Supervision : Jean-Charles FAUGÈRE
Co-supervision : WANG Dongming
Solving polynomial systems over finite fields
Polynomial systems, as generalization of linear systems, have been widely used in various scientific and engineering fields, and polynomial system solving is one of the major interests of the study of Symbolic Computations, an interdisciplinary area of Computer Science and Mathematics. Furthermore, solving polynomial systems over finite fields may lead to many potential applications to problems from Coding Theory and Cryptography, which are mainly embedded in the setting of finite fields, and therefore is of particular importance.
This thesis aims at a relatively comprehensive study on the subject of solving polynomial systems over finite fields, including algorithms designing, program implementation, and its applications to practical problems. The two main methods for polynomial system solving, namely Gröbner bases and triangular sets, are both considered. Several algorithms are designed and implemented to perform the change of ordering of Gröbner bases and the squarefree decomposition of triangular sets over finite fields. The applications of these solving methods to Biology and Coding Theory are also investigated.
Defence : 05/29/2013
- J.‑Ch. Faugère, Ch. Mou : “Sparse FGLM algorithms”, Journal of Symbolic Computation, vol. 80 (3), pp. 538-569, (Elsevier) (2017)
- Ch. Mou : “Solving polynomial systems over finite fields”, thesis, defence 05/29/2013, supervision Faugère, Jean-Charles, rapporteurs : WANG Dongming (2013)
- Ch. Mou, D. Wang, X. Li : “Decomposing polynomial sets into simple sets over finite fields: The positive-dimensional case”, Theoretical Computer Science, vol. 468, pp. 102-113, (Elsevier) (2013)
- Ch. Mou : “Design of termination criterion of BMS algorithm for lexicographical ordering”, Journal of Computer Applications, vol. 32 (11), pp. 2977-2980, (Science Press) (2012)
- X. Li, Ch. Mou, W. Niu, D. Wang : “Stability Analysis for Discrete Biological Models Using Algebraic Methods”, Mathematics in Computer Science, vol. 5 (3), pp. 247-262, (Springer) (2011)
- J.‑Ch. Faugère, Ch. Mou : “Fast algorithm for change of ordering of zero-dimensional Gröbner bases with sparse multiplication matrices”, Proceedings of the 36th international symposium on Symbolic and algebraic computation, San Jose, United States, pp. 115-122, (ACM) (2011)
- D. Wang, Ch. Mou, X. Li, J. Yang, M. Jin, Y. Huang : “Polynomial Algebra (in Chinese)”, 1-374 pages, (Higher Education Press), (ISBN: 9787040316988) (2011)
- X. Li, Ch. Mou, D. Wang : “Decomposing polynomial sets into simple sets over finite fields: The zero-dimensional case”, Computers & Mathematics with Applications, vol. 60 (11), pp. 2983-2997, (Elsevier) (2010)
- X. Li, Ch. Mou, W. Niu, D. Wang : “Stability Analysis for Discrete Biological Models Using Algebraic Methods”, Proceedings of the Joint Conference of ASCM 2009 and MACIS 2009, vol. 22, COE Lecture Note Series, Fukuoka, Japan, pp. 382-385, (Kyushu University) (2009)