Abstract
In this note we initiate the probabilistic study of the critical points of polynomials of large degree with a given distribution of roots. Namely, let f be a polynomial of degree n whose zeros are chosen IID from a probability measure μ on \(\mathbb{C}\). We conjecture that the zero set of f ′ always converges in distribution to μ as n → ∞. We prove this for measures with finite one-dimensional energy. When μ is uniform on the unit circle this condition fails. In this special case the zero set of f ′ converges in distribution to that of the IID Gaussian random power series, a well known determinantal point process.
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Pemantle, R., Rivin, I. (2013). The Distribution of Zeros of the Derivative of a Random Polynomial. In: Kotsireas, I., Zima, E. (eds) Advances in Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30979-3_14
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