Lower bounds for zero-dimensional projections

Brownawell, W. Dale; Yap, Chee

doi:10.1145/1576702.1576716

Cited by 9 publications

(29 citation statements)

References 31 publications

(24 reference statements)

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“…They are comparable to those in [5] on the absolute value of root coordinates, but they are an improvement when expressed using mixed volumes. It seems nontrivial to apply sparse elimination theory to the approach of [5]. More importantly, our result is extended to positive-dimensional systems, thus addressing a problem that has only been examined very recently in [5].…”

Section: Introductionmentioning

confidence: 60%

“…One application is to the bitsize of the eigenvalues and eigenvectors of an integer matrix, which also yields a new proof that the problem is polynomial. We also compare against recent lower bounds on the absolute value of the root coordinates by Brownawell and Yap [5], obtained under the hypothesis there is a 0-dimensional projection. Our bounds are in general comparable, but exploit sparseness; they are also tighter when bounding the value of a positive polynomial over the simplex.…”

mentioning

confidence: 90%

“…It seems nontrivial to apply sparse elimination theory to the approach of [5]. More importantly, our result is extended to positive-dimensional systems, thus addressing a problem that has only been examined very recently in [5].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The DMM bound

Emiris

Mourrain

Tsigaridas

2010

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

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In this paper we derive aggregate separation bounds, named after Davenport-Mahler-Mignotte (DMM), on the isolated roots of polynomial systems, specifically on the minimum distance between any two such roots. The bounds exploit the structure of the system and the height of the sparse (or toric) resultant by means of mixed volume, as well as recent advances on aggregate root bounds for univariate polynomials, and are applicable to arbitrary positive dimensional systems. We improve upon Canny's gap theorem [7] by a factor of O(d n−1 ), where d bounds the degree of the polynomials, and n is the number of variables. One application is to the bitsize of the eigenvalues and eigenvectors of an integer matrix, which also yields a new proof that the problem is polynomial. We also compare against recent lower bounds on the absolute value of the root coordinates by Brownawell and Yap [5], obtained under the hypothesis there is a 0-dimensional projection. Our bounds are in general comparable, but exploit sparseness; they are also tighter when bounding the value of a positive polynomial over the simplex. For this problem, we also improve upon the bounds in [2,16]. Our analysis provides a precise asymptotic upper bound on the number of steps that subdivision-based algorithms perform in order to isolate all real roots of a polynomial system. This leads to the first complexity bound of Milne's algorithm [22] in 2D.

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Section: Introductionmentioning

confidence: 60%

mentioning

confidence: 90%

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The DMM bound

Emiris

Mourrain

Tsigaridas

2010

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

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“…Using the results of [20] (see also [8,12,24]) we have the following Proposition 9 [20] Let P ∈ Z[X, Y ] be a square-free polynomial of total degree d and integer coefficients of bitsize bounded by τ . Supposing that, for every real root α of D, deg(gcd(P (α, Y )), ∂Y P (α, Y )) is known, there is an algorithm with bit complexityÕ(d 5 τ + d 6 ) for − computing a set of special boxes …”

Section: Cylindrical Algebraic Decompositionmentioning

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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“…The multivariate separation bounds that we use, were introduced in [17]. For other multivariate separation bounds we refer the reader to [3,5,37]. Proposition 1.…”

Section: The Dmm Boundmentioning

confidence: 99%

Univariate real root isolation in multiple extension fields

Strzeboński

Tsigaridas²

2012

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

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We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in Bα ∈ L[y], where L = ( Q(α1, . . . , α ) is an algebraic extension of the rational numbers. Our bounds are single exponential in and match the ones presented in [34] for the case = 1. We consider two approaches. The first, indirect approach, using multivariate resultants, computes a univariate polynomial with integer coefficients, among the real roots of which are the real roots of Bα. The Boolean complexity of this approach is OB(N 4 +4 ), where N is the maximum of the degrees and the coefficient bitsize of the involved polynomials. The second, direct approach, tries to solve the polynomial directly, without reducing the problem to a univariate one. We present an algorithm that generalizes Sturm algorithm from the univariate case, and modified versions of well known solvers that are either numerical or based on Descartes' rule of sign. We achieve a Boolean complexity of OB(min{N 4 +7 , N 2 2 +6 }) and OB(N 2 +4 ), respectively. We implemented the algorithms in C as part of the core library of mathematica and we illustrate their efficiency over various data sets.

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Lower bounds for zero-dimensional projections

Cited by 9 publications

References 31 publications

The DMM bound

The DMM bound

On the computation of the topology of plane curves

Univariate real root isolation in multiple extension fields

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