# The Zariski cancellation problem for Poisson algebras

## TL;DR

This paper investigates the Zariski cancellation problem for Poisson algebras, establishing positive results for specific classes and introducing new invariants to analyze broader cases.

## Contribution

It provides affirmative solutions for certain classes of Poisson algebras and introduces Poisson analogues of invariants to address the cancellation problem more generally.

## Key findings

- Confirmed Zariski cancellation for connected graded Poisson algebras without degree one Poisson central elements.
- Proved cancellation for Poisson integral domains of Krull dimension two with nontrivial brackets.
- Developed Poisson versions of the Makar-Limanov invariant and discriminant for broader analysis.

## Abstract

We study the Zariski cancellation problem for Poisson algebras asking whether $A[t]\cong B[t]$ implies $A\cong B$ when $A$ and $B$ are Poisson algebras. We resolve this affirmatively in the cases when $A$ and $B$ are both connected graded Poisson algebras finitely generated in degree one without degree one Poisson central elements and when $A$ is a Poisson integral domain of Krull dimension two with nontrivial Poisson bracket. We further introduce Poisson analogues of the Makar-Limanov invariant and the discriminant to deal with the Zariski cancellation problem for other families of Poisson algebras.