Reduction theory of binary forms

TL;DR

This paper introduces the reduction theory of binary forms, covering quadratic and Hermitian forms, and discusses a reduction algorithm over integers based on recent research.

Contribution

It presents a comprehensive overview of reduction methods for binary forms, including the Julia quadratic and an algorithm over integers.

Findings

Introduction of Julia quadratic for binary forms

Survey of a reduction algorithm over integers

Connection to recent work by Cremona and Stoll

Abstract

In these lectures we give an introduction to the reduction theory of binary forms starting with quadratic forms with real coefficients, Hermitian forms, and then define the Julia quadratic for any degree $n$ binary form. A survey of a reduction algorithm over $\mathbb Z$ is described based on recent work of Cremona and Stoll.