Topology of zero sets of polynomials with square discriminant

Abstract

Let $\mathcal{N} \neq \{0\}$ be a fixed set of integers, closed under multiplication, closed under negation, or containing $\{\pm 1\}$. We prove that any zero of a polynomial in $\mathbf{Z}[X]$ whose coefficients lie in $\mathcal{N}$ can be approximated in $\mathbf{C}$ to arbitrary precision by a zero of a polynomial in $\mathbf{Z}[X]$ with square discriminant whose coefficients also lie in $\mathcal{N}$. Hence the topology of the closure in $\mathbf{C}$ of the set of zeros of all such polynomials is insensitive to the discriminant being a square, in contrast to the Galois groups of the polynomials.