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چكيده انگليسي :

Application of stable normal form in solving polynomial systems Afsaneh Ebrahimi Amroabadi a ebrahimiamroabadi@math iut ac ir January 6 2019 Master of Science thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Associate Professor Amir Hashemi amir hashemi@cc iut ac irAdvisor Assistant Professor Azam Etemad ae110mat@cc iut ac ir2010MSC 13P10 68W30Keywords Gr bner basis Border basis Monomial ordering Algorithm Normal form Stability of the basis AbstractOver recent decades solving polynomial system becomes the cornerstone of many computations in robotics geo metric modeling signal processing chromatology structural molecular biology etc The Gr bner bases have beenconsidered as one of the known and important methods in this field This concept was introduced by Buchbergerin his PhD thesis with the help of Gr bner who was his supervisor After Buchberger many mathematicians suchas Lazard Moller and Mora went on to investigate this concept and finally in 2002 Faug re introduced the F5algorithm which is the fatest method for calculating the Gr bner basis Today many books and essays have beenwritten about Gr bner basis its computation and applications The Gr bner basis algorithm is implemented in manycomputer algebra systems such as Singular Maple Mathematica Magma and so on Although the Gr bner basis helps us to solve polynomial systems it also has some weaknesses for example Gr bnerbasis is suitable for an ideal algebra study with exact coefficients and fractions But Auzinger Stettter Moller Ten berg and Mourrian used the basis so callled border basis to solve polynomial systems having finitely many solutions which even sometime has floating point coefficients Border basis plays a key role in numerical polynomial algebrabeacause it behaves numerically better than Gr bner basis In previous papers Robbiano and his co authors laid afoundation for the algebraic theory of border basis Here we address the question of how to compute a border basisof a zero dimensional ideal from its set of generators This general border basis algorithm weakens the monomial ordering requirment for Gr bner basis computations Alsoin the calculation of Gr bner basis if there is a small change in the input coefficient there may be a lot of changesin the output of the algorithm which indicates that Gr bner basis is not numerically stable A border basis study wascontucted on algebraic study by Kehrein and others Due to the structure and features of border basis it has appropriate stability Another advantage of border basis ratherthan Gr bner basis is that for calculating Gr bner basis we need monomial ordering but for calculating the borderbasis this condition is not necessary Mourrian and Trebuchet weakened this condition they provided a comprehen sive algorithm to construct a new type of presentation of quotient algebra It should be mentioned that for introducingthe Gr bner basis we use a concept that is called monomial ordering But for introducing the border basis the conceptof the order ideal is replaced In this thesis we use a new concept that is called connected to one rather than the conceptof order ideal A generative set for syzygy modules is another fundamental concept that we use in this thesis which is one of theapplications of Gr bner basis Therefore first we define the structure of an R module such as syzygy and then byintroducing some of the module monomial orderings we also define and calculate the Gr bner basis for a syzygymodule by using the Schreyer s ordering After that we define Schreyer s theorem for border bases and the compu tation of the border basis of a syzygy module The following is a brief introduction to the various chapters of this thesis In the first chapter we talk about the Gr b ner basis Buchberger s algorithm and also the application of Gr bner basis for studying the syzygy of polynomials In the second chapter we introduce the border basis in addition its calc

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كاربرد شكل متعارف پايدار در حل دستگاه هاي چند جمله اي

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