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arXiv · 2609.37886

Smyth's Conjecture on the Mahler Measure of non-reciprocal trinomials of height $1$

Paul M Voutier
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Abstract

We prove a conjecture of Smyth on the Mahler measure of non-reciprocal trinomials of height $1$, determining exactly when their Mahler measures are greater than or less than $M(x+y+1)=ρ=1.381356\ldots$, conjectured to be the smallest limit point of $M(P)$ for non-reciprocal polynomials. Conjecturally, our result gives the full spectrum of $M(P)<ρ$ for all non-reciprocal polynomials, not just for trinomials.

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Paul M Voutier. 2026-09-29. Smyth's Conjecture on the Mahler Measure of non-reciprocal trinomials of height $1$. https://arxiv.org/abs/2609.37886

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