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

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• 作者：Toshio Sumi^a ; ^{sumi@artsci.kyushu-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Mitsuhiro Miyazaki^b ; ^{g53448@kyokyo-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Toshio Sakata^c ; ^{sakatastat@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A69 ; 13C40 ; 14M12 ; 14P10
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：471
• 期：Complete
• 页码：409-453
• 全文大小：701 K

文摘

Let m,n≥3$m,n\ge 3$, (m−1)(n−1)+2≤p≤mn$(m-1)(n-1)+2\le p\le mn$, and u=mn−p$u=mn-p$. The set 76e15895afa623c0faba03" title="Click to view the MathML source">R^u×n×m${\mathbb{R}}^{u\times n\times m}$ of all real tensors with size u×n×m$u\times n\times m$ is one to one corresponding to the set of bilinear maps R^m×Rⁿ→R^u${\mathbb{R}}^m\times {\mathbb{R}}^n\to {\mathbb{R}}^u$. We show that R^m×n×p${\mathbb{R}}^{m\times n\times p}$ has plural typical ranks p   and 76d6" title="Click to view the MathML source">p+1$p+1$ if and only if there exists a nonsingular bilinear map R^m×Rⁿ→R^u${\mathbb{R}}^m\times {\mathbb{R}}^n\to {\mathbb{R}}^u$. We show that there is a dense open subset 6b6bc5a13e438fac" title="Click to view the MathML source">O$\mathcal{O}$ of 76e15895afa623c0faba03" title="Click to view the MathML source">R^u×n×m${\mathbb{R}}^{u\times n\times m}$ such that for any Y∈O$Y\in \mathcal{O}$, 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$\mathcal{T}$ of R^n×p×m${\mathbb{R}}^{n\times p\times m}$ and continuous surjective open maps ν:O→R^u×p$\nu :\mathcal{O}\to {\mathbb{R}}^{u\times p}$ and σ:T→R^u×p$\sigma :\mathcal{T}\to {\mathbb{R}}^{u\times p}$, where R^u×p${\mathbb{R}}^{u\times p}$ is the set of 76c10f8ef01bcdf6bedf99013c" title="Click to view the MathML source">u×p$u\times p$ matrices with entries in 7614754697d5bdc24a1b766c83" title="Click to view the MathML source">R$\mathbb{R}$, such that if b25ca5" title="Click to view the MathML source">ν(Y)=σ(T)$\nu (Y)=\sigma (T)$, then 6b2">$\mathrm{rank}T=p$ if and only if the ideal of maximal minors of the matrix defined by Y is a real prime ideal.