Random Kac polynomials have discriminant |Δ(f_n)| = n^{2n} e^{-D_* n(1+o(1))} with an explicit universal constant D_* ≈ 5.92947.
Counting (skew-)reciprocal Littlewood polynomials with square discriminant
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A Littlewood polynomial is a single-variable polynomial all of whose coefficients lie in $\{ \pm 1\}$. We establish the leading term asymptotics of the number of reciprocal or skew-reciprocal Littlewood polynomials with square discriminant. This relates to a bounded-height analogue of the Van der Waerden conjecture on Galois groups of random polynomials. As a byproduct, we establish the asymptotics of certain Gaussian-weighted counts of Pythagorean triples.
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Law of large numbers for the discriminant of random polynomials
Random Kac polynomials have discriminant |Δ(f_n)| = n^{2n} e^{-D_* n(1+o(1))} with an explicit universal constant D_* ≈ 5.92947.