# On the Expansion Coefficients of KP Tau Function

## TL;DR

This paper generalizes the Giambelli formula for KP hierarchy tau functions to cases where the function vanishes at the origin, providing a new characterization of solutions.

## Contribution

It introduces a generalized Giambelli formula applicable when the tau function vanishes at the origin, expanding the understanding of KP hierarchy solutions.

## Key findings

- Generalized Giambelli formula for vanishing tau functions
- Characterization of KP solutions with zero at the origin
- Extension of classical results to new solution classes

## Abstract

We study the expansion coefficients of the tau function of the KP hierarchy. If the tau function does not vanish at the origin, it is known that the coefficients are given by Giambelli formula and that it characterizes solutions of the KP hierarchy. In this paper, we find a generalization of Giambelli formula to the case when the tau function vanishes at the origin. Again it characterizes solutions of the KP hierarchy.