Reciprocal Dynamics in Polynomial Congruence Jumping: Integral Orbits and Valuation Cones
Abstract
Alternating reciprocal transformations of polynomial divisibility systems couple an orbit of integer points to a rational dynamical system on polynomial coefficients. For fixed positive degrees n,m, we prove universal four-step return to rational gauge classes. Weighted primitive representatives preserve integral continuation through nondegenerate jumps using only exact division and coefficient gcds; the stronger requirement of monic integral representatives is governed by endpoint p-adic valuation defects. After one integral block, constant-term valuations evolve by an explicit unimodular matrix whose trace depends only on nm. This gives finite-order behavior for nm<4, parabolic dilations for nm=4, and hyperbolic dynamics for nm>4. In the hyperbolic case, invariant thresholds, irrational eigencones, and discrete valley formulas give finite criteria for forward and bi-infinite integral coefficient orbits, although failure can be delayed arbitrarily long. Every positive-constant canonical monic coefficient chain admits an explicit positive rational four-phase gauge. At nm=4, geometric gauges reduce nondegenerate compatible rational point orbits to two-periodic affine recurrences, hence to sequences satisfying linear recurrences with constant coefficients. Polynomial identities of degree at most h on each invariant conic admit certificates from 2h+1 initial phase states, while integral unimodular recurrence models yield periodic modular obstructions. For nm>4, eventually positive normalized point orbits tending to infinity have hyperbolic logarithmic growth and cannot satisfy such linear recurrences. Finally, separated auxiliary-relation transport reduces rational rank to at most one after two alternating jumps; quadratic specialization gives exact local saturation criteria and persistence results across periodic, parabolic, and hyperbolic orbits.
Related
- A Congruence Conjecture of Z.-W. Sun for Reciprocal Central Binomial Sums
- Multivariable Congruence Jumping: Periodic States, Integral Realizations, and Descent
- Congruence Jumping in Polynomial Divisibility Systems: Arithmetic Structure and Solution Sets
- Deterministic Polynomial-Time Primality Testing with a Single Binomial Congruence
- Counting Integral Matrices with Given Cubic Characteristic Polynomial
BibTeX
@misc{alekseyev2026reciprocal,
title = {{Reciprocal Dynamics in Polynomial Congruence Jumping: Integral Orbits and Valuation Cones}},
author = {Max A. Alekseyev and Dmitry I. Khomovsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20176},
eprint = {2610.20176},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20176},
}
RIS
TY - UNPB
TI - Reciprocal Dynamics in Polynomial Congruence Jumping: Integral Orbits and Valuation Cones
AU - Max A. Alekseyev
AU - Dmitry I. Khomovsky
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.20176
UR - https://arxiv.org/abs/2610.20176
AB - Alternating reciprocal transformations of polynomial divisibility systems couple an orbit of integer points to a rational dynamical system on polynomial coefficients. For fixed positive degrees n,m, we prove universal four-step return to rational gauge classes. Weighted primitive representatives preserve integral continuation through nondegenerate jumps using only exact division and coefficient gcds; the stronger requirement of monic integral representatives is governed by endpoint p-adic valuation defects. After one integral block, constant-term valuations evolve by an explicit unimodular matrix whose trace depends only on nm. This gives finite-order behavior for nm<4, parabolic dilations for nm=4, and hyperbolic dynamics for nm>4. In the hyperbolic case, invariant thresholds, irrational eigencones, and discrete valley formulas give finite criteria for forward and bi-infinite integral coefficient orbits, although failure can be delayed arbitrarily long. Every positive-constant canonical monic coefficient chain admits an explicit positive rational four-phase gauge. At nm=4, geometric gauges reduce nondegenerate compatible rational point orbits to two-periodic affine recurrences, hence to sequences satisfying linear recurrences with constant coefficients. Polynomial identities of degree at most h on each invariant conic admit certificates from 2h+1 initial phase states, while integral unimodular recurrence models yield periodic modular obstructions. For nm>4, eventually positive normalized point orbits tending to infinity have hyperbolic logarithmic growth and cannot satisfy such linear recurrences. Finally, separated auxiliary-relation transport reduces rational rank to at most one after two alternating jumps; quadratic specialization gives exact local saturation criteria and persistence results across periodic, parabolic, and hyperbolic orbits.
ER -