Stratification of the fourth secant variety of Veronese variety via the symmetric rank

TL;DR

This paper provides a detailed stratification of the fourth secant variety of Veronese varieties based on symmetric rank, offering a comprehensive classification of minimal decompositions of symmetric tensors and homogeneous polynomials.

Contribution

It introduces a complete stratification of the fourth secant variety of Veronese varieties via symmetric rank, linking tensor and polynomial decompositions.

Findings

Classifies all possible ranks in minimal decompositions for border rank 4 tensors and polynomials.

Provides a complete stratification of the fourth secant variety of Veronese varieties.

Connects tensor decompositions with polynomial representations in a unified framework.

Abstract

If $X\subset \mathbb{P}^n$ is a projective non degenerate variety, the $X$-rank of a point $P\in \mathbb{P}^n$ is defined to be the minimum integer $r$ such that $P$ belongs to the span of $r$ points of $X$. We describe the complete stratification of the fourth secant variety of any Veronese variety $X$ via the $X$-rank. This result has an equivalent translation in terms both of symmetric tensors and homogeneous polynomials. It allows to classify all the possible integers $r$ that can occur in the minimal decomposition of either a symmetric tensor or a homogeneous polynomial of $X$-border rank 4 (i.e. contained in the fourth secant variety) as a linear combination of either completely decomposable tensors or powers of linear forms respectively.