Magic functions and the equidistribution of zeros of polynomials

Date: 2021-05-31

Time: 11:00 - 11:40

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In 1950, Erdös and Turán established an interesting inequality for the discrepancy on angular equidistribution of the zeros of a given polynomial $P$. In qualitative terms, assuming that the polynomial is monic, it roughly says that if the size of $P$ in the unit disk is small, and the constant coefficient is not too small, then the angles of the zeros tend to equidistribute as the degree grows. The quantitative discrepancy inequality of Erdös and Turán was later refined three times: by Ganelius in 1954, by Mignotte in 1992, and by Soundararajan in 2019. I  want to present in this talk a recent approach, based on a Fourier optimization problem involving the Hilbert transform, that leads to a new non-trivial refinement of this inequality.