发表机构
The George Washington University（乔治华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文研究整数系数多项式整除系统 $y\mid A(x), x\mid B(y)$ 的非零整数解，提出同余跳跃方法，建立互反多项式规则并分类二次情形，构造非单位循环四周期，给出解集有限化判据。
AI 中文摘要
设 $A$ 和 $B$ 为整数系数多项式。我们研究满足 $$y\mid A(x),\qquad x\mid B(y)$$ 的非零整数解，使用可能改变多项式对的商变换。我们将此过程称为\emph{同余跳跃}。我们建立了一个一般的互反多项式规则来支配此类跳跃，无需互素、首一或单位假设，并给出了所得多项式状态何时可整体规范化的精确判据。伴随曲面恒等式提供了一种构造无限整体商链的机制，包括一个显式的混合次数例子。二次情形要严格得多。我们获得了有限链的定量分母界，对具有非整体圆锥参数的非恒定单侧无限链进行了分类，并表明改变的二次状态归结为固定圆锥上的普通 Vieta 动力学。我们进一步构造了一个真正的非单位循环四周期，具有无穷多个正整点，并表明其动力学允许统一的 Pell 型线性化。一个具有激励作用的非单位整除系统通过 Pell 轨道和变状态阶梯进行分析。最后，一个独立的关系格判据将某些解集归结为有限除数搜索。
英文摘要
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.
