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Hermite equivalence of polynomials

Volume 209 / 2023

Manjul Bhargava, Jan-Hendrik Evertse, Kálmán Győry, László Remete, Ashvin A. Swaminathan Acta Arithmetica 209 (2023), 17-58 MSC: Primary 11C08. DOI: 10.4064/aa211113-12-11 Published online: 28 February 2023

Abstract

We compare results of Birch and Merriman (1972), Győry (1973, 1974) and Evertse and Győry (1991, 2017), which imply in a precise form that polynomials in $\mathbb {Z} [X]$ of given discriminant lie in finitely many ${\rm GL}_2(\mathbb {Z} )$-equivalence classes, with a forgotten theorem of Hermite (1854, 1857) which implies that polynomials in $\mathbb {Z} [X]$ with given discriminant lie in finitely many equivalence classes with respect to a weaker notion of equivalence, henceforth called “Hermite equivalence”. We show that ${\rm GL}_2(\mathbb {Z})$-equivalent polynomials are Hermite equivalent and that for polynomials of degree $2$ or $3$ the converse is also true. On the other hand, for every $n\geq 4$ we give an infinite class of examples of polynomials $f,g\in \mathbb {Z} [X]$ of degree $n$ that are Hermite equivalent but not ${\rm GL}_2(\mathbb {Z} )$-equivalent. One of the constructions (see Theorem 5.1) uses an irreducibility result for a certain class of polynomials which may be of independent interest.

We point out that the results of Birch and Merriman, Győry, and Evertse and Győry are in general much more precise than Hermite’s theorem. As a consequence, we correct a faulty reference occurring in Narkiewicz’ excellent book (2019), where ${\rm GL}_2 (\mathbb {Z})$-equivalence and Hermite equivalence of polynomials were mixed up.

Authors

  • Manjul BhargavaDepartment of Mathematics
    Princeton University
    Princeton, NJ 08540, USA
    e-mail
  • Jan-Hendrik EvertseDepartment of Mathematics
    Leiden University
    Leiden, the Netherlands
    e-mail
  • Kálmán GyőryInstitute of Mathematics
    University of Debrecen
    H-4002 Debrecen, Hungary
    e-mail
  • László RemeteInstitute of Mathematics
    University of Debrecen
    H-4002 Debrecen, Hungary
    e-mail
  • Ashvin A. SwaminathanDepartment of Mathematics
    Harvard University
    Cambridge, MA 02138, USA
    e-mail

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