同余跳跃中的互反动力学：积分轨道与赋值锥
Reciprocal Dynamics in Polynomial Congruence Jumping: Integral Orbits and Valuation Cones
AI总结：
该研究针对固定次数n,m的多项式对，定义同余跳跃中的互反离散动力系统，通过SL₂(ℤ)矩阵演化常数项赋值，分类不同s=nm regime下的积分轨道，给出局部准则与周期轨道结果。
摘要：
对于固定的正次数n,m，首一交替互反变换定义了多项式对的有限维系数空间上的有理离散动力系统。该系统源于多项式整除问题中的同余跳跃，其中商移动改变多项式状态，但其系数动力学可独立研究。我们证明，所有非零常数项的状态经过四次交替更新后会回到其有理规范类，并分类该返回轨道何时在ℤ上实现。局部障碍由端点p进赋值缺陷编码。在第一个积分块后，常数项赋值由显式矩阵M_{n,m}∈SL₂(ℤ)演化，其迹仅依赖于s=nm。这产生了s<4时的有限阶椭圆型 regime、s=4时的抛物型膨胀 regime，以及s>4时的双曲型 regime。在双曲情形中，不变阈值、无理特征锥与离散谷公式给出了前向和双向无限积分轨道的有限准则；然而，失败可被任意长时间延迟。逐点层面上，分离的辅助关系通过跳跃变量投影传输。一次跳跃保持可见支撑秩，而两次交替跳跃将其坍缩至至多一维；饱和无法恢复秩。在二次特例中，秩为2的p进关系格给出了精确的局部消去深度公式与周期Vieta轨道的有限准则。我们分类在不可约六环上无限饱和的固定辅助关系，并在抛物型与固定双曲轨道上构造任意长的持续性。
英文摘要：
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.
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