The Asymptotic Number of Simple Singular Vector Tuples of a Cubical Tensor

TL;DR

This paper proves a conjecture about the asymptotic number of simple singular vector tuples in cubical tensors, providing precise dominant terms and suggesting a general phenomenon for subdominant effects.

Contribution

It confirms a conjecture on the asymptotic behavior of singular vector tuples in cubical tensors and identifies the exact dominant asymptotic term including the constant.

Findings

Proved the asymptotic conjecture for cubical tensors.

Identified the dominant asymptotic term and constant.

Suggested the generality of subdominant effects using differential approximants.

Abstract

S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic $m_1 \times \cdots \times m_d$ tensor has an elegant rational form involving elementary symmetric functions, and provided a partial conjecture for the asymptotic behavior of the cubical case $m_1 = \cdots = m_d$. We prove this conjecture and further identify completely the dominant asymptotic term, including the multiplicative constant. Finally, we use the method of differential approximants to conjecture that the subdominant connective constant effect observed by Ekhad and Zeilberger for a particular case in fact occurs more generally.