An upper bound for the smallest limit point of the absolute Mahler measure of totally positive algebraic integers

Let Ω(α) = M(α)1/deg α be the absolute Mahler measure of an algebraic number α, and let ℓ+ be the smallest limit point of the values of Ω on totally positive algebraic integers. Smyth's work gives ℓ+ ≤ ℓ2 = 1.72731…, where ℓ = 1.31427… is the limit point that he found for totally real algebraic integers. We prove ℓ+ ≤ 1.7239501, and that infinitely many totally positive algebraic integers satisfy Ω(α) < 1.7239501. Together with the lower bound ℓ+ > 1.7234615 of a companion paper, this places ℓ+ in an interval of length less than 4.9·10−4. The proof exhibits a measure μ on [0, 9] with a piecewise linear density on 3481 rational nodes, together with a lower bound for its logarithmic potential by a combination of log|Q| for 21 irreducible polynomials Q, namely x and polynomials taken from Flammang's work. By a criterion of Smith, μ, normalized to mass 1, is a limit of distributions of conjugates of totally positive algebraic integers, and its integral of log+ x gives the bound. Two separately written programs verify the conditions in ball arithmetic. The certificate, the two checkers with their logs and controls, and the programs that produced the certificate are archived at doi:10.5281/zenodo.23529778 and available at github.com/tejasvi/totally-positive-mahler-upper.