Geometrija kubičnih polinoma

Abstract

U ovom radu opisujemo vezu između nultočaka kompleksnog kubičnog polinoma i nultočaka njegove derivacije. U prvom poglavlju prezentiramo osnovne rezultate o polinomima s realnim i kompleksnim koeficijentima. Drugo poglavlje započinje prezentiranjem dvaju kompleksnih analogona Rolleovog teorema za polinome proizvoljnog stupnja: Gauss–Lucasov teorem koji kaže da stacionarne točke polinoma leže u konveksnoj ljusci njegovih nultočaka; te Jensenov teorem o raspodjeli ne-realnih stacionarnih točaka kompleksnog polinoma s realnim koeficijentima. Zatim dokazujemo Sendov–Iliefovu slutnju o vezi između nultočaka nekih posebnih vrsta kompleksnih polinoma i njihovih stacionarnih točaka. U radu je posebna pažnja posvećena kubičnim polinomima. Fina veza između nultočaka kubičnog polinoma i nultočaka njegove derivacije slijedi iz Steinerovog geometrijskog rezultata: postoji jedinstvena elipsa upisana trokutu koja dira stranice tog trokuta u njihovim polovištima. Dokazujemo Mardenov teorem koji kaže: ako su nultočke kubičnog polinoma vrhovi trokuta, onda su njegove stacionarne točke fokusi Steinerove elipse upisane tom trokutu. Prezentiramo Saff i Twomeyev rezultat o položaju stacionarnih točaka familije kubičnih polinoma $\mathcal{P}(a), (|a| \leq 1)$, koji imaju sve nultočke u zatvorenom jediničnom krugu i barem jednu nultočku u točki $a$. Proučavamo strukturu stacionarnih točaka familije kubičnih polinoma s nultočkom 1, kada ostale dvije nultočke pomičemo duž jedinične kružnice. Pokazujemo da stacionarna točka svakog takvog polinoma gotovo uvijek određuje polinom jedinstveno.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