Comptes Rendus
Article de recherche - Systèmes dynamiques, Théorie des nombres
Topology of zero sets of polynomials with square discriminant
[Topologie des ensembles de zéros des polynômes à discriminant carré]
David Hokken1
1Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, The Netherlands
Comptes Rendus. Mathématique, Volume 364 (2026), pp. 101-106

Let $\mathcal{N} \ne \lbrace 0\rbrace $ be a fixed set of integers, closed under multiplication, closed under negation, or containing $\lbrace \pm 1\rbrace $. 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.

Soit $\mathcal{N} \ne \lbrace 0\rbrace $ un ensemble fixe d’entiers, stable par multiplication, stable par passage à l’opposé, ou contenant $\lbrace \pm 1\rbrace $. Nous démontrons que tout zéro d’un polynôme de $\mathbf{Z}[X]$ dont les coefficients appartiennent à $\mathcal{N}$ peut être approximé dans $\mathbf{C}$, avec une précision arbitraire, par un zéro d’un polynôme de $\mathbf{Z}[X]$ à discriminant carré dont les coefficients appartiennent également à $\mathcal{N}$. Ainsi, la topologie de l’adhérence dans $\mathbf{C}$ de l’ensemble des zéros de tous ces polynômes est insensible au fait que le discriminant soit un carré, contrairement aux groupes de Galois de ces polynômes.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.824
Classification : 30C15, 11C08, 28A80
Keywords: Zeros of polynomials, self-similar fractals, square discriminant
Mots-clés : Zéros de polynômes, fractales autosimilaires, discriminant carré

David Hokken  1

1 Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, The Netherlands
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Topology of zero sets of polynomials with square discriminant},
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David Hokken. Topology of zero sets of polynomials with square discriminant. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 101-106. doi: 10.5802/crmath.824

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