Equations for the Fifth Secant Variety of Segre Products of Projective Spaces

It was known classically that $2 \times 2 \times 2 \times 2 $ tensors are defective in rank 3 -- the generic tensor of that format and rank has infinitely many decompositions. 

 

Catalisano, Geramita and Gimigliano showed that binary tensors for $5$ or more factors are not defective, so tensors of rank less than the generic rank have finitely many decompositions. 

 

Bocci and Chiantini showed that $2 \times 2 \times 2 \times 2 \times 2$ tensors are not identifiable in rank 5 ---  the generic tensor of that format has exactly 2 decompositions up to symmetry.   

 

Bocci, Chiantini and Ottaviani showed that for $n \geq 6$ factors, binary $n$-factor tensors are almost always $k$-identifiable.

 

Bernd Sturmfels asked in his ``algebraic fitness session'' at the Simon's Institute (Fall 2014) to study the boundary case of binary 5 factor tensors of rank 5 and to find its minimal defining equations. 

 

In this talk I will describe a computational proof that the fifth secant variety of the Segre product of five copies of the projective line is a codimension 2 complete intersection of equations of degree 6 and 16. Our computations rely on pseudo-randomness, and numerical accuracy, so parts of our proof are only valid “with high probability”.

 

This is joint work with Steven Sam.