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master's thesis
Gauss-Lucasov teorem i njemu srodni rezultati u geometriji
Zagreb: University of Zagreb, Faculty of Science, 2014. urn:nbn:hr:217:043651
Milić, Matija
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Milić, M. (2014). Gauss-Lucasov teorem i njemu srodni rezultati u geometriji (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:043651

Milić, Matija. "Gauss-Lucasov teorem i njemu srodni rezultati u geometriji." Master's thesis, University of Zagreb, Faculty of Science, 2014. https://urn.nsk.hr/urn:nbn:hr:217:043651

Milić, Matija. "Gauss-Lucasov teorem i njemu srodni rezultati u geometriji." Master's thesis, University of Zagreb, Faculty of Science, 2014. https://urn.nsk.hr/urn:nbn:hr:217:043651

Milić, M. (2014). 'Gauss-Lucasov teorem i njemu srodni rezultati u geometriji', Master's thesis, University of Zagreb, Faculty of Science, accessed 01 November 2024, https://urn.nsk.hr/urn:nbn:hr:217:043651

Milić M. Gauss-Lucasov teorem i njemu srodni rezultati u geometriji [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2014 [cited 2024 November 01] Available at: https://urn.nsk.hr/urn:nbn:hr:217:043651

M. Milić, "Gauss-Lucasov teorem i njemu srodni rezultati u geometriji", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2014. Available at: https://urn.nsk.hr/urn:nbn:hr:217:043651

Metadata
TitleGauss-Lucasov teorem i njemu srodni rezultati u geometriji
AuthorMatija Milić
MentorJuraj Šiftar (mentor)
Committee memberJuraj Šiftar (predsjednik povjerenstva)
Committee memberVedran Krčadinac (član povjerenstva)
Committee memberVlado Volenec (član povjerenstva)
Committee memberNela Bosner (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2014-09-24, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U ovom diplomskom radu su proučavani odnosi između položaja nultočaka polinoma p i nultočaka derivacije p' tog polinoma. Najpoznatiji teorem iz tog područja je Gauss - Lucasov teorem, koji kaže da se nultočke derivacije p' nalaze u konveksnoj ljusci skupa nultočaka od p. Također smo pokazali neke srodne rezultate, poput Siebeckova teorema. U njemu je dana geometrijska veza između nultočaka kubnog polinoma i nultočaka njegove derivacije. Teorem tvrdi da ako su nultočke od p nekolinearne točke kompleksne ravnine tada su nultočke derivacije p' žarišta jednoznačno određene elipse upisane trokutu nultočaka koja dodiruje stranice trokuta u polovištima. U dokazima su korištene afine transformacije, pa smo naveli i dokazali neke činjenice o afinim i linearnim transformacijama, posebno u kompleksnoj ravnini.
Abstract (english)
The relationship between the location of the zeros of a complex polynomial p and those of its derivative has been studied in this graduate thesis. The best-known theorem in this area is the Gauss-Lucas Theorem, that the zeros of p' lie in the convex hull of the zeros of p. A special case, Siebeck's theorem is also interesting. That theorem describes geometric connection between the zeros of a cubic polynomial p with complex coefficients and the zeros of the polynomial's derivate p'. To wit, if the zeros of p are noncollinear points z_1, z_2 and z_3 in the complex plane, the zeros of p' are the foci of the unique ellipse inscribed in triangle z_1z_2z_3 and tangent to the sides at their midpoints. Our proofs makes use of affine transforma tions, so we recall and prove the necessary facts about linear and affine transformations, especially in the complex plane.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:043651
Study programmeTitle: Mathematical Statistics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsAccess restricted to students and staff of home institution
Terms of useLicence In copyright icon
Created on2019-01-24 12:28:14