On schemes evinced by generalized additive decompositions and their regularity

TL;DR

This paper explores schemes derived from generalized additive decompositions of homogeneous polynomials, analyzing their regularity and establishing conditions under which they exhibit certain regularity properties.

Contribution

It explicitly constructs schemes from GADs, investigates their regularity, and identifies when minimal or irredundant schemes are regular.

Findings

Irredundant schemes to F need not be d-regular unless evinced by special GADs.

Tangential decompositions of minimal length are always d-regular.

Irredundant apolar schemes of length at most 2d+1 are d-regular.

Abstract

We define and explicitly construct schemes evinced by generalized additive decompositions (GADs) of a given $d$-homogeneous polynomial $F$. We employ GADs to investigate the regularity of $0$-dimensional schemes apolar to $F$, focusing on those satisfying some minimality conditions. We show that irredundant schemes to $F$ need not be $d$-regular, unless they are evinced by special GADs of $F$. Instead, we prove that tangential decompositions of minimal length are always $d$-regular, as well as irredundant apolar schemes of length at most $2d+1$.