A Note on Nahm's Conjecture in Rank 2 Case

TL;DR

This paper classifies certain positive definite matrices with integer entries related to Nahm's conjecture in rank 2, focusing on solutions to specific algebraic equations and their reality properties.

Contribution

It provides a complete list of matrices for which all solutions to the system are real, advancing understanding of Nahm's conjecture in rank 2.

Findings

Identifies all matrices with the specified properties.

Establishes conditions under which solutions are real.

Contributes to the validation of Nahm's conjecture in rank 2.

Abstract

The aim of this paper is to get a complete list of positive definite symmetric matrices with integer entries $\a&b\b&d\$ such that all complex solutions to the system of equations $1-x_1=x_1^ax_2^b\ 1-x_2=x_1^bx_2^d$ are real. This result is related to Nahm's conjecture in rank 2 case.