Intersective Polynomials and Universal Separation of Divosor Profiles

Technical Summary: Intersective Polynomials and Universal Separation of Divisor Profiles

Problem Statement
The paper addresses the classification of "universal divisor-profile separation" for pairs of integer polynomials $F, G \in \mathbb{Z}[x]$. For a subset $A \subset \mathbb{N}$ and an integer $m$, let $d_A(m)$ denote the number of members of $A$ that divide $m$. The property $P(F, G)$ is defined as the condition that for every infinite set $A \subset \mathbb{N}$, the difference between the divisor counts of the polynomial values is unbounded:
$$\limsup_{n \to \infty} \delta_A(F(n), G(n)) = \infty$$
where $\delta_A$ is an extended difference function handling cases where arguments vanish. The central question, extending prior work by Sárkőzy and Ding on linear pairs, is to determine exactly which pairs of polynomials satisfy $P(F, G)$, particularly when the polynomials share common factors or have arbitrary degrees.

Methodology
The proofs rely on a combination of analytic number theory, algebraic number theory, and combinatorial arguments:

1. Uniform Almost-Prime Values: The author utilizes a classical saturation theorem by Halberstam and Richert. A key technical contribution is Lemma 3.1, which establishes a uniform bound on the number of prime factors ($\Omega$) for integer-valued polynomials on specific arithmetic progressions (root progressions), independent of the polynomial's coefficients.
2. Adaptive Local-Routing: To handle common factors, the paper employs an "adaptive local-routing" mechanism. This involves constructing arithmetic progressions where specific divisors are forced to divide one polynomial while being controlled (kept small or non-dividing) for the other, effectively routing divisors through linear branches where valuations can be managed.
3. Galois Theory and Chebotarev Density: For polynomials sharing factors, the analysis shifts to the Galois group of the splitting field. The existence of roots modulo primes is linked to the action of Frobenius elements. The paper uses the Chebotarev density theorem to relate the existence of roots modulo almost all primes (prime-covering) to the absence of derangements in the Galois group's action on the roots.
4. Local-Global Analysis: The distinction between "intersective" (roots modulo every integer) and "prime-covering" (roots modulo almost every prime) is resolved by analyzing finite sets of "bad" primes and their $p$-adic root profiles, particularly for low-degree polynomials.

Key Contributions and Results

1. Coprime Classification (Theorem 1.1):
   For coprime nonzero polynomials $F, G \in \mathbb{Z}[x]$, the property $P(F, G)$ holds if and only if at least one of the polynomials is intersective (has a root modulo every positive integer). This result generalizes previous linear classifications to arbitrary degrees.

2. Simultaneous Dominance (Theorem 1.2):
   The paper proves a stronger simultaneous statement: if $F$ has a nonconstant intersective divisor $H$ coprime to a finite set of opponents $G_1, \dots, G_s$, then $F(n)$ eventually dominates the divisor counts of all $G_j(n)$ simultaneously for any infinite set $A$.

3. Common-Support Obstruction (Theorem 1.3 & Corollary 1.4):
   When $F$ and $G$ share factors, let $C$ be the product of common irreducible factors, and $U, V$ be the products of factors unique to $F$ and $G$, respectively.

   • A necessary condition for $P(F, G)$ is that the product $UV$ must have a root modulo all but finitely many primes (i.e., $UV$ is prime-covering).
   • Equivalently, the Galois group of $UV$ acting on its roots must contain no derangements (elements fixing no roots).

4. Complete Degree-Two Classification (Corollary 1.6):
   For pairs with degree at most two, the paper resolves the remaining "finite $p$-adic boundary." The property $P(F, G)$ holds if and only if:

   • $UV$ is prime-covering (Galois condition), and
   • At least one of $F$ or $G$ is intersective.
     The criterion is shown to be invariant under contents and factor multiplicities.

5. Three Linear Support Factors (Theorem 1.5):
   For a specific family involving three linear support factors ($L, U, V$) with arbitrary positive multiplicities, the paper provides a complete classification. $P(F, G)$ holds if and only if $\gcd(|u|, |v|) = 1$ and $\min\{\gcd(|\ell|, |u|), \gcd(|\ell|, |v|)\} = 1$. This corresponds to the condition that $UV$ and at least one of $F, G$ are intersective.

6. Intersectivity Testing (Proposition 5.2):
   The paper provides a complete, finite test for the intersectivity of polynomials of degree at most three, detailing the conditions on discriminants and local solvability for quadratic and cubic factors.

Significance and Claims
The paper claims to provide the first complete classification of universal divisor-profile separation for coprime polynomial pairs of arbitrary degree. It isolates the "quotient support" ($UV$) as the governing factor for the common-factor obstruction, distinguishing between the global Galois condition (prime-covering) and the local $p$-adic condition (intersectivity).

The author notes that while the Galois condition (no derangements) is necessary, it is not sufficient for the common-factor case due to the "finite $p$-adic boundary." The paper successfully closes this gap for all pairs of degree at most two and for specific higher-degree configurations involving three linear factors. The work extends the scope of polynomial recurrence and difference-set problems, moving from linear neighbors to arbitrary polynomial opponents and higher degrees.

The paper explicitly states that it does not provide a complete finite local-Galois criterion for the general case where $UV$ is prime-covering but neither $U$ nor $V$ is intersective (Question 6.2), identifying this as an open problem requiring a combination of permutation-group covering and $p$-adic branch analysis.