Intersective Polynomials and Universal Separation of Divosor Profiles

Abstract

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.