Tropical Implicitization and Mixed Fiber Polytopes

Sturmfels, Bernd; Yu, Josephine

doi:10.1007/978-0-387-78133-4_7

Cited by 29 publications

(41 citation statements)

References 18 publications

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“…Then the vector p of coefficients of the implicit equation is in the kernel of M . This idea was used in [11,13,19,23]; it is also the starting point of this paper.…”

Section: Previous Workmentioning

confidence: 99%

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Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

Emiris

Kalinka

Konaxis

et al. 2013

Computer-Aided Design

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We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points. Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring). All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface. For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system. This yields an efficient, output-sensitive algorithm for computing the discriminant polynomial.

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“…Then the vector p of coefficients of the implicit equation is in the kernel of M . This idea was used in [11,13,19,23]; it is also the starting point of this paper.…”

Section: Previous Workmentioning

confidence: 99%

“…There are methods for the computation of the implicit polytope based on tropical geometry [22,23], see also [5]. Our method relies on sparse elimination theory.…”

Section: Support Prediction -The Software Respolmentioning

confidence: 99%

Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

Emiris

Kalinka

Konaxis

et al. 2013

Computer-Aided Design

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show abstract

“…In this paper, we continue the study of the following two lines of research and in particular advance their connections. Firstly, tropical elimination as developed in [18,19,20] (see also [6,7]) can be seen as tropical analog of elimination theory. Secondly, a natural way to handle an ideal is by means of a basis, i.e., a finite set of generators.…”

Section: Introductionmentioning

confidence: 99%

“…As a basic ingredient, we start by studying the bases from the viewpoint of tropical elimination and its relations to mixed fiber polytopes as recently developed by Sturmfels, Tevelev and Yu [18,19,20]; see also [6,7]. In these papers, it is shown that in various situations the Newton polytope of the polynomial generating this hypersurface is affinely isomorphic to a mixed fiber polytope.…”

Section: Introductionmentioning

confidence: 99%

Projections of tropical varieties and their self-intersections

Hept¹,

Theobald²

2012

Advances in Geometry

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Abstract. We study algebraic and combinatorial aspects of (classical) projections of m-dimensional tropical varieties onto (m + 1)-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work of the authors on projection-based tropical bases, we characterize algebraic properties of the relevant ideals and provide a characterization of the dual subdivision (as a subdivision of a fiber polytope). This dual subdivision naturally leads to the issue of self-intersections of a tropical variety under projections. For the case of curves, we provide some bounds for the (unweighted) number of self-intersections of projections onto the plane and give constructions with many self-intersections.

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“…The purpose of this paper is to revisit this approach from the computational point of view, with the goal to provide an explicit and constructive description of the resulting tropical bases. Specifically, we apply tropical elimination on a particular class of ideals; for a general treatment of tropical elimination see the recent papers of Sturmfels, Tevelev, and Yu [12,13].…”

Section: Introductionmentioning

confidence: 99%

Tropical bases by regular projections

Hept

Theobald

2009

Proc. Amer. Math. Soc.

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Abstract. We consider the tropical variety T (I) of a prime ideal I generated by the polynomials f 1 , . . . , f r and revisit the regular projection technique introduced by Bieri and Groves from a computational point of view. In particular, we show that I has a short tropical basis of cardinality at most r + codim I + 1 at the price of increased degrees, and we provide a computational description of these bases.

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Tropical Implicitization and Mixed Fiber Polytopes

Cited by 29 publications

References 18 publications

Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

Projections of tropical varieties and their self-intersections

Tropical bases by regular projections

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