For Kac polynomials and related random polynomials, the limiting zero distribution of the N_n-th derivative depends on the limit of N_n/n, with a phase transition and rescaling when N_n/n tends to 1.
Feng , Zeros of derivatives of Gaussian random polynomials on plane domain, preprint
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Zeros of repeated derivatives of random polynomials
For Kac polynomials and related random polynomials, the limiting zero distribution of the N_n-th derivative depends on the limit of N_n/n, with a phase transition and rescaling when N_n/n tends to 1.