Reducing quadratic forms by kneading sequences

TL;DR

This paper introduces a new operation called 'kneading' on sequences, establishing a bijection with certain quadratic forms and linking sequence invariants to form class invariants, offering new insights into form reduction.

Contribution

The paper defines the kneading operation, proves its invariance properties, and connects sequence invariants to quadratic form classification, providing a novel perspective on form reduction.

Findings

Kneading preserves sequence invariants such as length parity, sum, and alternant.

A bijection between sequences and Zagier-reduced forms is established.

The sum of a sequence acts as a class invariant for the associated quadratic form.

Abstract

We introduce an invertible operation on finite sequences of positive integers and call it "kneading". Kneading preserves three invariants of sequences -- the parity of the length, the sum of the entries, and one we call the "alternant". We provide a bijection between the set of sequences with alternant $a$ and parity $s$ and the set of Zagier-reduced indefinite binary quadratic forms with discriminant $a^2 + (-1)^s \cdot 4$, and show that kneading corresponds to Zagier reduction of the corresponding forms. It follows that the sum of a sequence is a class invariant of the corresponding form. We conclude with some observations and conjectures concerning this new invariant.