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Effective reduction theory of integral polynomials of given non-zero discriminant and its applications

Jan-Hendrik Evertse, Kálmán Győry

TL;DR

This survey traces the evolution of the effective reduction theory for integral polynomials with a fixed nonzero discriminant, connecting classical results of Lagrange and Hermite to Birch–Merriman, Győry, and Evertse–Győry, and then to contemporary, Baker-based methods for unit equations. It clarifies the relationships between Hermite equivalence, Z- and GL_2(Z)-equivalence, showing that modern notions provide far stronger finiteness and computability than Hermite’s original framework. The work then explicates the far-reaching consequences for algebraic number theory, including effective monogenicity criteria, index form equations, and algorithms to determine power integral bases and canonical number systems, with extensions to number fields and finitely generated domains. It concludes by outlining generalizations, conjectural improvements under abc-type hypotheses, and applications to Diophantine equations and canonical number systems, while noting open questions about the multiplicity of monogenicity and invariant orders.

Abstract

We give a survey on the general effective reduction theory of integral polynomials and its applications. We concentrate on results providing the finiteness for the number of `$\mathbb{Z}$-equivalence classes' and `$GL_2(\mathbb{Z})$-equivalence classes' of polynomials of given discriminant. We present the effective finiteness results of Lagrange from 1773 and Hermite from 1848, 1851 for quadratic resp. cubic polynomials. Then we formulate the general ineffective finiteness result of Birch and Merriman from 1972, the general effective finiteness theorems of Győry from 1973, obtained independently, and of Evertse and Győry from 1991, and a result of Hermite from 1857 not discussed in the literature before 2023. We briefly outline our effective proofs which depend on Győry's effective results on unit equations, whose proofs involve Baker's effective theory of logarithmic forms. Then we focus on our joint paper with Bhargava, Remete and Swaminathan from 2023, where Hermite's finiteness result from 1857 involving `Hermite equivalence classes' is compared with the above-mentioned modern results involving $\mathbb{Z}$-equivalence and $GL_2(\mathbb{Z})$-equivalence, and where it is confirmed that Hermite's result from 1857 is much weaker than the modern results mentioned. The results of Győry from 1973 and Evertse and Győry from 1991 together established a general effective reduction theory of integral polynomials with given non-zero discriminant, which has significant consequences and applications, including Győry's effective finiteness theorems from the 1970's on monogenic orders and number fields. We give an overview of these in our paper. We also give an overview of bounds on the number of times a given order is monogenic or rationally monogenic. In the Appendix we discuss related topics not strictly belonging to the reduction theory of integral polynomials.

Effective reduction theory of integral polynomials of given non-zero discriminant and its applications

TL;DR

This survey traces the evolution of the effective reduction theory for integral polynomials with a fixed nonzero discriminant, connecting classical results of Lagrange and Hermite to Birch–Merriman, Győry, and Evertse–Győry, and then to contemporary, Baker-based methods for unit equations. It clarifies the relationships between Hermite equivalence, Z- and GL_2(Z)-equivalence, showing that modern notions provide far stronger finiteness and computability than Hermite’s original framework. The work then explicates the far-reaching consequences for algebraic number theory, including effective monogenicity criteria, index form equations, and algorithms to determine power integral bases and canonical number systems, with extensions to number fields and finitely generated domains. It concludes by outlining generalizations, conjectural improvements under abc-type hypotheses, and applications to Diophantine equations and canonical number systems, while noting open questions about the multiplicity of monogenicity and invariant orders.

Abstract

We give a survey on the general effective reduction theory of integral polynomials and its applications. We concentrate on results providing the finiteness for the number of `-equivalence classes' and `-equivalence classes' of polynomials of given discriminant. We present the effective finiteness results of Lagrange from 1773 and Hermite from 1848, 1851 for quadratic resp. cubic polynomials. Then we formulate the general ineffective finiteness result of Birch and Merriman from 1972, the general effective finiteness theorems of Győry from 1973, obtained independently, and of Evertse and Győry from 1991, and a result of Hermite from 1857 not discussed in the literature before 2023. We briefly outline our effective proofs which depend on Győry's effective results on unit equations, whose proofs involve Baker's effective theory of logarithmic forms. Then we focus on our joint paper with Bhargava, Remete and Swaminathan from 2023, where Hermite's finiteness result from 1857 involving `Hermite equivalence classes' is compared with the above-mentioned modern results involving -equivalence and -equivalence, and where it is confirmed that Hermite's result from 1857 is much weaker than the modern results mentioned. The results of Győry from 1973 and Evertse and Győry from 1991 together established a general effective reduction theory of integral polynomials with given non-zero discriminant, which has significant consequences and applications, including Győry's effective finiteness theorems from the 1970's on monogenic orders and number fields. We give an overview of these in our paper. We also give an overview of bounds on the number of times a given order is monogenic or rationally monogenic. In the Appendix we discuss related topics not strictly belonging to the reduction theory of integral polynomials.
Paper Structure (43 sections, 86 theorems, 182 equations)

This paper contains 43 sections, 86 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction theory of integral quadratic and cubic polynomials of given non-zero discriminant
  4. Hermite's attempt (1857) to extend the reduction results of polynomials of degree $\leq 3$ to polynomials of arbitrary degree
  5. $\mathbf{GL_n(\mathbb{Z} )}$-equivalence of decomposable forms
  6. Hermite equivalence of polynomials and Hermite's finiteness theorem
  7. Comparison between Hermite equivalence and $GL_2(\mathbb{Z})$-equival-ence and $\mathbb{Z}$-equivalence
  8. Reduction theory of integral polynomials of given non-zero discriminant and of arbitrary degree
  9. The theorems of Birch and Merriman (1972), Győry (1973) and Evertse and Győry (1991a)
  10. Explicit versions of theorems of Győry (1973) and Evertse and Győry (1991a)
  11. Consequences of Theorems \ref{['thm:3.9']}, \ref{['thm:3.8']} and Theorem \ref{['prop:2.3']} for Hermite equivalence classes
  12. Unit equations and $S$-unit equations
  13. A brief sketch of the proof of a less precise version of Theorem \ref{['thm:3.2']}
  14. A brief sketch of the proof of a less precise version of Theorem \ref{['thm:3.4']}
  15. Conjectural improvements (partly joint work with von Känel)
  16. ...and 28 more sections

Key Result

Theorem 2.1

For any quadratic polynomial $f\in\mathbb{Z}[X]$ of discriminant $D\ne 0$, there exists $g\in\mathbb{Z}[X]$, $GL_2(\mathbb{Z})$-equivalent to $f$, such that $H(g)\leq c(D)$ with some effectively computable constant $c(D)$ depending only on $D$.

Theorems & Definitions (119)

  • Theorem 2.1: Lagrange, 1773; Gauss, 1801
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4: Hermite, 1848, 1851
  • Theorem 2.5: Delone, 1930; Nagell, 1930
  • Theorem 3.1: Hermite, 1851
  • Theorem 3.2: Hermite, 1854, 1857
  • Theorem 3.3: BEGyRS, 2023
  • proof
  • Theorem 3.4: BEGyRS, 2023
  • ...and 109 more