Computation of the Primary Decomposition of Polynomial Ideals Using Gröbner Bases

2021-8-06
Tolgay, Betül
In this thesis, we investigate algorithms for computing primary decompositions of ideals in polynomial rings. Every ideal in a polynomial ring over a Noetherian commutative ring with identity has a primary decomposition, that is, it can be expressed as the intersection of primary ideals (in a unique way or not). The existence of primary decompositions in such polynomial rings is a result of the ascending chain condition and the existence proof does not suggest any construction method for the primary components of the ideal. In the first part of the thesis, we investigate the algorithms developed by Gianni et al. [13] for the computation of a primary decomposition of a given ideal in a polynomial ring. The main tool used in these algorithms is Gröbner basis techniques for the computation of certain operations on ideals. We give a complete discussion and analysis of the theorems and algorithms developed by Gianni et al. in [13] here. The second part of the thesis presents another approach to the problem of computation of primary decomposition developed by Eisenbud et al. in [4]. This method avoids the projection of an ideal to a polynomial subring with one less variable which was used for reduction in the algorithms developed by Gianni et al. [13]. We give an outline of the algorithms developed by Eisenbud et al. in [4] here. The algorithms developed by both Gianni et al. [13] and Eisenbud et al. [4] make it possible to compute primary components and associated primes of a given ideal, hence also the radical of the ideal.
Citation Formats
B. Tolgay, “Computation of the Primary Decomposition of Polynomial Ideals Using Gröbner Bases,” M.S. - Master of Science, Middle East Technical University, 2021.