Multiplicity of zeros of polynomials

In 1940 Paul Erdős and Paul Turán proved their basic discrepancy estimate stating that if a monic polynomial of degree $n$ with real zeros has supremum norm $\|P_n\|_{[-1,1]}\le M_n2^{-n}$ on $[-1,1]$, then the normalized distribution of their zeros is closer than $C\sqrt {(\log M_n)/n}$ to the arcsine (Chebyshev) distribution. This implies that the highest multiplicity of the zeros is $\le C\sqrt {(\log M_n) n}$. In particular, if the norm is $O(1)2^{-n}$, then the highest multiplicity is $O(\sqrt{n})$, and Erdős and Turán conjectured that $\sqrt n$ order multiplicity can be achieved under this condition.

In this talk sharp bounds are given for the highest multiplicity of zeros of polynomials in terms of their norm on smooth Jordan curves and arcs. The results solve the just mentioned problem of Erdős and Turán. In addition, their bound on the multiplicities for the case $[-1,1]$ will be made more precise by comparing the norm not to $2^{-n}$ but to the theoretical minimum $2^{1-n}$ (note that, by Chebyshev's theorem, $M_n\ge 2$, so from the Erdős–Turán estimate one cannot get a smaller bound for the multiplicity than $O(\sqrt n)$, while if one can prove the bound $O(\sqrt {(\log N_n) n})$ from $\|P_n\|_{[-1,1]}\le N_n2^{1-n}$, then this gives $o(\sqrt{n})$ as soon as $N_n=1+o(1)$). We shall also discuss why there are no such results on non-smooth curves and arcs, or on sets consisting of more than one component. The proofs use potential theory and a Faber-type modification of a result of G. Halász.