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Absolutely Nonsingular Tensors and Determinantal Polynomials

  • Toshio Sakata
  • Toshio Sumi
  • Mitsuhiro Miyazaki
Chapter
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Abstract

In this chapter, we define absolute nonsingularity for 3-tensors over \(\mathbb {R}\) of format \(n\times n\times m\). An \(n\times n\times m\) tensor \(T=(T_1;\ldots ;T_m)\) is said to be absolutely nonsingular if \(\sum _{k=1}^m a_kT_k\) is nonsingular except for the case where \(a_1=\cdots =a_m=0\). In terms of the determinantal polynomial, T is absolutely nonsingular if the zero locus of the determinantal polynomial defined by T is \(\{(0,\ldots ,0)\}\). We state a criterion for the existence of an \(n\times n\times m\) absolutely nonsingular tensor in terms of Hurwitz–Radon numbers. In order to prove this criterion, we use the notion of Hurwitz–Radon family. As for the determinantal polynomial, we show that the principal ideal generated by a determinantal polynomial is almost prime and is a real prime ideal or the zero locus of that ideal is \(\{(0, \ldots , 0)\}\). Since a square matrix is nonsingular if and only if it is full column rank, we define, by generalizing the notion of an absolutely nonsingular tensor, the notion of an Absolutely full column rank tensor. Some facts about Absolutely full column rank tensors and the existence of Absolutely full column rank tensors are discussed.

Keywords

Open Subset Prime Ideal Homogeneous Polynomial Inverse Image Principal Ideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Toshio Sakata
    • 1
  • Toshio Sumi
    • 2
  • Mitsuhiro Miyazaki
    • 3
  1. 1.Faculty of DesignKyushu UniversityFukuokaJapan
  2. 2.Faculty of Arts and ScienceKyushu UniversityFukuokaJapan
  3. 3.Department of MathematicsKyoto University of EducationKyotoJapan

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