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A certified lower bound for the essential minimum of the Zhang-Zagier height

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

Let h be the absolute logarithmic Weil height. We prove that the essential minimum of the Zhang-Zagier height h(a)+h(1-a) is at least 0.2497815: for every d > 0, only finitely many algebraic numbers a satisfy h(a)+h(1-a) < 0.2497815 - d. The previous best lower bound was 0.24874 (Flammang, 2018). The proof combines Smyth's auxiliary-function method, with the polynomials of Doche and of Flammang, and the condition that the discriminant of the minimal polynomial is a nonzero integer, in the potential-theoretic form used by Smith and by Orloski, Talebizadeh Sardari and Smith for the trace problem. The infimum of the resulting auxiliary function is bounded below rigorously by monotone interval bounds and adaptive bisection, from a stored certificate. The statement concerns the essential minimum and is weaker in nature than the pointwise bounds of Doche and Flammang.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

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Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 03, 2026