Base loci of linear systems and the Waring problem

TL;DR

This paper investigates the uniqueness of additive decompositions of homogeneous forms into powers of linear forms, using geometric methods related to base loci of linear systems with specified singularities.

Contribution

It refines previous work by providing conditions for uniqueness of Waring decompositions under a divisibility assumption, translating algebraic questions into geometric ones.

Findings

Provides criteria for uniqueness of Waring decompositions

Relates algebraic decomposition to geometric base loci analysis

Extends understanding of singularities in linear systems

Abstract

Waring problem for homogeneus forms asks for additive decomposition of a form $f$ into powers of linear forms. A classical problem is to determine when such a decomposition is unique. In this note I refine the work in arXiv:math/0406288v1 and answer this question under a divisibility assumption. To do this I translate the algebraic statement into a geometric one concerning the base loci of linear systems of $P^n$ with assigned singularities.