Reuben Wheeler

The structure of the roots of real polynomials

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.