The local root correlations of Kac polynomials with i.i.d. mean-zero finite-variance coefficients converge at scale 1/n to those of the Gaussian analytic function, proven by direct comparison using basic complex analysis and Esseen's anti-concentration bound.
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A simple proof of local universality for roots of Kac polynomials
The local root correlations of Kac polynomials with i.i.d. mean-zero finite-variance coefficients converge at scale 1/n to those of the Gaussian analytic function, proven by direct comparison using basic complex analysis and Esseen's anti-concentration bound.