Hypergraph LSS-ideals and coordinate sections of symmetric tensors

TL;DR

This paper explores the algebraic properties of Lovasz-Saks-Schrijver ideals associated with hypergraphs and their connection to symmetric tensor rank, providing new insights into irreducibility and primality conditions.

Contribution

It establishes a link between LSS-ideals and tensor coordinate sections, deriving new results on primality, complete intersections, and irreducibility using hypergraph combinatorics.

Findings

Results on primality and complete intersection properties of LSS-ideals.

Bounds on when LSS-ideals are prime based on hypergraph decompositions.

Insights into irreducibility of coordinate sections of symmetric tensors.

Abstract

Let K be a field, [n]= {1,...,n} and H=([n],E) be a hypergraph. For an integer d >= 1 the Lovasz-Saks-Schrijver ideal (LSS-ideal) L_H^K (d) in K[y_{ij}~:~(i,j) \in [n] x [d]] is the ideal generated by the polynomials $f^{(d)}_{e}= \sum\limits_{j=1}^{d} \prod\limits_{i \in e} y_{ij}$ for edges e of H. In this paper for an algebraically closed field 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 rank <= 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^K (d). We then use the combinatorial concept of positive matching decomposition of a hypergraph H to provide bounds on when L_H^K(d) turns prime to provide results on the irreducibility of coordinate sections of S_{n, k}^d.