Topology of zero sets of polynomials with square discriminant

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>≠</mml:mo> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> be a fixed set of integers, closed under multiplication, closed under negation, or containing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . We prove that any zero of a polynomial in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="bold">Z</mml:mi> <mml:mo>[</mml:mo> <mml:mi>X</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> whose coefficients lie in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒩</mml:mi> </mml:math> can be approximated in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">C</mml:mi> </mml:math> to arbitrary precision by a zero of a polynomial in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="bold">Z</mml:mi> <mml:mo>[</mml:mo> <mml:mi>X</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> with square discriminant whose coefficients also lie in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒩</mml:mi> </mml:math> . Hence the topology of the closure in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">C</mml:mi> </mml:math> 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.