Algebraic Integers With Small Trace And Positive Real Parts

In 1984 C. J. Smyth proposed the following problem: Let r≥0 be a given integer, one tries to find all totally positive algebraic integers a which satisfya) Tr(α) - deg(α) = r;b)α_i >0, i = 1,…,d,whereα_i are the conjugates ofα(setα₁ = a), Tr(α) = a₁ +α₂+…+α_d is the trace ofα, and d = deg(α) is the degree of its minimal polynomial. An algebraic integer is said to be totally positive, if all its conjugates are positive. For a totally positive algebraic integer, with degree d, C. L. Siegel [22] showed that Tr(α) > (?)d unlessα= 1 or (?). C. J. Smyth' improvement of C. L. Siegel's results is, Tr(α) > 1.7719d unless a has some special minimal polynomial. Furthermore, if a) is satisfied then d≤[1.2955r]. Based on this conclusion, C. J. Smyth enumerated the required polynomials (satisfy a), b), and 0≤r≤6) using a finite tree search, which was builded by him. It is shown from his results that the number of minimal polynomials increase exponentially by r. While, after that, since the difficulty of computation, there was not any authors discussing this problem as far as we know.In this work, we find the algebraic integer satisfying a) (r∈Z) and b') Re(α_i) > 0, i = 1,…, d, i.e., extending the problem to the complex plane.With this condition, the finite tree search cannot afford the corresponding problem. We try to explore an algorithm to get the algebraic integers satisfying the above-mentioned conditions so that more characteristics of algebraic integers with positive parts may displayed. There is not any references to refer to, thus our work is certainly original. The theorems of integer transfinite diameter, auxiliary function and semi-infinite linear programming are applied in our algorithm. We mainly consider algebraic integers whose arguments are in a fixed angle, i.e., adding the following condition to a:c) |Arg(α_i)|≤θ, i= l,2,…,d.Because of the huge size of the problem, in the numerical implementation, we setθbe an small angle, say,θ= (?), and get all the minimal polynomials of algebraic integers satisfying a), b'), c), 1≤d≤4 and 0≤r≤6. We find from the results that Tr(α)≥(?)d holds for all algebraic integers satisfying the provious conditions and d≤2.