On Amoebas and the Arithmetic Rigidity of the Mahler Measure
Abstract
We develop a geometric rigidity theory for Mahler measure based on the global structure of Ronkin functions and amoebas. Our starting point is the theorem of Ronkin describing the affine linearity of the Ronkin function on the complement of the amoeba and the identification of its gradients with lattice points of the Newton polytope. Combining this deep analytic input with convex duality and toric geometry, we show that the vanishing of the Mahler measure is an extremal phenomenon: it occurs if and only if the Ronkin function collapses to the piecewise linear support function of the Newton polytope. This collapse forces a degeneration of the amoeba onto the (n − 1)-skeleton of a toric fan and implies that the zero locus of the polynomial is contained in a translate of a codimension-one subtorus. As a first application of this rigidity principle, we recover the generalized Kronecker theorem characterizing Laurent polynomials of vanishing Mahler measure. Our approach bypasses the classical Boyd--Lawton limit formulas and replaces analytic projection arguments by a global convex-geometric and potential-theoretic mechanism rooted in the geometry of amoebas.
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BibTeX
@misc{hajli2026amoebas,
title = {{On Amoebas and the Arithmetic Rigidity of the Mahler Measure}},
author = {Mounir Hajli},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07181},
eprint = {2611.07181},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07181},
}
RIS
TY - UNPB
TI - On Amoebas and the Arithmetic Rigidity of the Mahler Measure
AU - Mounir Hajli
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07181
UR - https://arxiv.org/abs/2611.07181
AB - We develop a geometric rigidity theory for Mahler measure based on the global structure of Ronkin functions and amoebas. Our starting point is the theorem of Ronkin describing the affine linearity of the Ronkin function on the complement of the amoeba and the identification of its gradients with lattice points of the Newton polytope. Combining this deep analytic input with convex duality and toric geometry, we show that the vanishing of the Mahler measure is an extremal phenomenon: it occurs if and only if the Ronkin function collapses to the piecewise linear support function of the Newton polytope. This collapse forces a degeneration of the amoeba onto the (n − 1)-skeleton of a toric fan and implies that the zero locus of the polynomial is contained in a translate of a codimension-one subtorus. As a first application of this rigidity principle, we recover the generalized Kronecker theorem characterizing Laurent polynomials of vanishing Mahler measure. Our approach bypasses the classical Boyd--Lawton limit formulas and replaces analytic projection arguments by a global convex-geometric and potential-theoretic mechanism rooted in the geometry of amoebas.
ER -