# Absolutely Nonsingular Tensors and Determinantal Polynomials

## 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.