Computational Schemes for Subresultant Chains

Asadi, Mohammadali; Brandt, Alexander; Maza, Marc Moreno

doi:10.1007/978-3-030-85165-1_3

Cited by 3 publications

(5 citation statements)

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“…The modular method presented in Section 4 relies on the computations of subresultant chains of univariate polynomials as well as the computations of subresultant chains of bivariate polynomials. For the case of subresultant chains of univariate polynomials, we follow (a variant of) Algorithm 1 from [2]. Our code takes advantage of Maple's modp1 library for univariate polynomial arithmetic.…”

Section: Computation Of Subresultant Chainsmentioning

confidence: 99%

“…Finally, it is important to mention that there is large room for improvement. Indeed, using a modular method opens the door to using speculative algorithms for computing subresultants; see [2]. Speculative algorithms are asymptotically fast algorithms that:…”

Section: Computation Of Subresultant Chainsmentioning

confidence: 99%

“…We start with ℓ. . We note that using a probabilistic approach for computing resultants in modular fashion is common, see for instance [14,2]. In the sequel of the section, for simplicity of presentation, we focus on the probabilistic approach and we note that switching to the deterministic approach requires minor adjustments in our pseudo-code.…”

Section: The Modular Methodsmentioning

confidence: 99%

“…Details can be found in [8]. This paper also relies on the theory of subresultants and we refer the unfamiliar reader to the concise preliminaries section of [2]. Throughout this paper, let k be a perfect field, K be its algebraic closure, and k[𝑋 ] be the polynomial ring over k with 𝑛 ordered variables…”

Section: Preliminariesmentioning

confidence: 99%

“…When those assumptions are met, the modular algorithm relies on the computations of subresultants of index 0 and index 1, without requiring subresultants of higher indices. This strategy opens the door to using speculative algorithms for computing subresultants, an idea introduced and studied in [2]. Such algorithms are asymptotically fast algorithms that:…”

Section: Random Testmentioning

confidence: 99%

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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: Computation Of Subresultant Chainsmentioning

confidence: 99%

Section: Computation Of Subresultant Chainsmentioning

confidence: 99%

Section: The Modular Methodsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Random Testmentioning

confidence: 99%

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