Dominique Duval

Dynamic evaluation


Dynamic evaluation is a computational process that relies on a systematic use of case distinction, which has been introduced in computer algebra for computing exactly with algebraic numbers and with parameters.


Dynamic evaluation has applications to computer algebra, abstract algebra, geometry, logic:

Teo Mora. Solving Polynomial Equation Systems I. The Kronecker-Duval Philosophy.  Encyclopedia of Mathematics and its Applications 88, Cambridge University Press (2003). ISBN 978-0521811545.

Henri Lombardi, Claude Quitté. Algèbre commutative. Méthodes constructives. Calvage et Mounet (2011). ISBN 978-2-91-635221-3. Zentrallblatt Zbl 1242.13002

Henri Lombardi, Claude Quitté. Commutative Algebra: Constructive Methods. Finite Projective Modules. Springer Algebra and Applications (2015). ISBN 978-94-017-9943-0 .

Ihsen Yengui. Constructive Commutative Algebra. Projective Modules Over Polynomial Rings and Dynamical Gröbner Bases. Springer Lecture Notes in Mathematics 2138 (2015). ISBN 978-3-319-19493-6

François Boulier, François Lemaire, Marc Moreno Maza. Well known theorems on triangular systems and the D5 principle. Proc. Transgressive Computing 2006.

Thierry Coquand. Dynamical methods in algebra. Presented at LMPS 2003.

Michel Coste, Henri Lombardi, Marie-Françoise Roy. Dynamical method in algebra: Effective Nullstellensätze. Annals of Pure and Applied Logic, 111, 203-256 (2001).

Xavier Dahan, Marc Moreno Maza, Eric Schost, Yuzhen Xie. On the Complexity of the D5 Principle. Proc. Transgressive Computing 2006.

Gema M. Diaz-Toca, Henri Lombardi, Claude Quitté. Universal Decomposition Algebra. Proc. Transgressive Computing 2006.

Franz-Viktor Kuhlmann, Henri Lombardi, Hervé Perdry. Dynamic computations inside the algebraic closure of a valued field. Valuation Theory and its Applications. Vol 2. Fields Institute Comunications vol 33 (2003) 133-156.

Henri Lombardi. Dimension de Krull, Nullstellensätze et évaluation dynamique. Math. Zeitschrift, 242, (2002), 23-46.

Henri Lombardi. Structures algébriques dynamiques, espaces topologiques sans points et programme de Hilbert. Annals of Pure and Applied Logic 137 (2006).

Magma. Overview of Magma Features. Algebraically Closed Fields.

Montserrat Manubens, Antonio Montes. Improving DISPGB algorithm using the discriminant ideal. To be published at the special A3L issue of the Jour. Symb. Comp. (2006). Preprint arXiv: math.AC/0601763.

Antonio Montes. About the canonical discussion of polynomial systems with parameters. Preprint arXiv: math.AC/0601674 (modified in June 2006).

Masayuki Noro. Modular Dynamic Evaluation. Proc. Issac 2006.


Dynamic evaluation appears in 1985, under the name of "D5":

Jean Della Dora, Claire Dicrescenzo, Dominique Duval. About a new method for computing in algebraic number fields. In Proc. Eurocal'85, Springer LNCS (204) 289-290 (1985).

Dynamic evaluation has been used for computing exactly with algebraic numbers:

Claire Dicrescenzo, Dominique Duval. Algebraic computation on algebraic numbers. In Proc. Computers and computing, Masson-Wiley 54-61 (1985).

Claire Dicrescenzo, Dominique Duval. Algebraic extension and algebraic closure in Scratchpad. In Symbolic and algebraic computation, Springer LNCS (358) 440-446 (1989).

Dominique Duval. Algebraic numbers: an example of dynamic evaluation. Journal of Symbolic Computation (18) 429-445 (1994).

Dominique Duval. Evaluation dynamique et clôture algébrique en Axiom. Journal of Pure and Applied Algebra (99) 267-295 (1995).

The modularity of the dynamic evaluation method has been tested with the implementation of computations over a field of arbitrary characteristic:

Dominique Duval. Simultaneous computations in fields of arbitrary characteristic. In Computers and Mathematics 89, Springer 321-326 (1989)

There has been also a dynamic evaluation implementation of real algebraic numbers:

Dominique Duval, Laureano Gonzalez Vega. Dynamic evaluation and real closure. Mathematics and Computers in Simulation (42) 551-560 (1996).

Then dynamic evaluation has been used for computing exactly with parameters:

Teresa Gomez Diaz. Quelques applications de l'évaluation dynamique. Thèse, Université de Limoges (1994).

Dominique Duval, Teresa Gomez Diaz. A lazy method for triangularizing polynomial systems. Publication du LACO 1995-04 (1995).

Dynamic evaluation can be interpreted in terms of algebraic geometry:

Stéphane Dellière. Triangularisation de systèmes constructibles. Application à l'évaluation dynamique. Thèse, Université de Limoges (1999).

Stéphane Dellière, D.M. Wang. Simple systems and dynamic constructible closure. Rapport de Recherche du LACO 2000-16.

Stéphane Dellière. A first course to D7 with examples. Rapport de Recherche du LACO 2000-17.

Stéphane Dellière. Pgcd de deux polynômes à paramètres: approche par la clôture constructible dynamique et généralisation de la méthode de S.A. Abramov, K.Yu. Kvashenko. Rapport de recherche de l'INRIA - Sophia Antipolis RR-3882 (2000).

Stéphane Dellière. On the links between triangular sets and dynamical constructible closure. Journal of Pure and Applied Algebra 163 49-68 (2001).

A formal framework for dynamic evaluation can be expressed thanks to sketch theory:

Dominique Duval, Jean-Claude Reynaud. Sketches and computations over fields. Mathematics and Computers in Simulation (42) 363-373 (1996).

Dominique Duval, Jean-Claude Reynaud. Sketches and computation (Part I): Basic Definitions and Static Evaluation. Mathematical Structures in Computer Science (4) 185-238 (1994).

Dominique Duval, Jean-Claude Reynaud. Sketches and computation (Part II): Dynamic Evaluation and Applications. Mathematical Structures in Computer Science (4) 239-271 (1994).