# Direct sum decomposability of polynomials and factorization of associated forms

## TL;DR

This paper establishes criteria for the direct sum decomposability of homogeneous polynomials, linking algebraic properties to factorization and providing algorithms for decomposition over various fields.

## Contribution

It introduces new criteria and algorithms for determining direct sum decomposability of polynomials, connecting it to factorization of associated forms and Jacobian ideals.

## Key findings

- Provides an if-and-only-if criterion for decomposability.
- Develops an algorithm for decomposition over any field.
- Shows many classes of polynomials are not decomposable.

## Abstract

We prove two criteria for direct sum decomposability of homogeneous polynomials. For a homogeneous polynomial with a non-zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of the Macaulay inverse system of its Milnor algebra. This leads to an if-and-only-if criterion for direct sum decomposability of such a polynomial, and to an algorithm for computing direct sum decompositions over any field, either of characteristic $0$ or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist. For homogeneous forms over algebraically closed fields, we interpret direct sums and their limits as forms that cannot be reconstructed from their Jacobian ideal. We also give simple necessary criteria for direct sum decomposability of arbitrary homogeneous polynomials over arbitrary fields and apply them to prove that many interesting classes of homogeneous polynomials are not direct sums.