Hypergraph LSS-ideals and coordinate sections of symmetric tensors

Abstract

Let $\mathbb{K}$ be a field, $[n]=\{1,\dots ,n\}$ and $H=([n],E)$ be a hypergraph. For an integer $d\ge 1$ the Lovász-Saks-Schrijver ideal (LSS-ideal) $L_H^{\mathbb{K}}(d)\subseteq \mathbb{K}[ y_{ij} : (i,j)\in [n]\times [d] ]$ is the ideal generated by the polynomials $f_e^{(d)}={\sum }_{j=1}^d\prod _{i\in e}{y}_{ij}$ for edges e of H.

In this paper for an algebraically closed field $\mathbb{K}$ and a k-uniform hypergraph $H=([n],E)$ we employ a connection between LSS-ideals and coordinate sections of the closure of the set $S_{n,k}^d$ of homogeneous degree k symmetric tensors in n variables of $\text{rank}\le d$ to derive results on the irreducibility of its coordinate sections. To this end we provide results on primality and the complete intersection property of $L_H^{\mathbb{K}}(d)$. We then use the combinatorial concept of positive matching decomposition of a hypergraph H to provide bounds on when $L_H^{\mathbb{K}}(d)$ turns prime to provide results on the irreducibility of coordinate sections of $S_{n,k}^d$.

Keywords:

• Lovasz-Saks-Schrijver ideals
• symmetric algebra postiive matching
• symmetric tensor
• tensor rank

2020 Mathematics Subject Classification:

• 05E40
• 05C62
• 13C40
• 13P10

Acknowledgments

The authors are grateful to Rashid Zaare-Nahandi for numerous suggestions and discussions.

Additional information

Funding

The first author acknowledges support from Ministry of Science, Research and Technology of Iran for her research visit to Marburg, where most of the work on the project was done.