PreprintCC BY 4.0arXiv
Asymptotic low slopes of toric exponential sums
Abstract
Let Δ be an integral Newton polytope containing the origin. We prove that the generic Newton slopes corresponding to Hodge slopes strictly smaller than one differ from their Hodge values by OΔ(p−1). No hypothesis is imposed on the semigroup generated by the lattice points of Δ. The proof combines an auxiliary prime and the Cartier operator to show that a coefficient of each required Dwork principal minor is nonzero. When the origin is interior to Δ, duality gives the same estimate for the corresponding high slopes and, in dimension two, for all slopes.
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BibTeX
@misc{ren2026asymptotic,
title = {{Asymptotic low slopes of toric exponential sums}},
author = {Rufei Ren},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25789},
eprint = {2610.25789},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25789},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Asymptotic low slopes of toric exponential sums
AU - Rufei Ren
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25789
UR - https://arxiv.org/abs/2610.25789
AB - Let Δ be an integral Newton polytope containing the origin. We prove that the generic Newton slopes corresponding to Hodge slopes strictly smaller than one differ from their Hodge values by O_Δ(p^−1). No hypothesis is imposed on the semigroup generated by the lattice points of Δ. The proof combines an auxiliary prime and the Cartier operator to show that a coefficient of each required Dwork principal minor is nonzero. When the origin is interior to Δ, duality gives the same estimate for the corresponding high slopes and, in dimension two, for all slopes.
ER -