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

TL;DR

This paper surveys the effective reduction theory of integral polynomials with a given non-zero discriminant, highlighting historical and modern finiteness results, their proofs, and applications in number theory.

Contribution

It compares classical and modern finiteness results, clarifies their relationships, and discusses the implications for algebraic number theory and monogenic orders.

Findings

Hermite's 1857 finiteness result is weaker than modern results.

Gy"H{o}ry's and Evertse-Gy"H{o}ry's theorems establish effective finiteness for polynomial classes.

The modern theory has significant applications in number fields and monogenic 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 Gy\H{o}ry from 1973, obtained independently, and of Evertse and Gy\H{o}ry 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 Gy\H{o}ry'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 Gy\H{o}ry from 1973 and Evertse and Gy\H{o}ry from 1991 together established a general effective reduction theory of integral polynomials with given non-zero discriminant, which has significant consequences and applications, including Gy\H{o}ry'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.