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Explicit auxiliary functions can be used in the "Schur-Siegel-Smyth trace problem". In the previous works, these functions were constructed only with polynomials having all their roots positive. Here, we use several polynomials with complex roots, which are found with Wu's algorithm and we improve the known lower bounds of the trace of totally positive algebraic integers. This improvement has a consequence for the search of Salem numbers that have a negative trace. The same method also gives a small improvement of the upper bound for the integer transfinite diameter of [0,1].

Abstract. In this work, we show how suitable generalizations of the integer transfinite diameter of some compact sets in C give very good bounds for coefficients of polynomials with small Mahler measure. By this way, we give the list of all monic irreducible primitive polynomials of Z[X] of degree at most 36 with Mahler measure less than 1. 324... and of degree 38 and 40 with Mahler measure less than 1. 31.

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Let α be an algebraic integer whose all conjugates lie in a sector | arg z| ≤ θ, 0 ≤ θ < 90 •. Using the method of auxiliary functions, we first improve the known lower bounds of the absolute length of totally positive algebraic integers, i.e., when θ is equal to 0. Then, for 0 < θ < 90 • , we compute the greatest lower bound c(θ) of the absolute length of α, for θ belonging to eight subintervals of [0, 90 •). Moreover, we have a complete subinterval, i.e., an interval on which the function c(θ) describing the minimum on the sector | arg(z)| ≤ θ is constant, with jump discontinuities at each end. Finally, we obtain an upper bound for the integer transfinite diameter of the interval [0, 1] from the lower bound of the absolute length. The polynomials involved in the auxiliary functions are found by our recursive algorithm.