### The structure of the roots of real polynomials

Analysis and Geometry Seminar

9th December 2021, 2:00 pm – 3:00 pm

Fry Building, 2.06

We consider real polynomials,

\[\Psi(t) = xt + y_1 t^{ k_1 } + \ldots + y_Lt^{k_L},\]

with exponents taken from a fixed set of size L+1. We carry out a structural analysis of their roots. Extending work of Kowalski and Wright, we first determine the stratification of roots into up to L tiers, each containing roots of comparable sizes. By considering each tier separately, we then show that, for a suitable small parameter $\epsilon$, any ball $B(z,\epsilon |z|)$ contains at most L roots. These results were motivated by, and used to recover, a sharp oscillatory integral estimate of Hickman and Wright with applications to Fourier restriction.

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