an arbitrary positive integer L, the task is to approximate each root to L bits after the binary point. Abbott has proposed the quadratic interval refinement method (QIR for short), which is a variant of Regula Falsi combining the secant method and bisection. We formulate a variant of QIR, denoted AQIR ("Approximate QIR"), that considers only approximations of the polynomial coefficients and chooses a suitable working precision adaptively. It returns a certified result for any given real polynomial, whose roots are all simple. In addition, we propose several techniques to improve the asymptotic complexity of QIR: We prove a bound on the bit complexity of our algorithm in terms of the degree of the polynomial, the size and the separation of the roots, that is, parameters exclusively related to the geometric location of the roots. For integer coefficients, our variant improves, in theory and practice, the variant with exact integer arithmetic. Combining our approach with multipoint evaluation, we obtain near-optimal bounds that essentially match the best known theoretical bounds on root approximation as obtained by very sophisticated algorithms.

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“…For instance, we used this fact in [21] to derive considerably improved complexity bounds for the topology computation of algebraic plane curves.…”

Section: Discussionmentioning

confidence: 99%

Root refinement for real polynomials using quadratic interval refinement

Kerber

Sagraloff

2015

Journal of Computational and Applied Mathematics

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“…As a direct consequence, using our algorithm for computing a separating linear form directly yields a rational parameterization within the same overall complexity as our algorithm, both in the approach of Gonzalez-Vega et al [9,6] and in that of Bouzidi et al [4] for computing the alternative rational parameterization as defined in [13]. Moreover, this contribution is likely to impact the complexity of algorithms studying plane algebraic curves that require finding a shear that ensures the curves to be in "generic" position (such as [9,10]). In particular, it is hopeful that this result will improve the complexity of computing the topology of an algebraic plane curve.…”

Section: Introductionmentioning

confidence: 95%

“…We first outline a classical algorithm which is essentially the same as those proposed, for instance, in [6,Lemma 16] and [10,Thm. 24] 2 and whose complexity, in e OB(d 10 +d 9 τ ), is the best known so far for this problem.…”

Section: Overview and Organizationmentioning

confidence: 99%

“…Hence, a separating form can be found by computing, for every a in S, the degree of the squarefree part of R(T, a) and by choosing one a for which this degree is maximum. Indeed, for any (possibly non-separating) linear form X + aY , the number of distinct roots of R(T, a), which is the degree of its squarefree part, is always smaller than or equal to the number of distinct solutions of I, and equality is attained 2 The stated complexity of [10,Thm. 24] is e O B (d 9 τ ), but it seems the fact that the sheared polynomials have bitsize in e O(d+τ ) (see Lemma 5) instead of e O(τ ) has been overlooked in their proof.…”

Section: Known Ementioning

confidence: 99%

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Separating linear forms for bivariate systems

Bouzidi

Lazard

Pouget

et al. 2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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We present an algorithm for computing a separating linear form of a system of bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at distinct (complex) solutions of the system. In other words, a separating linear form defines a shear of the coordinate system that sends the algebraic system in generic position, in the sense that no two distinct solutions are vertically aligned. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and, moreover, the computation of a separating linear form is the bottleneck of these algorithms, in terms of worst-case bit complexity.Given two bivariate polynomials of total degree at most d with integer coefficients of bitsize at most τ , our algorithm computes a separating linear form in e OB(d2 ) bit operations in the worst case, where the previously known best bit complexity for this problem was e OB(d(where e O refers to the complexity where polylogarithmic factors are omitted and OB refers to the bit complexity).

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“…This is a classical problem in algorithmic real algebraic geometry with many applications in Computer Aided Geometric Design. It is extensively studied in the context of symbolic or semi-numerical computation (see for example [1,2,3,6,9,10,11,14,15,16,17,18,20,25] for recent references). Many papers are based on some variant of Cylindrical Decomposition : decompose the X-axis into a finite number of open intervals and points above which the curve has a cylindrical structure.…”

Section: Problem Description and Related Workmentioning

confidence: 99%

On the computation of the topology of plane curves

Diatta

Rouillier²,

Roy

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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International audienceLet P be a square free bivariate polynomial of degree at most d and with integer coefficients of bit size at most t. We give a deterministic algorithm for the computation of the topology of the real algebraic curve definit by P, i.e. a straight-line planar graph isotopic to the curve. Our main result is an algorithm for the computation of the local topology in a neighbourhood of each of the singular and critical points of the projection wrt the X axis in $\tilde{O} (d^6 t)$ bit operations where $\tilde{O}$ means that we ignore logarithmic factors in $d$ and $t$. Combined to state of the art sub-algorithms used for computing a Cylindrical Algebraic Decomposition, this result avoids a generic shear and gives a deterministic algorithm for the computation of the topology of the curve in $\tilde{O} (d^6 t + d^7)$ bit operations

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A worst-case bound for topology computation of algebraic curves

Cited by 28 publications

References 23 publications

Root refinement for real polynomials using quadratic interval refinement

Root refinement for real polynomials using quadratic interval refinement

Separating linear forms for bivariate systems

On the computation of the topology of plane curves

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