TejasviTomar
October 2026 · Preprint

A lower bound for the essential minimum of the Zhang–Zagier height

Tejasvi Singh Tomar

In plain words

The height of an algebraic number measures its complexity: roughly, the logarithm of the size of the simplest whole-number polynomial with it as a root, divided by its degree. Zhang proved in 1992 that only finitely many algebraic numbers α make the height of α plus that of 1 − α smaller than some positive amount. The upper limit of such amounts is called the essential minimum. Flammang showed in 2018 that it is at least 0.24874. Constructions cap it at 0.2537. This paper raises the lower bound to 0.24982933. Two separately written programs check its certificate.

Abstract

For an algebraic number α let h(α) be its absolute logarithmic Weil height, and let hZ(α) = h(α) + h(1 − α). The essential minimum of hZ is the supremum of the numbers H for which only finitely many algebraic numbers satisfy hZ(α) ≤ H. Zagier showed that it is at least ½ log((1 + √5)/2) = 0.2406059…, and the best lower bound so far is 0.24874, due to Flammang. We prove that the essential minimum is at least 0.24982933. The proof combines Smyth's method of auxiliary functions, here with five integer polynomials in z(1 − z), with a logarithmic energy term relative to a measure on the circles |z| = 1 and |1 − z| = 1. Resultants and discriminants of minimal polynomials are nonzero integers. They bound the polynomial terms and the energy term, and a condition on the degrees absorbs the leading coefficients, so the argument applies to all algebraic numbers. A superharmonicity argument reduces the positivity of the auxiliary function to the unit circle, where two separately written programs verify it in ball arithmetic for a certificate with exact rational data.

A teal field with two large overlapping black circles side by side that form a lens where they cross. The shading glows white just inside the outer arcs of both circles and darkens towards the two centres, the two crossing points and the corners. Magenta bands of varying width lie along parts of the circles, thickest on the two arcs that bound the lens, and yellow dots with black outlines sit on or just beside the circles, at their centres and at six points just outside.
Illustration of the certificate's auxiliary function F on part of the complex plane, on a logarithmic scale: white where F is close to 0, darker teal where it is larger, with thin contour lines. The black circles are |z| = 1 and |1 − z| = 1, and the magenta bands along them show the density of the reference measure in the energy term, wider where it is denser. The yellow dots are the 34 zeros of the five polynomials Q_j(z(1 − z)), where F is infinite. The paper proves F ≥ 0 everywhere, which gives the lower bound 0.24982933; the picture is computed in floating point from the certificate's exact data.