On the number of Singular Vector Tuples of Hyper-Cubical Tensors

By Shalosh B. Ekhad and Doron Zeilberger

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(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org )

Written: April 29, 2016

Last update: May 23

NOTE (added May 23, 2016): The Conjecture at the end of the article has been proved (including the "extra credit" for the explicit value of the constant in front) by Jay Pantone. See his nice article, "The Asymptotic Number of Simple Singular Vector Tuples of a Cubical Tensor". A donation of \$125 to the OEIS Foundation, in honor of Jay Pantone has been made.

Bernd Sturmfels has given, last Friday, a fascinating colloquium talk here at Rutgers, and among many other intriguing things he mentioned an amazing and beautiful constant-term expression for the number of singular vector tuples of a general tensor due to Shmuel Friedland and Giorgio Ottaviani, proved in their beautiful paper, entitled "The number of singular vector tuples and uniqueness of best rank-one approximation of tensors", and "officially" published in the journal Found. Comput. Math. 14 (2014), no. 6, 1209-1242.

In that paper, (Theorem 1, Eq. (1.3)) they give a constant-term expression for the numbers that they call c(n1, n2, n3). They also compute a few values.

But what is

c(2000,2000,2000)?

Here it is .

This lead to sequence A271905 in the OEIS.

Even more interestingly, it found a fifth-order linear recurrence for the 3D case, and very precise asymptotics, that naturally leads to a conjectured asymptotic formula for the general case. One of us (DZ) is pledging 100 dollars donation to the OEIS foundation, in honor of the prover, for a proof of that conjecture.

The four-dimensional sequence lead to sequence A272551 in the OEIS.

Abstract: Shmuel Friedland and Giorgio Ottaviani's beautiful constant term expression for the number of singular vector tuples of generic tensors is used to derive a rational generating function for these numbers, that in turn, is used to obtain an asymptotic formula for the the number of such tuples for n by n by n three-dimensional tensors, and to conjecture an asymptotic formula for the general d-dimensional case. A donation of 100 dollars, in honor of the first prover, will be made to the On-line Encyclopedia of Integer Sequences.

# Maple package

• SVT.txt, a Maple package to study the sequences enumerating singular vector tuples of hyper-cubical tensors.

# Sample Input and Output Files for the Maple package SVT.txt

• If you want to see a fully computer-generated paper about the sequence enumerating the number of singular vector tuples of cubical tensors (i.e. in 3 dimensions), that gives the fifth-order recurrence, and asymptotics,

the input file generates the output file.

• If you want to see the first 70 terms of the sequence enumerating the number of singular vector tuples of four-dimensional hyper-cubical tensors, and estimated conjectured asymptotics,

the input file generates the output file.

• If you want to see the first 160 terms of the sequence enumerating the number of singular vector tuples of four-dimensional hyper-cubical tensors

the input file generates the output file.

• If you want to see the first 30 terms of the sequence enumerating the number of singular vector tuples of five-dimensional hyper-cubical tensors, and the first 20 terms for six-dimensional ones,

the input file generates the output file.

Added May 1, 2016: To get (much slower) a fully rigorous proof of the fifth-order linear recurrence equation with polynomial coefficients for C3(n)=c(n,n,n), using the Apagodu-Zeilberger algorithm, implemented in the package SMAZ.txt

the input file generates the the output file.

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger