Congruence Jumping in Polynomial Divisibility Systems: Arithmetic Structure and Solution Sets
Abstract
Let A and B be polynomials with integer coefficients. We study nonzero integer solutions of y| A(x), x| B(y), using quotient transformations that may change the polynomial pair. We call this process \emph{congruence jumping}. We establish a general reciprocal-polynomial rule governing such jumps, without coprimality, monicity, or unit assumptions, and give exact criteria for when the resulting polynomial states can be normalized integrally. Companion-surface identities provide a mechanism for constructing infinite integral quotient chains, including an explicit mixed-degree example. The quadratic case is considerably more rigid. We obtain a quantitative denominator bound for finite chains, classify the exceptional nonconstant one-sided infinite chains with nonintegral conic parameter, and show that changing quadratic states reduce to ordinary Vieta dynamics on a fixed conic. We further construct a genuinely nonunit recurrent four-cycle with infinitely many positive integral points and show that its dynamics admits a uniform Pell-type linearization. A motivating nonunit divisibility system is analyzed through Pell orbits and changing-state ladders. Finally, an independent relation-lattice criterion reduces certain solution sets to a finite divisor search.
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BibTeX
@misc{alekseyev2026congruence,
title = {{Congruence Jumping in Polynomial Divisibility Systems: Arithmetic Structure and Solution Sets}},
author = {Max A. Alekseyev and Dmitry I. Khomovsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00808},
eprint = {2610.00808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00808},
}
RIS
TY - UNPB
TI - Congruence Jumping in Polynomial Divisibility Systems: Arithmetic Structure and Solution Sets
AU - Max A. Alekseyev
AU - Dmitry I. Khomovsky
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00808
UR - https://arxiv.org/abs/2610.00808
AB - Let A and B be polynomials with integer coefficients. We study nonzero integer solutions of y| A(x), x| B(y), using quotient transformations that may change the polynomial pair. We call this process \emph{congruence jumping}. We establish a general reciprocal-polynomial rule governing such jumps, without coprimality, monicity, or unit assumptions, and give exact criteria for when the resulting polynomial states can be normalized integrally. Companion-surface identities provide a mechanism for constructing infinite integral quotient chains, including an explicit mixed-degree example. The quadratic case is considerably more rigid. We obtain a quantitative denominator bound for finite chains, classify the exceptional nonconstant one-sided infinite chains with nonintegral conic parameter, and show that changing quadratic states reduce to ordinary Vieta dynamics on a fixed conic. We further construct a genuinely nonunit recurrent four-cycle with infinitely many positive integral points and show that its dynamics admits a uniform Pell-type linearization. A motivating nonunit divisibility system is analyzed through Pell orbits and changing-state ladders. Finally, an independent relation-lattice criterion reduces certain solution sets to a finite divisor search.
ER -