[Paper Review] Distribution of reducible polynomials with a given coefficient set

Shane Chern|arXiv (Cornell University)|Feb 24, 2016

Advanced Combinatorial Mathematics8 references3 citations

TL;DR

This paper establishes an explicit upper bound for the number of reducible univariate polynomials with integer coefficients drawn from a fixed finite set 𝒮, using modular reduction and divisor function estimates. The key result bounds |ℛₙ*⁽𝒮⁾| ≤ 4(n−1)Mⁿ⁻²(∑ₐ∈𝒮* d(a))², where M ensures coefficient distinctness modulo M, and applies this to prove that the probability of random bivariate polynomials with coefficients in {0,±1} being reducible tends to zero as degree increases.

ABSTRACT

For a given set of integers $\mathcal{S}$, let $\mathcal{R}_n^*(\mathcal{S})$ denote the set of reducible polynomials $f(X)=a_nX^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$ over $\mathbb{Z}[X]$ with $a_i\in\mathcal{S}$ and $a_0a_n e 0$. In this note, we shall give an explicit bound of $|\mathcal{R}_n^*(\mathcal{S})|$. We also present an application of this bound to reducible bivariate polynomials over $\mathbb{Z}[X,Y]$.

Motivation & Objective

• To derive an explicit upper bound on the number of reducible univariate polynomials with coefficients restricted to a finite integer set 𝒮.
• To generalize Bary-Soroker and Kozma’s result on ±1 coefficient polynomials by extending the modular reduction method to arbitrary finite coefficient sets.
• To apply the bound to bivariate polynomials with coefficients in {0,±1}, proving that their reducibility probability tends to zero as degree increases.
• To demonstrate that the probability of reducibility vanishes asymptotically for such bivariate polynomials, using Euler’s identity and coefficient counting.

Proposed method

• Use modular reduction modulo M, where M ensures distinct residues for all elements in 𝒮, to injectively map polynomials into ℤ/Mℤ[X].
• Bound the number of factorizations by counting lifts of pairs (p̄, q̄) in ℤ/Mℤ[X] with fixed constant and leading coefficients.
• Apply the injectivity of the reduction map to show that each such pair lifts to at most one polynomial in ℛₙ*⁽𝒮⁾.
• Leverage the divisor function d(a) to count possible factorizations of constant and leading coefficients a₀ and aₙ.
• Use Euler’s identity ∏ₙ₌₀ᴺ⁻¹(x⁻³ⁿ + 1 + x³ⁿ) = ∑ₙ₌₋(³ᴺ⁻¹)/²^(³ᴺ⁻¹)/² xⁿ to show that substitution Y=3 yields all integer polynomials of bounded height.
• Combine the bound with asymptotic estimates ∑ₙ≤x d(n) ∼ x log x to show that the probability of reducibility decays as n⁻³/3ⁿ.

Experimental results

Research questions

• RQ1 What is the maximum number of reducible univariate polynomials with coefficients in a fixed finite set 𝒮 and non-zero constant and leading coefficients?
• RQ2 How can modular reduction techniques be extended to bound reducible polynomials over arbitrary finite coefficient sets?
• RQ3 Does the probability of reducibility in bivariate polynomials with coefficients in {0,±1} tend to zero as degree increases?
• RQ4 Can Euler’s identity be used to relate coefficient sets to the density of integer polynomials and aid in reducibility bounds?

Key findings

• An explicit upper bound is established: |ℛₙ*⁽𝒮⁾| ≤ 4(n−1)Mⁿ⁻²(∑ₐ∈𝒮* d(a))², where M is an integer such that all elements of 𝒮 are distinct modulo M.
• The bound is effective for any finite set 𝒮 ⊂ ℤ with 0 removed, and M can be chosen as max𝒮 − min𝒮 + 1 or smaller if residues are distinct modulo a smaller M.
• For bivariate polynomials F(X,Y) with coefficients in {0,±1}, the probability that F(X,3) is reducible decays as O(n³/3ⁿ) as n → ∞.
• The probability that F(X,Y) is reducible due to factorization F(X,Y) = f(X)g(Y) decays as O(3⁻ⁿ²), which is negligible compared to the main term.
• The combined probability of reducibility via any of the three mechanisms (F(X,3) reducible, F(3,Y) reducible, or F(X,Y) = f(X)g(Y)) tends to zero as n → ∞.
• Thus, limₙ→∞ ℙ(F reducible) = 0 for bivariate polynomials with coefficients in {0,±1}, extending Bary-Soroker and Kozma’s result to a broader coefficient set.