Examples of using dynamic constructible closure

Gómez-Díaz, Teresa

doi:10.1016/s0378-4754(96)80001-5

Cited by 9 publications

(11 citation statements)

References 9 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%

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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%

“…We give a sketch of an elementary decision algorithm, very near to dynamical evaluation in the dynamical constructible closure of a field (see [15]).…”

Section: Decision Algorithm and Constructive Nullstellensatzmentioning

confidence: 99%

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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“…Dynamic evaluation of an expression of /fL(a)// is an evaluation with 'automatic case distinction' according to the values of the parameter a (see Gomez-Diaz (1993)). We may view a as a 'parameter' (in the 'elementary school' sense).…”

Section: Simple Field Extensionsmentioning

confidence: 99%

“…The constructible closure allows us, for example, to solve polynomial equation systems and to perform some automatic proofs in elementary geometry (see Gomez-Diaz (1993)). The constructible closure allows us, for example, to solve polynomial equation systems and to perform some automatic proofs in elementary geometry (see Gomez-Diaz (1993)).…”

Section: Constructible Closurementioning

confidence: 99%

Sketches and computation – II: dynamic evaluation and applications

Duval¹,

Reynaud

1994

Math. Struct. Comp. Sci.

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In the first part of this paper (Duval and Reynaud 1994), we defined a categorical framework, based on the notion of sketch, for specification and evaluation in the senses of algebraic specifications and algebraic programming. Static evaluation in quasi-projective sketches was defined in Part I; in this paper, dynamic evaluation is introduced. It deals with more general structures, which may have no initial model. Until now, this process has not been used in algebraic specification systems, but computer algebra systems are beginning to use it as a basic tool. Finally, we give some applications of dynamic evaluation to computation in field extensions.

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Examples of using dynamic constructible closure

Cited by 9 publications

References 9 publications

Dynamical method in algebra: effective Nullstellensätze

Dynamical method in algebra: effective Nullstellensätze

Directed evaluation

Sketches and computation – II: dynamic evaluation and applications

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