From the Adler--Moser polynomials to the polynomial tau functions of KdV

TL;DR

This paper explicitly constructs the change of variables that transforms Adler-Moser polynomials into polynomial tau functions of the KdV hierarchy, clarifying their relationship.

Contribution

The paper provides an explicit formula for the change of variables linking Adler-Moser polynomials to KdV tau functions, which was previously known only to exist.

Findings

Explicit change of variables derived

Connection between Adler-Moser polynomials and KdV tau functions clarified

Enhances understanding of integrable systems and polynomial solutions

Abstract

In 1978, M. Adler and J. Moser proved that there exists a unique change of variables that transforms the Adler-Moser polynomials into the polynomial tau functions of the KdV hierarchy. In this paper we exhibit this change of variables.