Gröbnerove baze

Abstract

U ovom radu, glavni cilj nam je proučiti pojam Gröbnerove baze. Rad se sastoji od dva poglavlja. U prvom poglavlju upoznajemo osnovne algebarske strukture koje su nam bile potrebne u daljnjim razmatranjima. Upoznajemo grupe, prstene i ideale, te nešto više pažnje posvećujemo polinomima, koji su bitni u radu. Također definiramo Noetherin prsten, te jedno poglavlje posvećujemo monomima, monomijalnim uređajima, te leksikografskom uređaju. U drugom poglavlju smo konačno mogli definirati Gröbnerovu bazu, te navesti neka njena svojstva. Vrlo važan nam je bio Buchbergerov teorem, koji daje kriterij za određivanje je li neka baza Gröbnerova, te istoimeni algoritam za izračunavanje Gröbnerove baze.In this diploma thesis the main goal is to study the concept of Gröbner basis. The thesis consists of two chapters. In the first chapter we introduce the basic algebraic structures which were important in further considerations. We get to know groups, rings and ideals, and more attention was devoted to polynomials which are very important in our work. We also define Noetherian ring, and one chapter is devoted to monomials, monomial order and lexicographic order. In the second chapter we define the determine Gröbner basis, and prove some of its properties. Buchberger’s theorem was very important to us because it gives us criterion for determining whether the basis is Gröbner, and Buchberger’s algorithm gives us the Gröbner basis