Typical ranks of semi-tall real 3-tensors. Issue 9 (4th July 2021)

• Record Type: Journal Article
• Authors: Sumi, Toshio; Miyazaki, Mitsuhiro; Sakata, Toshio
• Is Part Of: Linear & multilinear algebra. Volume 69:Issue 9(2021)
• Page Start: 1734
• Page End: 1746
• Publication Date: 2021-07-04
• Subjects: Tensor rank -- typical rank -- tall tensor -- semi-tall tensor -- Bezout's theorem -- determinantal variety; 15A69 -- 14P10 -- 14M12 -- 13C40; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2019.1637811
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: Let m, n and p be integers with 3 ≤ m ≤ n and ( m − 1 ) ( n − 1 ) + 1 ≤ p ≤ ( m − 1 ) m . We showed in previous papers that if p ≥ ( m − 1 ) ( n − 1 ) + 2, then typical ranks of p × n × m -tensors over the real number field are p and p +1 if and only if there exists a non-singular bilinear map R m × R n → R m n − p . We also showed that the 'if' part is also valid in the case where p = ( m − 1 ) ( n − 1 ) + 1 . In this paper, we consider the case where p = ( m − 1 ) ( n − 1 ) + 1 and show that the typical ranks of p × n × m -tensors over the real number field are p and p +1 in several cases including the case where there is no non-singular bilinear map R m × R n → R m n − p . In particular, we show that the 'only if' part of the above mentioned fact is not valid for the case p = ( m − 1 ) ( n − 1 ) + 1 .