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master's thesis
Binarne kvadratne forme
Zagreb: University of Zagreb, Faculty of Science, 2018. urn:nbn:hr:217:978889
Rogar, Andreja
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Rogar, A. (2018). Binarne kvadratne forme (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:978889

Rogar, Andreja. "Binarne kvadratne forme." Master's thesis, University of Zagreb, Faculty of Science, 2018. https://urn.nsk.hr/urn:nbn:hr:217:978889

Rogar, Andreja. "Binarne kvadratne forme." Master's thesis, University of Zagreb, Faculty of Science, 2018. https://urn.nsk.hr/urn:nbn:hr:217:978889

Rogar, A. (2018). 'Binarne kvadratne forme', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:978889

Rogar A. Binarne kvadratne forme [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2018 [cited 2024 December 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:978889

A. Rogar, "Binarne kvadratne forme", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2018. Available at: https://urn.nsk.hr/urn:nbn:hr:217:978889

Metadata
TitleBinarne kvadratne forme
AuthorAndreja Rogar
MentorZrinka Franušić (mentor)
Committee memberZrinka Franušić (predsjednik povjerenstva)
Committee memberBoris Širola (član povjerenstva)
Committee memberIlja Gogić (član povjerenstva)
Committee memberVedran Krčadinac (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2018-07-17, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Izrazi oblika \[ ax^2+bxy+cy^2, \] gdje su \(a, b, c \in \mathbb{Z}\), nazivaju se binarnim kvadratnim formama i to su zapravo kvadratni homogeni polinomi u dvije varijable. Tijekom povijesti najviše su za proučavane forme oblika \(x^2+y^2\) i \(x^2-dy^2\), dok u današnje vrijeme raste važnost teorije binarnih kvadratnih formi zbog njihove primjene u kriptografiji. U ovom radu su predstavljene osnovne notacije, činjenice i svojstva koja vežemo uz kvadratne forme, poput ekvivalencije formi, njihovog matričnog zapisa, geometrijske interpretacije i diskriminante kvadratnih formi. Nadalje, bavimo se problem reprezentacije cijelog broja nekom kvadratnom formom. Kažemo da forma \(f(x,y)=ax^2+bxy+cy^2\) reprezentira cijeli broj \(n\) ako postoje \(x_{0}, y_{0}\in\mathbb{Z}\) takvi da je \(f(x_{0}, y_{0})=n.\)
Abstract (english)
Expressions such as \[ ax^2+bxy+cy^2, \] where \(a, b, c\in\mathbb{Z}\), are called binary quadratic forms. There are, in fact, quadratic homogeneous polinomials in two variables x and y. Throughout history quadratic forms such as \(x^2+y^2\)and \(x^2-dy^2\) have been studied the most, while today increases the importance of the theory of binary quadratic forms due to their application in cryptography. In this graduate thesis we expose basic notations, facts and properties associated with quadratic forms, such as equivalence of forms, the matrix representation, geometrical interpretations and the discriminant of a form. Furthermore, we deal with representations of integers by binary quadratic forms. We say that the form \(f(x,y)=ax^2+bxy+cy^2\) represents an integer if there exist \(x_{0}, y_{0}\in\mathbb{Z}\) such that \(f(x_{0}, y_{0})=n.\)
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:978889
Study programmeTitle: 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 resourceText
File originBorn digital
Access conditionsAccess restricted to students and staff of home institution
Terms of useLicence In copyright icon
Created on2018-08-30 12:21:11