Gauss-Lucasov teorem i njemu srodni geometrijski rezultati

Milić, Matija (2014) Gauss-Lucasov teorem i njemu srodni geometrijski rezultati. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

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.

Item Type: Thesis (Diploma thesis)
Supervisor: Šiftar, Juraj
Date: 2014
Number of Pages: 28
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 10 Jun 2015 09:57
Last Modified: 10 Jun 2015 09:57
URI: http://digre.pmf.unizg.hr/id/eprint/4030

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