The Absolute Length Of Algebraic Integers With Positive Real Parts

Let beαalgebraic integer of degree d, not 0 or a root of unity, all of whose conjugatesα_i are confined to a set S_θ= {α_i∈C : |arg(α_i)|≤θ}, 0 <θ< (?), i = 1,2,…, d. P(x) = a₀x^d + a₁x^d-1 +…+ a_d = a₀∏_i=1^d(x-α_i) is the minimal polynomial ofαThen, the length ofαis given by L(P) = |a₀| + |a₁| +…+ |a_d|=∑_i=0^d|a_i| and the absolute length ofαis given by L(P) = L(P)^?.We compute the greatest lower bound C(θ) of the absolute length L(P) ofα.As a result, We succeed in finding C(θ), 0 <θ< (?), exactly forθwith ten different values. For eachθ, the absolute length ofαall satisfy L(P)≥C(θ) except several particular algebraic integers.We use the integer transfinite diameter, auxiliary function, LLL algorithm and semiinfinitelinear programming in our computer method. All these can help us find the good values of C(θ).