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Standard Bases and Primary Decomposition in Polynomial Ring With Coefficients in Rings

Thesis Info

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Author

Sadiq, Afshan

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2006

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/1013

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727232879

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The theory of standard bases in polynomial rings with coefficients 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 coefficients in Euclidean rings is given. This algorithm computes successively a Gr ̈obner basis replacing the reduction of one single s-polynomial in Buchberger’s algorithm by the simultaneous reduction of several polynomials. And finally 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 finite fields, 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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