Typical ranks for 3-tensors, nonsingular bilinear maps and determinantal ideals

JOURNAL OF ALGEBRA

卷:471

Article

Sumi, Toshio¹  Miyazaki, Mitsnhiro²  Sakata, Toshio³ 

[1] Kyushu Univ, Fac Arts & Sci, Nishi Ku, 744 Motooka, Fukuoka 8190395, Japan

[2] Kyoto Univ, Dept Math, Fushimi Ku, 1 Fujinomori, Kyoto 6128522, Japan

[3] Kyushu Univ, Fukuoka, Japan

关键词: Tensor rank;    Typical rank;    Determinantal ideal;    Real radical;    Bilinear map;    Monomial preorder;   

DOI  :  10.1016/j.jalgebra.2016.09.028

来源: Elsevier

【 摘 要 】

Let m, n >= 3, (m - 1)(n - 1) + 2 <= p <= mn, and u = mn-p. The set R-uxnxm of all real tensors with size u x n x m is one to one corresponding to the set of bilinear maps R-m x R-n -> R-u. We show that R-mxnxp has plural typical ranks p and p + 1 if and only if there exists a nonsingular bilinear map R-m x R-n -> R-u. We show that there is a dense open subset O of R-uxnxm such that for any Y is an element of O, the ideal of maximal minors of a matrix defined by Y in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal. Further, we show that there is a dense open subset T of R-nxpxm and continuous surjective open maps nu: O -> R-uxp and sigma: T -> R-uxp, where R-uxp is the set of u x p matrices with entries in R, such that if nu(Y) = sigma(T), then rank T = p if and only if the ideal of maximal minors of the matrix defined by Y is a real prime ideal. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free