Multivariate Lucas Polynomials and Ideal Classes in Quadratic Number Fields

TL;DR

This paper introduces new polynomial families linked to Lucas and Fibonacci sequences using Pauli matrices, providing a geometric perspective on Gauss's class number problem in quadratic number fields.

Contribution

It presents four novel polynomial families with connections to classical sequences and offers a geometric interpretation of a longstanding number theory problem.

Findings

Two polynomial families are closely related to Lucas and Fibonacci sequences.

Provides a geometric interpretation of Gauss's class number problem.

Connects polynomial families to ideal class structures in quadratic fields.

Abstract

In this work, by using Pauli matrices, we introduce four families of polynomials indexed over the positive integers. These polynomials have rational or imaginary rational coefficients. It turns out that two of these families are closely related to classical Lucas and Fibonacci polynomial sequences and hence to Lucas and Fibonacci numbers. We use one of these families to give a geometric interpretation of the 200 years old class number problems of Gauss, which is equivalent to the study of narrow ideal classes in real quadratic number fields.