Title of Thesis

Algorithms for Primary Decomposition

Author(s)

Nazeran Idrees

Abstract
The primary decomposition methods of Eisenbud, Huneke and Vasconcelos are analysed in detail providing proofs of important theorems and all the corresponding algorithms are programmed in the language of Singular.MOreover, we investigated the parallelization of two modular algorithms. In fact, we consider the modular computation of Grobner bases (resp. standard bases) and the modular computation of the associated primes of a zero{dimensional ideal and describe their parallel implementation in Singular.The algorithms of Shimoyama and Yokoyama for primary decomposition of ideals are generalized to submodules of a free module over the polynomial ring in several variables with coe cients in a eld.The algorithms are implemented in Singular.

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1.1 Primary Modules and Primary Decomposition
1.2 Primary decomposition using the method of Gianni, Trager and Zacharias
1.3 The Methods of Eisenbud, Huneke and Vasconcelos

2.1 Computing Grobner bases with modular methods
2.2 A modular approach to primary decomposition
2.3 Examples and timings

3.1 Localization
3.2 Pseudo Primary Decomposition and Extraction
3.3 Criteria For Redundant Components
3.4 The Primary Decomposition Procedure
3.5 Pseudo Primary Decomposition
3.6 Extraction On A Pseudo Primary Submodule
3.7 Termination of the Procedure

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