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 $$y\mid A(x),\qquad x\mid B(y),$$ 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.