Primary Decomposition with Differential Operators

TL;DR

This paper introduces differential primary decompositions for ideals in commutative rings, characterizing ideal membership via differential conditions, and provides algorithms for minimal decompositions with applications to affine schemes.

Contribution

It generalizes Noetherian operators to a broader algebraic setting and offers a new, concise method for representing affine schemes and modules.

Findings

Differential primary decompositions characterize ideal membership.

Minimal decompositions are unique up to change of bases.

An algorithm for computing minimal decompositions in polynomial rings is implemented.

Abstract

We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential primary decompositions are unique up to change of bases. Our results generalize the construction of Noetherian operators for primary ideals in the analytic theory of Ehrenpreis-Palamodov, and they offer a concise method for representing affine schemes. The case of modules is also addressed. We implemented an algorithm in Macaulay2 that computes the minimal decomposition for an ideal in a polynomial ring.