Gauss-Lucasov teorem i njemu srodni rezultati u geometriji

master's thesis

• Author: Matija Milić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Committee member: Vlado Volenec (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-09-24, 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: Gauss - Lucasov teorem; Siebeckov teorem; afine transformacije; komleksna ravnina
• Keywords (english): Gauss-Lucas theorem; Siebeck's theorem; affine transformations; complex plane
• Language: croatian
• URN:NBN: urn:nbn:hr:217:043651
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2019-01-24 12:28:14