PreprintCC BY-NC-ND 4.0arXiv
Smyth's Conjecture on the Mahler Measure of non-reciprocal trinomials of height $1$
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…, 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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BibTeX
@misc{voutier2026smyths,
title = {{Smyth's Conjecture on the Mahler Measure of non-reciprocal trinomials of height $1$}},
author = {Paul M Voutier},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37886},
eprint = {2609.37886},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37886},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Smyth's Conjecture on the Mahler Measure of non-reciprocal trinomials of height $1$
AU - Paul M Voutier
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37886
UR - https://arxiv.org/abs/2609.37886
AB - 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…, 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.
ER -