Sketches and computation – II: dynamic evaluation and applications

Duval, Dominique; Reynaud, Jean-Claude

doi:10.1017/s096012950000044x

Cited by 19 publications

(15 citation statements)

References 11 publications

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“…Our theory has many connections with the following papers ( [6,7,8,9,10,11,12,13,15]), based on ( [1,3,4,14]), and with the theory of coherent toposes as well.…”

Section: Introductionmentioning

confidence: 79%

Dynamical method in algebra: effective Nullstellensätze

Coste

Lombardi

Roy

2001

Annals of Pure and Applied Logic

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We give a general method for producing various effective Null and Positivstellensätze, and getting new Positivstellensätze in algebraically closed valued fields and ordered groups. These various effective Nullstellensätze produce algebraic identities certifying that some geometric conditions cannot be simultaneously satisfied. We produce also constructive versions of abstract classical results of algebra based on Zorn's lemma in several cases where such constructive version did not exist. For example, the fact that a real field can be totally ordered, or the fact that a field can be embedded in an algebraically closed field. Our results are based on the concepts we develop of dynamical proofs and simultaneous collapse. MSC: 03F65, 06F15, 12J10, 12J15, 18B25

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“…Our theory has many connections with the following papers ( [6,7,8,9,10,11,12,13,15]), based on ( [1,3,4,14]), and with the theory of coherent toposes as well.…”

Section: Introductionmentioning

confidence: 79%

Dynamical method in algebra: effective Nullstellensätze

Coste

Lombardi

Roy

2001

Annals of Pure and Applied Logic

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“…One may regard dynamic evaluation as a computer algebra counterpart of the concept of non-deterministic evaluation from theoretical computer science. The strategy of dynamic evaluation has been extended to transcendental parameters [17,22], real algebraic numbers in [16], and more general algebraic structures [25, chapter 8].…”

Section: Previous Workmentioning

confidence: 99%

Directed evaluation

Hoeven

Lecerf

2020

Journal of Complexity

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Let be a fixed effective field. The most straightforward approach to compute with an element in the algebraic closure of is to compute modulo its minimal polynomial. The determination of a minimal polynomial from an arbitrary annihilator requires an algorithm for polynomial factorization over. Unfortunately, such algorithms do not exist over generic effective fields. They do exist over fields that are explicitly generated over their prime sub-field, but they are often expensive. The dynamic evaluation paradigm, introduced by Duval and collaborators in the eighties, offers an alternative algorithmic solution for computations in the algebraic closure of. This approach does not require an algorithm for polynomial factorization, but it still suffers from a non-trivial overhead due to suboptimal recomputations. For the first time, we design another paradigm, called directed evaluation, which combines the conceptual advantages of dynamic evaluation with a good worst case complexity bound.

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“…One can check that the multiplicity M of x * is 18. This can be computed as the dimension of the Q -algebra Q[[x 1 , x 2 , x 3 + 1]]/(F) thanks to the software Singular [32].…”

Section: Introductionmentioning

confidence: 99%

Quadratic Newton Iteration for Systems with Multiplicity

Lecerf¹

2002

Foundations of Computational Mathematics

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Newton's iterator is one of the most popular components of polynomial equation system solvers, either from the numeric or symbolic point of view. This iterator usually handles smooth situations only (when the Jacobian matrix associated to the system is invertible). This is often a restrictive factor. Generalizing Newton's iterator is still an open problem: How to design an efficient iterator with a quadratic convergence even in degenerate cases? We propose an answer for an m-adic topology when the ideal m can be chosen generic enough: compared to a smooth case we prove quadratic convergence with a small overhead that grows with the square of the multiplicity of the root.

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Sketches and computation – II: dynamic evaluation and applications

Cited by 19 publications

References 11 publications

Dynamical method in algebra: effective Nullstellensätze

Dynamical method in algebra: effective Nullstellensätze

Directed evaluation

Quadratic Newton Iteration for Systems with Multiplicity

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