Regular closedness off the real axis for root closures of bounded-height integer polynomials

Abstract

For each integer $h\ge1$, let $Z_h$ be the closure in the complex plane of the roots of nonzero integer polynomials whose coefficients have absolute value at most $h$, with no degree restriction. We prove that $Z_h$ is regular closed off the real axis: every non-real point of $Z_h$ is a limit of interior points. The root-closure identification of Espigulé and Juher, for which we include a self-contained proof, shows that the same closure arises from monic polynomials and that outside the closed unit disc it is the connectedness locus of $h+1$ equally spaced collinear similarities. The main step proves that this locus is regular closed off the real axis; this was known for $h=1$ by Calegari, Koch and Walker, for $h\ge19$, and inside an explicit lens. The symmetry $z\mapsto1/z$ then gives the full-plane statement. The proof adapts the method of traps of Calegari, Koch and Walker, using a neighboring-piece criterion, short chains through consecutive digits, and Rouché's theorem. We also classify the convex attractors.