  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.
