7117

6644

كارشناسي ارشد

رياضي محض﴿هندسه﴾

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1390

[هشت]، 171ص.: مصور، جدول، نمودار

ص.ع. به فارسي و انگليسي

امير هاشمي

قهرمان طاهريان

1/8/91

حسن دقيق، رضا رضاييان فراشاهي

1396/09/20

كتابنامه

علوم رياضي

رياضي

ID6644

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

Factorization of Polynomials over Finite Fields and its Application Ruholla Khodakaramian gilan r khodakaramiangilan@iut ac ir December 2011 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr A Hashemi amir hashemi@cc iut ac iradvisor Dr S G taherian taherian@cc iut ac ir2000 MSC 13P10 68W30 Key words Finite elds Gr bner bases Factorization of polynomials Berlekamp s algorithm Primary odecomposition Abstract Factorization of polynomials is a strong computational tool in algebraic geometry andcomputer algebra This plays an important role in industry and Mathematics especiallybecause of its application in solving polynomial systems In 1967 Berlekamp 3 has pre sented an algorithm for the factorization of polynomials over nite elds He has used theinvariant space of Frobenius s map over the quotient ring This method is very important because using the factorization over nite elds we can factor the polynomials in the morecomplicated polynomial rings like Z x Z x1 xn and GFq x1 xn where GFq denotesa nite eld of q elements Then Zassenhaus 48 has used Berlekamp s method and Hensel slemma to propose a new method for factorization of univariate polynomials over integers This algorithm rst reduces the factorization of a given univariate polynomial to the fac torization of the polynomial over a nite eld Fp for a convenient prime number p Afterfactoring this polynomial over Fp by Berlekamp s algorithm we can lift the factorization toFp2 FpN by Hensel s lemma for N enough large In 1975 using the same approach Wangand Rotchild 46 have described a new algorithm for factoring the multivariate polynomialsover integers This algorithm begins by substituting all the variable except one of them by some selected integers and we obtain a univariate polynomial Then we use Zassenhausmethod to factorize this new polynomial Finally we lift this factorization to the factoriza tion of the given multivariate polynomial by a generelization of Hensel s lemma In the rest of this thesis we deal with the concept of primary decomposition and its relationto factorization In this direction in 2002 Monico 37 has presented an applicable algorithmfor computing the primary decomposition of zero dimensional ideals over in nite elds It isworth noting that zero dimensional ideals are the most important ideals in practice becausethe corresponding system has a nite number of solutions and they have many applicationsin industry In 2009 Gao et al 18 has described a new algorithm by using Berlekamp salgorithm to compute a primary decomposition of a zero dimensional ideal over nite elds All the algorithm described in this thesis have been implemented in Maple 1

امير هاشمي

قهرمان طاهريان

حسن دقيق، رضا رضاييان فراشاهي