TejasviTomar
October 2026 · Preprint

Bounds for the smallest limit point of the areal Mahler measure of totally positive algebraic integers

Tejasvi Singh Tomar

In plain words

Totally positive algebraic integers are those whose polynomial has only positive roots. Pritsker's areal Mahler measure, taken per degree of that polynomial, gauges their size like the ordinary Mahler measure but averages over a disc rather than a circle. Flammang proved in 2015 that, apart from a few listed exceptions, it is at least 1.380047 for such integers. Using the methods of two companion papers on the ordinary measure, this paper places the smallest value at which these areal measures accumulate between 1.3830134 and 1.3842933. No earlier upper bound appears to be known.

Abstract

Pritsker's areal analog of the Mahler measure replaces the integral of log|P| over the unit circle by its integral over the unit disc with respect to normalized area. For an algebraic number α of degree n whose minimal polynomial P in Z[x] is primitive with positive leading coefficient, let N0(α) = ‖P‖01/n. Flammang proved that N0(α) ≥ 1.380047 for every totally positive algebraic integer α outside an explicit finite set. We prove that all but finitely many totally positive algebraic numbers satisfy N0(α) > 1.3830134, and that infinitely many totally positive algebraic integers satisfy N0(α) < 1.3842933. So the smallest limit point of the values of N0 on totally positive algebraic integers, and the essential minimum, lie between 1.3830134 and 1.3842933; we are not aware of an earlier upper bound. The lower bound comes from an auxiliary function with Flammang's polynomials and a logarithmic energy term, and the upper bound from a measure that satisfies a criterion of Smith, as in our two papers on the classical Mahler measure of totally positive algebraic integers. Both certificates are verified in ball arithmetic.

Two stacked panels on the same horizontal scale, with no axes or labels. Top: groups of thin magenta bars of uneven height on a black baseline, separated by gaps, with the widest gap a little left of the middle. Bottom: a teal region under a jagged black curve with many thin vertical spikes rising toward a row of small yellow dots, deep narrow dips that nearly reach the bottom, rounded arches where the spikes thin out, the largest arch a little left of the middle, and slopes rising at both ends.
Illustration of the two certificates on one logarithmic axis, from x = 0.1 to 20. Top: the measure μ of the upper bound, which has a piecewise linear density on 3170 nodes between 0.155 and 16; each bar is the mass of μ in one of 600 equal intervals of log x, divided by the interval's length, so that area is mass. By a criterion of Smith, μ, normalized to mass 1, is a limit of distributions of conjugates of totally positive algebraic integers, and the bound is e raised to the average over μ of g(x) = log⁺ x + (min(1, x)² − 1)/2, 1.38429329…, rounded up to 1.3842933. Bottom: log₁₀ F for the auxiliary function F of the lower bound, built from g, from 34 polynomials of Flammang's table and from the potential of a second measure; the panel runs from F = 10⁻⁶ at the bottom up to F = 10, where the curve is cut off. F rises to infinity at the 230 positive roots of these polynomials, marked by the yellow dots, and comes close to 0 in many places, with a smallest value of about 3·10⁻⁶ on the plotted grid. The paper proves F ≥ 0 for all x > 0 other than these roots, which gives the lower bound 1.3830134. The picture is computed in floating point from the certificates' exact data.