Grafički prikaz binarnih kvadratnih formi

master's thesis

• Author: Marina Vidolin
• Mentor: Tomislav Pejković (mentor)
• Committee member: Tomislav Pejković (predsjednik povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-02-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Binarna kvadratna forma je preslikavanje \(f:\mathbb{R}^2\rightarrow \mathbb{R}\) oblika \[f(x,y)=ax^2+bxy+cy^2, \quad a,b,c \in \mathbb{Z},\] tj. homogeni polinom dviju varijabli drugog stupnja s cjelobrojnim koeficijentima. U prvom poglavlju rada predstavljamo osnovne pojmove i rezultate o binarnim kvadratnim formama, poput ekvivalencije formi, njihovog matričnog zapisa i diskriminante. Također se bavimo problemom reprezentacije cijelog broja nekom kvadratnom formom i redukcijom pozitivno definitnih i indefinitnih formi. U drugom poglavlju opisujemo pojam topografa kojeg je osmislio engleski matematičar J. H. Conway. Topograf je izvanredan grafički pristup kvadratnim formama pomoću kojeg u trećem i četvrtom poglavlju opisujemo osnovna svojstva definitnih, semidefinitnih i indefinitnih kvadratnih formi s cjelobrojnim koeficijentima.

Abstract (english)

A binary quadratic form is a function \(f:\mathbb{R}^2\rightarrow \mathbb{R}\) of the general form \[f(x,y)=ax^2+bxy+cy^2, \quad a,b,c \in \mathbb{Z},\] In other words, binary quadratic form is a homogeneous polynomial of degree two in two variables with integer coefficients. In the first chapter of this master’s thesis we present the basic concepts and results on binary quadratic forms, such as the equivalence of forms, their matrix notation, and the discriminant. We also deal with the problem of the representation of an integer by some quadratic form and the reduction of positive definite and indefinite forms. In the second chapter, we describe the concept of topograph invented by the English mathematician J. H. Conway. A topograph is an extraordinary visual approach to quadratic forms by which we describe in the third and fourth chapter the basic properties of definite, semidefinite and indefinite quadratic forms with integer coefficients.

• Keywords: binarna kvadratna forma; topograf; Conway
• Keywords (english): binary quadratic form; topograph; Conway
• Language: croatian
• URN:NBN: urn:nbn:hr:217:810974
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-10-28 16:52:08