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.

Suggestions

Efficient and Accurate Electromagnetic Optimizations Based on Approximate Forms of the Multilevel Fast Multipole Algorithm
Onol, Can; Karaosmanoglu, Bariscan; Ergül, Özgür Salih (2016-01-01)
We present electromagnetic optimizations by heuristic algorithms supported by approximate forms of the multilevel fast multipole algorithm (MLFMA). Optimizations of complex structures, such as antennas, are performed by considering each trial as an electromagnetic problem that can be analyzed via MLFMA and its approximate forms. A dynamic accuracy control is utilized in order to increase the efficiency of optimizations. Specifically, in the proposed scheme, the accuracy is used as a parameter of the optimiz...
COMPUTATION OF PHYSICAL OPTICS INTEGRAL BY LEVIN'S INTEGRATION ALGORITHM
Durgun, Ahmet Cemal; Kuzuoğlu, Mustafa (2009-01-01)
In this paper, a novel algorithm for computing Physical Optics (PO) integrals is introduced. In this method, the integration problem is converted to an inverse problem by Levin's integration algorithm. Furthermore, the singularities, that are possible to occur in the applications of Levin's method, are handled by employing trapezoidal rule together with Levin's method. Finally, the computational accuracy of this new method is checked for some radar cross section (RCS) estimation problems performed on flat, ...
On the reduction of Gaussian inverse Wishart mixtures
Granström, Karl; Orguner, Umut (2012-09-12)
This paper presents an algorithm for reduction of Gaussian inverse Wishart mixtures. Sums of an arbitrary number of mixture components are approximated with single components by analytically minimizing the Kullback-Leibler divergence. The Kullback-Leibler difference is used as a criterion for deciding whether or not two components should be merged, and a simple reduction algorithm is given. The reduction algorithm is tested in simulation examples in both one and two dimensions. The results presented in the ...
Efficient Surface Integral Equation Methods for the Analysis of Complex Metamaterial Structures
Yla-Oijala, Pasi; Ergül, Özgür Salih; Gurel, Levent; Taskinen, Matti (2009-03-27)
Two approaches, the multilevel fast multipole algorithm with sparse approximate inverse preconditioner and the surface equivalence principle algorithm, are applied to analyze complex three-dimensional metamaterial structures. The efficiency and performance of these methods are studied and discussed.
Rigorous Analysis of Double-Negative Materials with the Multilevel Fast Multipole Algorithm
Ergül, Özgür Salih (2012-02-01)
We present rigorous analysis of double-negative materials (DNMs) with surface integral equations and the multilevel fast multipole algorithm (MLFMA). Accuracy and efficiency of numerical solutions are investigated when DNMs are formulated with two recently developed formulations, i.e., the combined tangential formulation (CTF) and the electric and magnetic current combined-field integral equation (JMCHE). Simulation results on canonical objects are consistent with previous results in the literature on ordin...
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.