# Waring rank of symmetric tensors, and singularities of some projective hypersurfaces

## TL;DR

This paper investigates the relationship between the Waring rank of homogeneous polynomials and the singularities of their associated projective hypersurfaces, revealing how the rank constrains singularity types and their combinatorial structure.

## Contribution

It establishes a link between Waring rank and hypersurface singularities, providing a complete description for rank n+1 and exploring special cases like plane curves with rank 5.

## Key findings

- Hypersurfaces with Waring rank n+1 have only isolated singularities.
- Singularity types are determined by hyperplane arrangement combinatorics.
- Special analysis for plane curves with Waring rank 5.

## Abstract

We show that if a homogeneous polynomial $f$ in $n$ variables has Waring rank $n+1$, then the corresponding projective hypersurface $f=0$ has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of $f$. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5.