

Title of Thesis
Standard Bases And Primary Decomposition In Polynomial Ring With
Coefficients In Rings 
Author(s)
Afshan Sadiq 
Institute/University/Department
Details Abdus Salam School of Mathematical Sciences / GC
University, Lahore 
Session 2011 
Subject Mathematics 
Number of Pages 100 
Keywords (Extracted from title, table of contents and
abstract of thesis) Bases, Standard, Singular,
Polynomial, Rings, Decomposition, Coefficients, Primary, Reduction,
Algorithm 
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 F4algorithm for polynomial
rings with coe cients in Euclidean rings is given.This algorithm
computes successively a Grobner basis replacing the reduction of
one single spolynomial 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 pseudoprimary ideals. A parallelized version of
the algorithm is implemented in Singular.

