 Finite-height polynomial root avoidance

Suppose $P({\xi }^{\prime },z)$ has degree $m$ in $z$ and a nonzero constant leading coefficient $a$. Among the $m+1$ heights $h_{\ell }=3\ell$, at least one satisfiesat each real ${\xi }^{\prime }$. Factor into $m$ linear factors with multiplicity. Each root's imaginary part can be within distance less than one of at most one candidate height. The pigeonhole principle leaves a height whose distance from every root is at least one, proving the product bound. The height can be chosen measurably: the set where a candidate succeeds is the closed intersection of $|P({\xi }^{\prime },q+i{h}_{\ell })|\ge |a|$ over rational $q$, and selecting the first successful candidate gives a Borel set partition. This avoids any need for continuous global root labels.