16059

1652 دكتري

دكتري

هندسه توپولوژي

اصفهان : دانشگاه صنعتي اصفهان

1399

نه، 145 ص. : مصور، جدول، نمودار

امير هاشمي

ورنر ام. زايلر

عبدالعلي بصيري، سجاد رحماني، رشيد زارع نهندي

1399/09/17

كتابنامه

رياضي محض

رياضي

1399/09/22

2655060

N ther Bases and its Applications Hossein Parnian Hossein Parnian@math iut ac ir November 18 2020 Doctor of Philosophy Thesis Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Amir Hashemi Amir Hashemi@cc iut ac ir Advisor Prof Werner M Seiler Seiler@mathematik uni kassel de Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Institut f r Mathematik Universit t Kassel Heinrich Plett Stra e 40 34132 Kassel Germany u aAbstractIn this thesis we rst introduce a new involutive division called D N ther division and thecorresponding notion of a N ther basis It is shown that an ideal is in N ther position ifand only if it possesses a nite N ther basis Furthermore we investigate upper bounds forthe maximum degree of the elements of any minimal Janet basis of an ideal generated by aset of homogeneous polynomials The presented bounds depend on the number of variablesand the maximum degree of the generating set of the ideal For this purpose by giving adeeper analysis of the method due to Dub 7 we improve and correct his bound on the edegrees of the elements of a reduced Gr bner basis By giving a simple proof it is shown othat this new bound is valid for Pommaret bases as well Finally following the approach by Dub 7 and by applying the Hilbert series method we eprovide an e cient algorithm to compute the Macaulay constants of a monomial ideal withoutcomputing any exact cone decomposition of the corresponding quotient ring Then based onthis construction and the method proposed by Mayr Ritscher 20 a new upper bound forthe maximum degree of the elements of any reduced Gr bner basis of an ideal generated by oa set of homogeneous polynomials is given The new bound depends on the Krull dimensionand the maximum degree of the generating set of the ideal Too we show that the presentedupper bound is sharper than the bounds proposed by Dub 7 and Mayr Ritscher 20 eKey Words Polynomial ideals Gr bner bases Involutive bases N ther position Pom omaret bases Quasi stable ideals Hilbert functions Degree upper bounds Exact cone decom positions Macaulay constants MSC 2010 13P10 13F20 14Q20 68W30 IntroductionThe concept of Gr bner bases along with the rst algorithm to compute them were introduced oby Bruno Buchberger in his PhD thesis 5 6 Since then many interesting applications ofthese bases have been found in Mathematics science and engineering For example we canpoint out their applications in the ideal membership problem computing the dimension of 1

امير هاشمي

ورنر ام. زايلر

عبدالعلي بصيري، سجاد رحماني، رشيد زارع نهندي