14639

1387 دكتري

دكتري

هندسه جبري محاسباتي

اصفهان : دانشگاه صنعتي اصفهان

1398

ده، 182ص. : مصور، جدول

امير هاشمي

مارتين كروزر

عبدالعلي بصيري، سجاد رحماني، محمود بهبودي

1398/03/01

كتابنامه

رياضي محض

رياضي

1398/03/04

2536763

Structure and E ciency of Computations in Border Bases Theory Samira Pourkhajouei s pourkhajooei@math iut ac ir 2019 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Amir Hashemi Amir Hashemi@cc iut ac ir Advisor Prof Martin Kreuzer Martin Kreuzer@uni passau de 2010 MSC 13P10 68W30 Keywords Polynomial ring Monomial ordering Gr bner basis Border basis oOrder ideal Buchberger M ller algorithm Ideal of points o1 AbstractIn 2006 Kehrein and Kreuzer 7 presented an algorithm referred to as the Border Basis algorithm for computing border bases The main objective of this dissertationis to improve this computation by applying new structures from the theory of Gr bner obases In doing so based on the method developed by Gao et al 5 a new variantof the BorderBasis algorithm is proposed to compute simultaneously a border basisand also a Gr bner basis for the syzygy module of the input polynomials this pair oof bases will be called a coupled border basis The new algorithm and also the orig inal form of the BorderBasis algorithm have been implemented in the computeralgebra systems Maple and ApCoCoA We compare the performance of our imple mentation of these algorithms via a set of benchmark polynomials Our experimentsshow that our algorithm performs more e ciently than the original BorderBasisalgorithm We consider the problem of computing all possible order ideals and alsosets connected to 1 and the corresponding border bases for the vanishing ideal ofa given nite set of points In this context two di erent approaches are discussed based on the Buchberger M ller algorithm 11 we rst propose a new algorithm to ocompute all possible order ideals and the corresponding border bases for an ideal ofpoints The second approach involves adapting the Farr Gao algorithm 4 for ndingall sets connected to 1 as well as the corresponding quasi border bases for an ideal ofpoints It should be noted that our algorithms are term ordering free Therefore theycan compute successfully all border bases for an ideal of points After that we present an application of these algorithms to an experimental designproblem in statistics We focus on presenting di erent models related to an experimentand show the role of our approaches in providing good statistical polynomial models

امير هاشمي

مارتين كروزر

عبدالعلي بصيري، سجاد رحماني، محمود بهبودي