Law of large numbers for the discriminant of random polynomials

2025/06/13 by Michelen, Marcus, Yakir, Oren
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2506.12206

Abstract

Let f_n be a random polynomial of degree n, whose coefficients are independent and identically distributed random variables with mean-zero and variance one. Let Δ(f_n) denote the discriminant of f_n, that is Δ(f_n) = A^2n-2∏_{i < j} (α_j - α_i)² where A is the leading coefficient of f_n and α₁,…α_n are its roots. We prove that with high probability |Δ(f_n)| = n²ⁿ e^-\sf D_∗ n(1+o(1)) as n→ ∞, for some explicit universal constant \sf D_∗>0. A key step in the proof is an analytic representation for the logarithm of the discriminant, which captures both the distributional reciprocal symmetry of the random roots and the cancellations this symmetry induces.

Citations

• Limit law for root separation in random polynomials
• Full Galois groups of polynomials with slowly growing coefficients
• Counting (skew-)reciprocal Littlewood polynomials with square discriminant
• Probabilistic Galois Theory -- The Square Discriminant Case
• Galois groups of random integer polynomials and van der Waerden's Conjecture
• Universality of the minimum modulus for random trigonometric polynomials
• The minimum modulus of Gaussian trigonometric polynomials
• Discriminant and root separation of integral polynomials

Related

• Discriminant and root separation of integral polynomials
• A strong law of large numbers for real roots of random polynomials
• On the distribution of polynomial discriminants: totally real case
• Large deviations for zeros of random polynomials with i.i.d. exponential\n coefficients
• Probabilistic Galois Theory -- The Square Discriminant Case