Tropical algebraic geometry in Maple: A preprocessing algorithm for finding common factors for multivariate polynomials with approximate coefficients

Adrovic, Danko; Verschelde, Jan

doi:10.1016/j.jsc.2010.08.011

Cited by 10 publications

(9 citation statements)

References 55 publications

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“…In [1,Proposition 5.3], it is stated that, given f i (X 1 , X 2 ) = p i (X 2 ) + P i (X 1 , X 2 ) for i = 1, 2, where p i have nonzero constant term and all terms in P i have a positive power in X 1 , and…”

Section: And Thementioning

confidence: 99%

“…For a system of two polynomials in two variables, in [1,Proposition 5.3] it is stated that, under certain hypotheses, if the second term in the Puiseux series expansion at a common root ξ can be computed, then there exists a curve of solutions for the original system; however, the result does not hold for arbitrary bivariate polynomial systems. In [2] the authors extend this result to the case of n variables and apply it successfully to produce exact representations for solution sets of the cyclic n-roots problem.…”

Section: Introductionmentioning

confidence: 99%

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Puiseux Expansions and Nonisolated Points in Algebraic Varieties

Herrero

Jeronimo

Sabia

2016

Communications in Algebra

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We consider the problem of deciding whether a common solution to a multivariate polynomial equation system is isolated or not. We present conditions on a given truncated Puiseux series vector centered at the point ensuring that it is not isolated. In addition, in the case that the set of all common solutions of the system has dimension 1, we obtain further conditions specifying to what extent the given vector of truncated Puiseux series coincides with the initial part of a parametrization of a curve of solutions passing through the point.

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Section: And Thementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Puiseux Expansions and Nonisolated Points in Algebraic Varieties

Herrero

Jeronimo

Sabia

2016

Communications in Algebra

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“…Once we have those roots, we can further grow the Puiseux series expansion symbolically, or apply numerical predictor-corrector methods to sample points along the solution curve. In Figure 4 (slightly adapted from [1]) we sketch the idea for computing this certificate.…”

Section: Tropisms and Initial Formsmentioning

confidence: 99%

Polyhedral methods in numerical algebraic geometry

Verschelde¹

2009

Interactions of Classical and Numerical Algebraic Geometry

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In numerical algebraic geometry witness sets are numerical representations of positive dimensional solution sets of polynomial systems. Considering the asymptotics of witness sets we propose certificates for algebraic curves. These certificates are the leading terms of a Puiseux series expansion of the curve starting at infinity. The vector of powers of the first term in the series is a tropism. For proper algebraic curves, we relate the computation of tropisms to the calculation of mixed volumes. With this relationship, the computation of tropisms and Puiseux series expansions could be used as a preprocessing stage prior to a more expensive witness set computation. Systems with few monomials have fewer isolated solutions and fewer data are needed to represent their positive dimensional solution sets.

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“…Our Contributions. This paper is a thorough revision of the unpublished preprint [2], originating in the dissertation of the first author [1], which extended [3] from the plane to space curves. In [4] we gave a tropical version of Backelin's Lemma in case n = m 2 , in this paper we generalize to the case n = ℓm 2 .…”

Section: Introductionmentioning

confidence: 99%

Polyhedral Methods for Space Curves Exploiting Symmetry Applied to the Cyclic n-roots Problem

Adrovic

Verschelde

2013

Computer Algebra in Scientific Computing

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We present a polyhedral algorithm to manipulate positive dimensional solution sets. Using facet normals to Newton polytopes as pretropisms, we focus on the first two terms of a Puiseux series expansion. The leading powers of the series are computed via the tropical prevariety. This polyhedral algorithm is well suited for exploitation of symmetry, when it arises in systems of polynomials. Initial form systems with pretropisms in the same group orbit are solved only once, allowing for a systematic filtration of redundant data. Computations with cddlib, Gfan, PHCpack, and Sage are illustrated on cyclic n-roots polynomial systems.

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Tropical algebraic geometry in Maple: A preprocessing algorithm for finding common factors for multivariate polynomials with approximate coefficients

Cited by 10 publications

References 55 publications

Puiseux Expansions and Nonisolated Points in Algebraic Varieties

Puiseux Expansions and Nonisolated Points in Algebraic Varieties

Polyhedral methods in numerical algebraic geometry

Polyhedral Methods for Space Curves Exploiting Symmetry Applied to the Cyclic n-roots Problem

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