پديد آورنده :
پورحقاني، آسيه
عنوان :
حل دستگاه هاي چند جمله اي پارامتري
مقطع تحصيلي :
كارشناي ارشد
گرايش تحصيلي :
رياضي محض﴿هندسه﴾
محل تحصيل :
اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي
يادداشت :
ص.ع.به فارسي و انگليسي
استاد راهنما :
امير هاشمي
توصيفگر ها :
مجموعه جبري , مجموعه نيمه جبري , مجموعه ساخت پذير , چند گوناي مبين , تجزيه جبري استوانه اي , پايه گروبنر
تاريخ نمايه سازي :
60/1/90
استاد داور :
حسين سبزرو، قهرمان طاهريان
چكيده فارسي :
به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي
چكيده انگليسي :
Solving Parametric Polynomial Systems Asieh Pourhaghani asieh pourhaghani@math iut ac ir October 20 2010 Master of Science Thesis in Persian Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Amir Hashemi amir hashemi@cc iut ac ir 2010 MSC 13P10 Key words Parametric polynomial system Algebraic set Semi algebraic set Constructible set Discrimi nant variety Cylindrical algebraic decomposition Gr bner basis o Abstract Many problems in science and engineering such as biology chemistry computer vision robotics and so on can be reduced to solve a parametric polynomial system Such a sys tem has two di erent unknowns variables and parameters Solving parametric polynomial system which is the main purpose of this thesis means nding the values of variables w r t the di erent values of parameters In the other words the main obstacle in solving such a system is to describe the structure of the solution set in dependence of the parameters In this thesis we reduce the space of computations to the space of parameters Then by cylindrical algebraic decomposition CAD method which had been discovered by Collins in 6 we decompose whole the space of parameters into the cells on which the initial polyno mial system has constant sign But this method requires too much computation time and its complexity is doubly exponential Lazard and Rouillier in 16 by introducing the concept of minimal discriminant variety have reduced the space of computations to the parameter space and then they have used CAD to decompose the space of parameters into the cells To describe the main result of this thesis more precisely let us consider the constructible semi algebraic set C x Cn p1 x ps x 0 f1 x 0 f x 0 where pi fi Q U X U u1 ud is the set of parameters and X xd 1 xn is the set of variables If U denotes the canonical projection onto the parameter s space then 1 computing C is reduced to the computation of a subspace U Cd such that U U C U is a covering space of U in this case we say that U has the U C covering property This guarantees that the cardinality of 1 u C is constant on a neighborhood of u that U 1 U U C is a nite collection of sheets and that U is local di eomorphism from each of these sheets onto U We show that the complement in U C the closure of U C for the usual topology of n C of the union of all the open subsets of U C which have the U C covering property is a Zariski closed set which is called the minimal discriminant variety of C w r t U denoted as WD We propose an algorithm to compute WD e ciently The variety WD can then be used to solve the parametric system C as long as one can describe U C WD by using CAD In this thesis as a new result we modify Lazard Rouillier algorithm to compute WD and we show that with our improvements the output of our algorithm is despite of Lazard Rouillier algorithm always minimal and it does not need to compute the radical of ideals 1
استاد راهنما :
امير هاشمي
استاد داور :
حسين سبزرو، قهرمان طاهريان
