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Quantum Pick's Theorem in two dimensions
Abstract
Consider an n-dimensional sum of the form Sk=βqf(bf z) over lattice points bf z which lie in Kπ«, the K scaling of an n-dimensional rational polytope π«. We formulate a conjecture on the structure of SK considered for K prime, as a power series in h=qβ1 modulo the Kth cyclotomic polynomial, and in particular the existence of a decomposition in powers of the Gauss sum πΎ, up to the nth, whose coefficients are almost independent of K, where the leading coefficient is related to the volume of π« and the coefficient of πΎr has contributions from the codimension r skeleton of π«. The conjecture is proved for n=1,2.
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BibTeX
@misc{bardavid2026quantum,
title = {{Quantum Pick's Theorem in two dimensions}},
author = {Moris Metin Bardavid and Ruth Lawrence},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15234},
eprint = {2610.15234},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15234},
}
RIS
TY - UNPB
TI - Quantum Pick's Theorem in two dimensions
AU - Moris Metin Bardavid
AU - Ruth Lawrence
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15234
UR - https://arxiv.org/abs/2610.15234
AB - Consider an n-dimensional sum of the form S_k=βq^f(bf z) over lattice points bf z which lie in Kπ«, the K scaling of an n-dimensional rational polytope π«. We formulate a conjecture on the structure of S_K considered for K prime, as a power series in h=qβ1 modulo the K^th cyclotomic polynomial, and in particular the existence of a decomposition in powers of the Gauss sum πΎ, up to the n^th, whose coefficients are almost independent of K, where the leading coefficient is related to the volume of π« and the coefficient of πΎ^r has contributions from the codimension r skeleton of π«. The conjecture is proved for n=1,2.
ER -