AI 中文总结
该研究针对任意次数的互素多项式对及所有次数不超过2的多项式对，分类了除数剖面的通用分离问题，得出互素多项式对通用无界分离的充要条件，并解决了次数不超过2的多项式对的有限p进边界问题，证明结合了根进展的一致殆素数值与自适应局部路由论证。
AI 中文摘要
我们对任意次数的互素多项式对以及所有次数不超过2的多项式对的通用除数剖面分离进行了分类。对于\footnotesize A\footnotesize \normalsize⊂\footnotesize N\footnotesize \normalsize和\footnotesize m\footnotesize \normalsize∈\footnotesize Z\footnotesize \normalsize，令\footnotesize d_A(m)\footnotesize \normalsize计数\footnotesize A\footnotesize \normalsize中整除\footnotesize m\footnotesize \normalsize的元素个数。对于互素非零的\footnotesize F,G\footnotesize \normalsize∈\footnotesize Z[x]\footnotesize \normalsize，\footnotesize d_A(F(n))\footnotesize \normalsize与\footnotesize d_A(G(n))\footnotesize \normalsize之间的通用无界分离发生当且仅当\footnotesize F,G\footnotesize \normalsize中的一个是相交多项式，即对每个正整数都有模根。更一般地，从有限个多项式对手中分离出一个相交因子会产生对所有这些对手的同时单侧优势。对于有公共不可约因子的多项式对，设\footnotesize U,V\footnotesize \normalsize分别为仅出现在两边的因子的乘积。通用分离迫使\footnotesize UV\footnotesize \normalsize在几乎每个素数处都有模根；等价地，其伽罗瓦作用没有无不动点置换。我们对所有次数不超过2的多项式对解决了剩余的有限\footnotesize p\footnotesize \normalsize进边界问题：分离恰好当\footnotesize UV\footnotesize \normalsize和\footnotesize F,G\footnotesize \normalsize中至少一个是相交多项式时成立，且该判据不随内容或因子重数变化。对于具有任意正重数的三个线性支撑因子，该判据在任意次数下均成立。证明结合了根进展上的一致殆素数值与自适应局部路由论证。
英文摘要
We classify universal divisor-profile separation for coprime polynomial pairs of arbitrary degree and for all pairs of degree at most two. For \(A\subset\N\) and \(m\in\Z\), let \(d_A(m)\) count the members of \(A\) dividing \(m\). For coprime nonzero \(F,G\in\Z[x]\), universally unbounded separation between \(d_A(F(n))\) and \(d_A(G(n))\) occurs if and only if one of \(F,G\) is intersective, that is, has a root modulo every positive integer. More generally, an intersective factor separated from finitely many polynomial opponents yields simultaneous one-sided dominance against all of them. For pairs with common irreducible factors, let \(U,V\) be the products of the factors occurring only on the two respective sides. Universal separation forces \(UV\) to have a root modulo almost every prime; equivalently, its Galois action has no derangement. We resolve the remaining finite \(p\)-adic boundary for all pairs of degree at most two: separation holds exactly when \(UV\) and at least one of \(F,G\) are intersective, and the criterion is unchanged by contents or factor multiplicities. The same criterion holds, in arbitrary degree, for three linear support factors with arbitrary positive multiplicities. The proofs combine uniform almost-prime values on root progressions with an adaptive local-routing argument.
