A family of Salem polynomials breaking the Lehmer record

We present a family of symmetrized polynomials Sk(λ) that yield new irreducible Salem polynomials with Mahler measures decreasing below the classical Lehmer number L = 1.1762808183 for k ≥ 6. The smallest value obtained so far is M31 = 1.03878370673 (degree 162), which improves L by 11.71%. All computations were performed with SageMath using 200-bit precision and verified multiple times.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-03-05
DOI
https://doi.org/10.5281/zenodo.18874127
Primary Topic
Advanced Mathematical Identities
Type
article
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article

A family of Salem polynomials breaking the Lehmer record

Emma Helmdach
Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Identities
article

A family of Salem polynomials breaking the Lehmer record

Emma Helmdach
article en

Abstract

We present a family of symmetrized polynomials Sk(λ) that yield new irreducible Salem polynomials with Mahler measures decreasing below the classical Lehmer number L = 1.1762808183 for k ≥ 6. The smallest value obtained so far is M31 = 1.03878370673 (degree 162), which improves L by 11.71%. All computations were performed with SageMath using 200-bit precision and verified multiple times.

Zenodo (CERN European Organization for Nuclear Research)
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Advanced Mathematical Identities
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