Algorithmic problems for differential polynomial algebras

TL;DR

This paper proves that certain fundamental algorithmic problems are undecidable in differential polynomial algebras with multiple derivations, highlighting limitations in computational approaches for these algebraic structures.

Contribution

It establishes the undecidability of ideal and subalgebra membership problems in differential polynomial algebras with two or more derivations, advancing understanding of their computational complexity.

Findings

Ideal membership problem is undecidable.

Subalgebra membership problem is undecidable.

Results apply to algebras with at least two derivations.

Abstract

We prove that the ideal membership problem and the subalgebra membership problem are algorithmically undecidable for differential polynomial algebras with at least two basic derivation operators.