# Waring, tangential and cactus decompositions

## TL;DR

This paper revises and generalizes an algorithm for symmetric tensor decomposition, extending it to detect decompositions involving tangential points and providing an algorithm for cactus rank and decomposition with support detection.

## Contribution

It introduces a generalized algorithm for tensor decomposition that includes tangential and cactus decompositions, along with support detection of minimal apolar schemes.

## Key findings

- Revised the classical symmetric tensor decomposition algorithm.
- Extended the algorithm to detect tangential decompositions.
- Developed an algorithm for cactus rank and decomposition with support detection.

## Abstract

(EN) We revise the famous algorithm for symmetric tensor decomposition due to Brachat, Comon, Mourrain and Tsidgaridas. Afterwards, we generalize it in order to detect possibly different decompositions involving points on the tangential variety of a Veronese variety. Finally, we produce an algorithm for cactus rank and decomposition, which also detects the support of the minimal apolar scheme and its length at each component. (FR) Nous revenons sur le fameux algorithme de Brachat, Comon, Mourrain et Tsidgaridas pour la d\'composition des tenseurs sym\'etriques. Ensuite, nous le g\'en\'eralisons afin de d\'etecter de possibles d\'ecompositions diff\'erentes impliquant des points sur la vari\'et\'e tangentielle d'une vari\'et\'e de Veronese. Enfin, nous proposons un algorithme pour le rang et la d\'ecomposition cactus, qui, lui aussi, d\'etecte le support du sch\'ema apolaire minimal ainsi que sa longueur sur chaque composante.