Least common multiples of polynomial values: averaging, ray classes, and Galois statistics
Abstract
We study analogues over number fields and the probabilistic congruence input that enters averaged versions of this problem. We prove an asymptotic formula for the least common multiple of ideals in a fixed ray class, record local equidistribution estimates for random Sn-polynomials, and give a conditional reduction of the averaged higher degree problem to explicit dispersion and large prime collision estimates. We also formulate an individual quadratic number field analogue and reduce it to a prime ideal quadratic root equidistribution statement together with a geometric pair counting estimate. The Galois group counting input is updated using Bhargava's theorem over ℚ and the determinant method estimates of https://arxiv.org/abs/2610.35030 .
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BibTeX
@misc{viglino2026least,
title = {{Least common multiples of polynomial values: averaging, ray classes, and Galois statistics}},
author = {Ilaria Viglino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39911},
eprint = {2610.39911},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39911},
note = {CC0},
}
RIS
TY - UNPB
TI - Least common multiples of polynomial values: averaging, ray classes, and Galois statistics
AU - Ilaria Viglino
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39911
UR - https://arxiv.org/abs/2610.39911
AB - We study analogues over number fields and the probabilistic congruence input that enters averaged versions of this problem. We prove an asymptotic formula for the least common multiple of ideals in a fixed ray class, record local equidistribution estimates for random S_n-polynomials, and give a conditional reduction of the averaged higher degree problem to explicit dispersion and large prime collision estimates. We also formulate an individual quadratic number field analogue and reduce it to a prime ideal quadratic root equidistribution statement together with a geometric pair counting estimate. The Galois group counting input is updated using Bhargava's theorem over ℚ and the determinant method estimates of https://arxiv.org/abs/2610.35030 .
ER -