# Geometrija kubičnih polinoma

Kukec, Petra (2015) Geometrija kubičnih polinoma. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In this thesis, we describe a relationship between the roots of a complex cubic polynomial and the roots of its derivative. In the first chapter, we present basic results on polynomials with real and complex coefficients. The second chapter begins by presenting two complex analogues of Rolle’s theorem for polynomial of arbitrary degree: Gauss–Lucas theorem which states that the critical points of any polynomial lie in the convex hull of its roots; and Jensen’s theorem on the distribution of non-real critical points of a complex polynomial with real coefficients. Then we prove the Sendov–Ilieff conjecture on a relationship between the roots of some special types of complex polynomials and their critical points. In this work, special attention is paid to cubic polynomials. A sophisticated connection between the roots of a cubic polynomial and those of a derivative follows from a lovely geometric result of Steiner: there is the unique ellipse that is inscribed in the triangle and tangent to the sides at their midpoints. We prove Marden’s theorem which states: if the roots of a cubic polynomial are the vertices of the triangle, then its critical points are foci of the Steiner ellipse that is inscribed in the triangle. We present Saff and Twomey’s result on the location of the critical points of the family of cubic polynomials $\mathcal{P}(a), (|a| \leq 1)$, which have all of their roots in the closed unit disk and at least one root at the point $a$. We study the structure of the critical points of the family of cubic polynomials with a root 1, when the other two roots move around the unit circle. We show that a critical point of each such polynomial almost always determines the polynomial uniquely.