MOHR Rafael

PhD student at Sorbonne University
Team : PolSys
https://lip6.fr/Rafael.Mohr

Supervision : Mohab SAFEY EL DIN
Co-supervision : HORN Max, TU Kaiserslautern

Fast Groebner Bases Algorithms for Ideal Theoretic Computations in Commutative Algebra

An algebraic set is the common zero locus of a set of multivariate polynomials. When such an algebraic set does not consist of finitely many points, a frequent algorithmic task is to decompose this algebraic set into equidimensional algebraic sets, this comes up, for example, in robotics, in enumerative geometry and in critical point computations in real algebraic geometry. This thesis presents a series of algorithms, based on the classical tool of Gröbner bases in symbolic computation, related to this problem of algebraic set decomposition. Three of these algorithms, partially based on joint work with Christian Eder, Pierre Lairez and Mohab Safey El Din, deal with the problem of equidimensional decomposition directly, i.e. with the problem of partitioning a given algebraic set dimension by dimension. Another algorithm, based on joint work with Jérémy Berthomieu, combines classical Gröbner basis algorithms with Hensel lifting techniques in a novel way in order to compute Gröbner bases of parametric polynomial systems. Such computations can again be used for equidimensional decomposition but also for decomposing an algebraic set into irreducible components. A final algorithm, based on joint work with Martin Helmer (NC State University), is given for the computation of so-called Whitney stratification which can be understood as certain decomposition arising in singularity theory. This algorithm simplifies a key step in a previous algorithm by Helmer by using techniques similar to that of equidimensional decomposition. All the algorithms designed in this thesis come with software implementations written in the programming language julia. These implementations are used to demonstrate the practical efficiency of our algorithms compared to state-of-the-art computer algebra systems. Some of our implementations are available in the public julia package AlgebraicSolving.jl.


Phd defence : 10/25/2024

Jury members :

Michael Stillman, Cornell University [rapporteur]
Gilles Villard, CNRS, ENS Lyon [rapporteur]
Anna M. Bigatti, Università degli Studi di Genova
Christian Eder, Co-Supervisor, RPTU Kaiserslautern-Landau
Claus Fieker, RPTU Kaiserslautern-Landau
Stef Graillat, Sorbonne Université
Pierre Lairez, Co-Supervisor, Inria Saclay
Anton Leykin, Georgia Tech University

Departure date : 10/31/2024

2023-2024 Publications