On Solving Parametric Polynomial Systems

Maza, Marc Moreno; Xia, Bican; Xiao, Rong

doi:10.1007/s11786-012-0136-3

Cited by 7 publications

(3 citation statements)

References 26 publications

(14 reference statements)

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“…We call 𝐵 𝑇 ,𝐻 the border polynomial of [𝑇 , 𝐻 ]. Proposition 1 follows from the specialization property of sub-resultants and states a fundamental property of border polynomials [13].…”

Section: Specialization and Border Polynomialmentioning

confidence: 99%

A Maple implementation of a modular algorithm for computing the common zeros of a polynomial and a regular chain

Trochez

Calder²,

Maza

et al. 2022

Maple Trans.

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One of the core commands in the RegularChains library isTriangularize. The underlying decomposes the solution set of anpolynomial system into geometrically meaningful components representedby regular chains. This algorithm works by repeatedly calling aprocedure, called Intersect, which computes the common zeros of apolynomial p and a regular chain T.As the number of variables of p and T, as well as their degrees,increase, the call Intersect(p, T) becomes more and morecomputationally expensive. It was observed in (C. Chen an M. MorenoMaza, JSC 2012) that when the input polynomial system iszero-dimensional and T is one-dimensional then this cost can besubstantially reduced. The method proposed by the authors is aprobabilistic algorithm based on evaluation and interpolationtechniques. This is the type of method which is typically challengingto implement in a high-level language like Maple's language, as asharp control of computing resources (in particular memory) is needed.In this paper, we report on a successful Maple implementation of thisalgorithm. We take advantage of Maple's modp1 function which offersfast arithmetic for univariate polynomials over a prime field.The method avoids unlucky specialization and the probabilistic aspectonly comes from the fact that non-generic solutions are notcomputed.

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Section: Specialization and Border Polynomialmentioning

confidence: 99%

A Maple implementation of a modular algorithm for computing the common zeros of a polynomial and a regular chain

Trochez

Calder²,

Maza

et al. 2022

Maple Trans.

Self Cite

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show abstract

“…One of the listed approaches consists of using comprehensive Gröbner bases, where the GC algorithm can be classified. Besides discriminant varieties, the notion of border polynomial should be cited as an alternative way to discuss parametric systems (see [45] for the relations between both concepts for triangular systems).…”

Section: Using Gröbner Systems To Detect Degenerate Componentsmentioning

confidence: 99%

Automatic deduction in (dynamic) geometry: Loci computation

Botana

Abánades

2014

Computational Geometry

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A symbolic tool based on open source software that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction is presented. The prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction, namely with the open source dynamic geometry system GeoGebra or using the common file format for dynamic geometry developed by the Intergeo project. Locus computation algorithms based on Automatic Deduction techniques are recalled and presented as basic for an efficient treatment of advanced methods in dynamic geometry. Moreover, an algorithm to eliminate extraneous parts in symbolically computed loci is discussed. The algorithm, based on a recent work on the Gröbner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Several examples are shown in detail.

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“…In the context of real solving, basic objects such as discriminant variety [21] and border polynomial [40] were introduced to establish conditions for parametric polynomial or semi-algebraic systems to have constant numbers of solutions. Such objects have been studied and used to partition the parameter space into connected regions described by parametric constraints under which the solutions depend continuously on the parameters and to solve the systems by means of isolating or classifying their real or complex solutions [24]. Related to this study there is a large amount of work devoted to solving semi-algebraic systems via cylindrical algebraic decomposition, Gröbner bases, and triangular decomposition [7].…”

Section: Introductionmentioning

confidence: 99%

Comprehensive Characteristic Decomposition of Parametric Polynomial Systems

Dong

Mou

et al. 2021

Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation

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This paper presents an algorithm that decomposes an arbitrary set F of multivariate polynomials involving parameters into finitely many sets Γ i of (lexicographical) Gröbner bases G i j such that associated with each Γ i there is a system A i of parametric constraints, all the A i 's partition the parameter space, all the G i j 's in each Γ i remain Gröbner bases under specialization of the parameters satisfying the constraints in A i , and for each i the Gröbner bases G i j together with their corresponding W-characteristic sets form a normal characteristic decomposition of F . The sets of Gröbner bases computed by the algorithm provide a comprehensive characteristic decomposition of F that is structure-invariant under the associated constraints and possesses many algebraic and geometric properties, including most of the notable properties on comprehensive Gröbner systems and comprehensive triangular decomposition. Some of these properties are highlighted in the paper and advantages of the proposed algorithm are discussed briefly from the aspects of methodological simplicity and computational performance using illustrative examples and preliminary experiments. CCS CONCEPTS• Computing methodologies → Symbolic and algebraic algorithms; Algebraic algorithms.

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On Solving Parametric Polynomial Systems

Cited by 7 publications

References 26 publications

A Maple implementation of a modular algorithm for computing the common zeros of a polynomial and a regular chain

A Maple implementation of a modular algorithm for computing the common zeros of a polynomial and a regular chain

Automatic deduction in (dynamic) geometry: Loci computation

Comprehensive Characteristic Decomposition of Parametric Polynomial Systems

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