Thesis (B.A.)--Haverford College, Dept. of Mathematics, 2004. / Includes bibliographical references.

Thesis (Ph. D.)--York University, 2001. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 170-177). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ66344.

In the study of mathematics, one of the most useful, relevant topics explored in secondary mathematics remains the zeros of polynomials. This paper will present various ways to explore this topic while preserving the fundamental concept as a whole. In addition, this paper will reveal some distinct relationships between roots and their behavior within the different branches of mathematics. The purpose of this paper is to show how this topic can be inserted at key points in the developmental curriculum to preserve the autonomy of this vital mathematical concept, allowing students to experience the behavior and value of this topic in a variety of contexts. / text

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 17-18).

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1925. / Vita.

This dissertation is about the Weil height of algebraic numbers and the Mahler measure of polynomials in one variable. We investigate connections between the normalizer of a stabilizer and lower bounds for the Weil height of algebraic numbers. In the Archimedean case we extend a result of Schinzel [Sch73] and in the non-archimedean case we establish a result related to work of Amoroso and Dvornicich [Am00a]. We establish that amongst all polynomials in Z[x] whose splitting fields are contained in dihedral Galois extensions of the rationals, x³-x-1, attains the lowest Mahler measure different from 1. / text

published_or_final_version / Mathematics / Master / Master of Philosophy

In many mathematical investigations such as determination of degree of a field extension, determination of the Galois group etc. the knowledge of ineducibility of a polynomial f(x),or if reducible the nature of the irreducible factors of f(x)are desired. We wish to give here a brief survey of the polynomial domain, factorisation in such a domain and the criteria by which non factorisability of a polynomial in such a domain can be determined. We shall try to make the contents self supporting and self explanatory as much as possible within the scope of our present work. [...]

This thesis is devoted to polynomial inequalities with constraints. We present a history of the development of this subject together with recent progress. In the first part, we solve an analog of classical Markov's problem for monotone polynomials. More precisely, if ∆n denotes the set of all monotone polynomials on [-1,1] of degree n, then for Pn ϵ ∆n and x ϵ [-1,1] the following sharp inequalities hold: │P’n(x)│≤ 2 max(Sk(-x),Sk(-x))║Pn║, for n = 2k + 2, k ≥ 0, and │P'n(x)│ ≤ 2 max (Fk(x), Hk(x))║Pn║, for n = 2k + 1, k ≥ 0, where Sk(x) := (1+x)∑_(l=0 )^k▒(J_l (0,1)(x^2)) ; S_k (x) &:=(1+x)\sum\limits_{l=0}^{k} (J^{(0,1)}_l (x))^2;\\ H_k (x) &:=(1-x^2)\sum\limits_{l=0}^{k-1} (J_l ^{(1,1)} (x))^2;\\ F_k(x) &:=\sum\limits_{l=0}^{k} (J_l ^{(0,0)} (x))^2, \end{align*} and $J_l^{(\alpha,\beta)}(x),$ $l\ge 1$ are the Jacobi polynomials. Let ∆n(1) be the set of all monotone nonnegative polynomials on $[-1,1]$ of degree $n.$ In the second part, we investigate the asymptotic behavior of the constants $$M_{q,p}^{(1)}(n,1):=\sup_{P_n\in\triangle^{(1)}_n}\frac{\|P'_n\|_{L_q [-1,1]}}{\|P_n\|_{L_p [-1,1]}},$$ in constrained Markov-Nikolskii type inequalities. Our conjecture is that \[M^{(1)}_{q,p} (n,1)\asymp \left\{ \begin{array}{ll} n^{2+2/p-2/q} , & \mbox{\rm if } 1>1/q-1/p ,\\ \log{n} , & \mbox{\rm if } 1=1/q-1/p, \\ 1 , & \mbox{\rm if } 1< 1/q - 1/p . \end{array} \right. \] We prove this conjecture for all values of p,q > 0, except for the case 0 < q < 1, 1/2 ≤ 1/q- 1/p ≤ 1, p ≠ 1

This thesis is devoted to polynomial inequalities with constraints. We present a history of the development of this subject together with recent progress. In the first part, we solve an analog of classical Markov's problem for monotone polynomials. More precisely, if ∆n denotes the set of all monotone polynomials on [-1,1] of degree n, then for Pn ϵ ∆n and x ϵ [-1,1] the following sharp inequalities hold: │P’n(x)│≤ 2 max(Sk(-x),Sk(-x))║Pn║, for n = 2k + 2, k ≥ 0, and │P'n(x)│ ≤ 2 max (Fk(x), Hk(x))║Pn║, for n = 2k + 1, k ≥ 0, where Sk(x) := (1+x)∑_(l=0 )^k▒(J_l (0,1)(x^2)) ; S_k (x) &:=(1+x)\sum\limits_{l=0}^{k} (J^{(0,1)}_l (x))^2;\\ H_k (x) &:=(1-x^2)\sum\limits_{l=0}^{k-1} (J_l ^{(1,1)} (x))^2;\\ F_k(x) &:=\sum\limits_{l=0}^{k} (J_l ^{(0,0)} (x))^2, \end{align*} and $J_l^{(\alpha,\beta)}(x),$ $l\ge 1$ are the Jacobi polynomials. Let ∆n(1) be the set of all monotone nonnegative polynomials on $[-1,1]$ of degree $n.$ In the second part, we investigate the asymptotic behavior of the constants $$M_{q,p}^{(1)}(n,1):=\sup_{P_n\in\triangle^{(1)}_n}\frac{\|P'_n\|_{L_q [-1,1]}}{\|P_n\|_{L_p [-1,1]}},$$ in constrained Markov-Nikolskii type inequalities. Our conjecture is that \[M^{(1)}_{q,p} (n,1)\asymp \left\{ \begin{array}{ll} n^{2+2/p-2/q} , & \mbox{\rm if } 1>1/q-1/p ,\\ \log{n} , & \mbox{\rm if } 1=1/q-1/p, \\ 1 , & \mbox{\rm if } 1< 1/q - 1/p . \end{array} \right. \] We prove this conjecture for all values of p,q > 0, except for the case 0 < q < 1, 1/2 ≤ 1/q- 1/p ≤ 1, p ≠ 1