Apolarity and direct sum decomposability of polynomials

TL;DR

This paper investigates criteria for decomposing homogeneous polynomials into direct sums using apolar ideals, establishing a link between minimal generators and limits of direct sums.

Contribution

It introduces a new criterion based on apolar ideals for determining when a polynomial is a limit of direct sums.

Findings

Apolar ideal of degree d polynomial has a minimal generator of degree d iff it is a limit of direct sums.

Provides a characterization of direct sum decomposability via apolar ideals.

Establishes a connection between polynomial decomposability and algebraic properties of apolar ideals.

Abstract

A polynomial is a direct sum if it can be written as a sum of two non-zero polynomials in some distinct sets of variables, up to a linear change of variables. We analyze criteria for a homogeneous polynomial to be decomposable as a direct sum, in terms of the apolar ideal of the polynomial. We prove that the apolar ideal of a polynomial of degree $d$ strictly depending on all variables has a minimal generator of degree $d$ if and only if it is a limit of direct sums.