10088

9318

كارشناسي ارشد

رياضي محض﴿هندسه﴾

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1393

هشت، 122ص.: مصور، جدول

ص.ع. به فارسي و انگليسي

1396/09/27

كتابنامه

علوم رياضي

رياضي

ID9318

Solving Symmetric Polynomial Systems Fateme Ahmadi fateme ahmadi@math iut ac ir Date 2014 12 6 Depatrment of Mathematical Sciences Isfahan University of Tecnology Isfahan 84156 83111 IranSupervisor Dr Amir Hashemi Amir Hashemi@cc iut ac irAdvisor Dr Azam etemad ae110mat@cc iut ac ir2010 MSC Primary 13P10 Secondary 68W30Keywords Invariant polynomial ring primary invariant secondary invariant SAGBI Gr bner ba sis invariant Gr bner basis Hironaka decomposition Cohen Macaulley ring Hilbert series Molianseries Abstract One of the classical problems in mathematics is to solve systems of polynomial equations in severalunknowns Today polynomial models are ubiquitous and widely applied across the sciences Theyarise in robotics coding theory statistics machine learninig control theory computer vision imageprocessing and numerous other areas Solving polynomial systems is also an important tool in computer algebra It is possible to solvepolynomial equation systems using Gr bner bases For this purpose we can compute the Gr bnerbasis w r t lexicographical ordering and from this basis we can compute simpler the solutions ofthe initial system On the other hand some of the polynomail equation systems that are important inapplication have symmetries In this thesis we study an efficient method to solve polynomial systems whose equations are leftinvariant by the action of a finite group G For this purpose we shall consider first an invariantpolynomial ring i e the ring of polynomials which are invariant under the action of the given group Such a polynomial ring can be represented by the primary and secondary invariants For solvingsymmetric polynomial system we use the truncated form of SAGBI Gr bner bases a generalizationof Gr bner bases to ideals of sub algebras of a given polynomial ring and a Gr bner bases in theinvariant ring K 1 n where i for each i is the i th elementary symmetric polynomial In fact to solve an invariant system we provide two main algorithms first from an algorithm like

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