LE Huu Phuoc
Team : PolSys
Arrival date : 09/08/2018
- Sorbonne Université - LIP6
Boîte courrier 169
Couloir 26-00, Étage 3, Bureau 338
4 place Jussieu
75252 PARIS CEDEX 05
Tel: +33 1 44 27 71 30, Huu-Phuoc.Le (at) nulllip6.fr
Supervision : Mohab SAFEY EL DIN
On solving parametric polynomial systems and quantifier elimination over the reals : algorithms, complexity and implementations
Solving polynomial systems is an active research area located between computer sciences and mathematics. It finds many applications in various fields of engineering and sciences (robotics, biology, cryptography, imaging, optimal control). In symbolic computation, one study and design-efficient algorithms that compute exact solutions to those applications, which could be very delicate for numerical methods because of the non-linearity of the given systems.
Most applications in engineering are interested in the real solutions to the system. The development of algorithms to deal with polynomial systems over the reals is based on the concepts of effective real algebraic geometry in which the class of semi-algebraic sets constitutes the main objects.
This thesis focuses on three problems below, which appear in many applications and are widely studied in computer algebra and effective real algebraic geometry:
- Classify the real solutions of a parametric polynomial system according to the values of the parameters;
- One-block quantifier elimination, which is also the computation of the projection of a semi-algebraic set
- Computation of the isolated points of a semi-algebraic set.
- H. Le : “Faster algorithms for computing real isolated points of an algebraic hypersurface”, (2022)
- A. Ferguson, H. Le : “Finer complexity estimates for the change of ordering of Gröbner bases for generic symmetric determinantal ideals”, (2022)
- H. Le, M. Safey El Din : “Solving parametric systems of polynomial equations over the reals through Hermite matrices”, Journal of Symbolic Computation, vol. 112, pp. 25-61, (Elsevier) (2022)
- H. Le, D. Manevich, D. Plaumann : “Computing totally real hyperplane sections and linear series on algebraic curves”, Le Matematiche, (Università degli Studi di Catania) (2022)
- H. Le, M. Safey El Din : “Faster One Block Quantifier Elimination for Regular Polynomial Systems of Equations”, Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation (ISSAC '21), Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation, Saint Petersburg, Russian Federation, pp. 265–272 (2021)
- H. Le, M. Safey El Din, T. De Wolff : “Computing the real isolated points of an algebraic hypersurface”, ISSAC '20: International Symposium on Symbolic and Algebraic Computation, Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, Kalamata, Greece, pp. 297–304 (2020)