# On the Characteristic Polynomial of the Gross Regulator Matrix

## TL;DR

This paper proposes a conjectural formula for the characteristic polynomial of Gross's regulator matrix, linking it to the Eisenstein cocycle and refining the Gross--Stark conjecture.

## Contribution

It introduces a new conjectural formula for principal minors and the characteristic polynomial of the regulator matrix, extending previous work and connecting to the Eisenstein cocycle.

## Key findings

- Conjectural formula for the principal minors of the regulator matrix.
- Verification of the determinant case using recent results.
- New intermediate cases providing a refinement of the Gross--Stark conjecture.

## Abstract

We present a conjectural formula for the principal minors and the characteristic polynomial of Gross's regulator matrix associated to a totally odd character of a totally real field. The formula is given in terms of the Eisenstein cocycle, which was defined and studied earlier by the authors and collaborators. For the determinant of the regulator matrix, our conjecture follows from recent work of Kakde, Ventullo and the first author. For the diagonal entries, our conjecture overlaps with the conjectural formula presented in our prior work. The intermediate cases are new and provide a refinement of the Gross--Stark conjecture.