Topology of zero sets of polynomials with square discriminant

TL;DR

This paper shows that the set of zeros of polynomials with coefficients in a certain integer set is topologically similar whether or not the discriminant is a perfect square, highlighting a surprising insensitivity in their zero set topology.

Contribution

It proves that zeros of polynomials with coefficients in a fixed set can be approximated by zeros of polynomials with square discriminant, revealing topological invariance.

Findings

Zeros can be approximated arbitrarily closely by polynomials with square discriminant.

The topology of zero sets is unaffected by the discriminant being a square.

The result contrasts with the behavior of Galois groups of the polynomials.

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.