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We establish two new results in this dissertation. Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give lower bounds on the Weil height of an algebraic number [alpha] under certain assumptions on [alpha]. We prove a theorem which introduces an auxiliary polynomial for giving lower bounds on the height of any algebraic number. In particular, we prove the following theorem. [Mathematical equations] / text

Let K(q) be the finite field with q elements and characteristic p. Let f(x) be a monic polynomial of degree d with coefficients in K(q). Let C(f) denote the number of distinct values of f(x) as x ranges over K(q). It is easy to show that C(f) ≤ [|(q - 1)/d|] + 1. Now, there is a characterization of polynomials of degree d < √q for which C(f) = [|(q - 1)/d|] +1. The main object of this work is to give a characterization for polynomials of degree d < ⁴√q for which C(f) < 2q/d. Using two well known theorems: Hurwitz genus formula and Andre Weil's theorem, the Riemann Hypothesis for Algebraic Function Fields, it is shown that if d < ⁴√q and C(f) < 2q/d then f(x) - f(y) factors into at least d/2 absolutely irreducible factors and f(x) has one of the following forms: (UNFORMATTED TABLE FOLLOWS) f(x - λ) = D(d,a)(x) + c, d|(q² - 1), f(x - λ) = D(r,a)(∙ ∙ ∙ ((x²+b₁)²+b₂)²+ ∙ ∙ ∙ +b(m)), d|(q² - 1), d=2ᵐ∙r, and (2,r) = 1 f(x - λ) = (x² + a)ᵈ/² + b, d/2|(q - 1), f(x - λ) = (∙ ∙ ∙((x²+b₁)²+b₂)² + ∙ ∙ ∙ +b(m))ʳ+c, d|(q - 1), d=2ᵐ∙r, f(x - λ) = xᵈ + a, d|(q - 1), f(x - λ) = x(x³ + ax + b) + c, f(x - λ) = x(x³ + ax + b) (x² + a) + e, f(x - λ) = D₃,ₐ(x² + c), c² ≠ 4a, f(x - λ) = (x³ + a)ⁱ + b, i = 1, 2, 3, or 4, f(x - λ) = x³(x³ + a)³ +b, f(x - λ) = x⁴(x⁴ + a)² +b or f(x - λ) = (x⁴ + a) ⁱ + b, i = 1,2 or 3, where D(d,a)(x) denotes the Dickson’s polynomial of degree d. Finally to show other polynomials with small value set, the following equation is obtained C((fᵐ + b)ⁿ) = αq/d + O(√q) where α = (1 – (1 – 1/m)ⁿ)m and the constant implied in O(√q) is independent of q.

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

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The all terminal reliability of a graph G is the probability that at least a spanning tree is operational, given that vertices are always operational and edges independently operate with probability p in [0,1]. In this thesis, an investigation of all terminal reliability is undertaken. An open problem regarding the non-existence of optimal graphs is settled and analytic properties, such as roots, thresholds, inflection points, fixed points and the average value of the all terminal reliability polynomial on [0,1] are studied. A new reliability problem, the k -clique reliability for a graph G is introduced. The k-clique reliability is the probability that at least a clique of size k is operational, given that vertices operate independently with probability p in [0,1] . For k-clique reliability the existence of optimal networks, analytic properties, associated complexes and the roots are studied. Applications to problems regarding independence polynomials are developed as well.