Title of Thesis

Standard Bases And Primary Decomposition In Polynomial Ring With Coefficients In Rings

Author(s)

Afshan Sadiq

Abstract
The theory of standard bases in polynomial rings with coe cients in a ring A with respect to local orderings is developed.A is a commutative Noetherian ring with 1 and we assume that linear equations are solvable in A.
Then the generalization of Faug ere F4-algorithm for polynomial rings with coe cients in Euclidean rings is given.This algorithm computes successively a Grobner basis replacing the reduction of one single s-polynomial in Buchberger's algorithm by the simultaneous reduction of several polynomials.
And nally we present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over nite elds, and the idea of Shimoyama{Yokoyama resp. Eisenbud{Hunecke{Vasconcelos to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in Singular.

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1.1 Monomial Ordering
1.2 Normal Form and Standard Bases
1.3 The F4 Algorithm
1.4 Primary Decomposition

2.1 Basic De nitions
2.2 Computing Standard Bases By Using Homogenization
2.3 Normal Form
2.4 Computing Standard Bases
2.5 Standard Bases Over Principal Ideal Domains
2.6 Standard Bases In The Formal Power Series Rings

3.1 F4 Algorithm
3.2 Example

4.1 Basic de nitions and results
4.2 The algorithms
4.3 Examples and timings

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