Root structures of polynomials with sparse exponents

TL;DR

This paper investigates the root structures of sparse polynomials with fixed exponents, establishing a stratification of roots and demonstrating how roots cluster and relate to polynomial factorisations.

Contribution

It introduces a stratification of roots into tiers and shows how roots are contained within separated balls, linking root structures to polynomial factorisation.

Findings

Roots can be grouped into tiers of comparable sizes.

Each root is contained in a small ball with at most L roots.

Root structures of the polynomial and its factorisation are closely related.

Abstract

For real polynomials with (sparse) exponents in some fixed set, \[ \Psi(t)=x+y_1t^{k_1}+\ldots +y_L t^{k_L}, \] we analyse the types of root structures that might occur as the coefficients vary. We first establish a stratification of roots into tiers, each containing roots of comparable sizes. We then show that there exists a suitable small parameter $\epsilon>0$ such that, for any root $w\in \mathbb{C}$, $B(w,\epsilon|w|)$ contains at most $L$ roots, counted with multiplicity. Our analysis suggests the consideration of a rough factorisation of the original polynomial and we establish the closeness of the corresponding root structures: there exists a covering of the roots by balls wherein a) each ball contains the same number of roots of the original polynomial and of its rough factorisation and b) the balls are strongly separated.