[Paper Review] Is a bivariate polynomial with plus minus 1 coefficients irreducible? Very likely!

Lior Bary‐Soroker, Gady Kozma|arXiv (Cornell University)|Feb 21, 2016

Analytic Number Theory Research3 references3 citations

TL;DR

This paper investigates the irreducibility of bivariate polynomials with coefficients restricted to ±1. Using probabilistic methods and a variant of Rivin's argument on coefficient reduction modulo M, the authors prove that as the degree r increases, the probability of such a polynomial being reducible tends to zero—demonstrating that these polynomials are almost surely irreducible for large degrees.

ABSTRACT

We prove that a random bivariate polynomial with plus minus 1 coefficients is irreducible with high probability.

Motivation & Objective

• To investigate the likelihood of reducibility in bivariate polynomials with coefficients restricted to ±1.
• To extend known results on univariate polynomials with ±1 coefficients to the bivariate case.
• To show that for large degrees, such bivariate polynomials are overwhelmingly likely to be irreducible.
• To provide a rigorous probabilistic bound on the reducibility probability using modular arithmetic and divisor counting.

Proposed method

• Use a variant of Rivin’s argument that conditions on leading and constant coefficients modulo M to bound reducible polynomials.
• Define the set Ω(r,N) of polynomials with odd coefficients in [−(2N−1), 2N−1], ensuring injective reduction modulo M=2N+1.
• Bound the number of reducible polynomials by counting factorizations via coefficient decomposition and modular image constraints.
• Apply the bound ∑|d|≤N 1/|d| ≪ log N to derive a (log N)² factor in the probability estimate.
• Use the fact that F(X,Y) reducible implies F(2,Y) or F(X,2) reducible or F(X,Y)=f(X)g(Y), enabling case analysis.
• Estimate each case’s probability using the main bound and show decay as r→∞.

Experimental results

Research questions

• RQ1 What is the asymptotic probability that a bivariate polynomial with ±1 coefficients is reducible as the degree increases?
• RQ2 Can the irreducibility of such polynomials be established using probabilistic methods despite the difficulty in the univariate case?
• RQ3 How does the inclusion of a second variable affect the likelihood of reducibility compared to univariate ±1 polynomials?
• RQ4 What role does modular reduction play in bounding the number of reducible polynomials with odd coefficients?
• RQ5 Can the structure of bivariate factorization be leveraged to prove almost-sure irreducibility?

Key findings

• The probability that a bivariate polynomial F(X,Y) of degree r with ±1 coefficients is reducible decays to zero as r→∞.
• The bound on reducibility probability is O(r³ / 2ʳ), which tends to zero exponentially fast.
• The main technical tool is a refined version of Rivin’s argument, yielding a bound of C·r(log N)² / N · (1 + 1/(2N))ʳ for polynomials with odd coefficients in [−(2N−1), 2N−1].
• For F(2,Y) and F(X,2), the coefficient sizes grow as 2ʳ, so setting N=2ʳ gives the decay rate r³ / 2ʳ.
• The case F(X,Y)=f(X)g(Y) contributes negligibly, with probability at most 2⁻ʳ², which is asymptotically negligible.
• The combined probability of reducibility across all cases tends to zero, proving that such polynomials are almost surely irreducible for large r.