Good slopes, congruence preservation, and gaps over residue rings
Abstract
For n≥2, a good slope of a permutation f of ℤn=ℤ/nℤ is an element m∈ℤn for which x↦ f(x)+mx is a permutation. We call f congruence-preserving if it preserves congruences modulo every divisor of n. For odd prime powers n, we classify all permutations with more than n/2 good slopes. Among permutations that are not congruence-preserving, we determine the maximum number of good slopes for n=3e with e≥2, and for n=25, $125$ and $625$. Every permutation of ℤn has at most φ(n) good slopes, where φ is Euler's totient function. For several families of moduli with at least two distinct prime divisors, we prove a sharp upper bound for the number of good slopes of the permutations with fewer than φ(n) good slopes. These families include 9q and 3qe for every prime q≥5 and every e≥1. Together with a companion paper, this settles the bound for every odd composite modulus below $105$.
Related
- Extremal permutations of ℤn with many good slopes and applications to quantum Latin squares
- On the variation of slopes over moduli stacks of Galois representations
- Reducible handle additions along separating slopes
- Asymptotic low slopes of toric exponential sums
- Extended Tropical Characteristic Fibers and Slopes of Binomial D-Modules
BibTeX
@misc{bhunia2026good,
title = {{Good slopes, congruence preservation, and gaps over residue rings}},
author = {Dipak K. Bhunia},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04551},
eprint = {2611.04551},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04551},
}
RIS
TY - UNPB
TI - Good slopes, congruence preservation, and gaps over residue rings
AU - Dipak K. Bhunia
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04551
UR - https://arxiv.org/abs/2611.04551
AB - For n≥2, a good slope of a permutation f of ℤ_n=ℤ/nℤ is an element m∈ℤ_n for which x↦ f(x)+mx is a permutation. We call f congruence-preserving if it preserves congruences modulo every divisor of n. For odd prime powers n, we classify all permutations with more than n/2 good slopes. Among permutations that are not congruence-preserving, we determine the maximum number of good slopes for n=3^e with e≥2, and for n=25, $125$ and $625$. Every permutation of ℤ_n has at most φ(n) good slopes, where φ is Euler's totient function. For several families of moduli with at least two distinct prime divisors, we prove a sharp upper bound for the number of good slopes of the permutations with fewer than φ(n) good slopes. These families include 9q and 3q^e for every prime q≥5 and every e≥1. Together with a companion paper, this settles the bound for every odd composite modulus below $105$.
ER -