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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si1.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=54472ac072b078e80039b295b8568b88" title="Click to view the MathML source">m,n≥3class="mathContainer hidden">class="mathCode">$m,n\ge 3$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si2.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=02bc52de695f77a7f939780026741465" title="Click to view the MathML source">(m−1)(n−1)+2≤p≤mnclass="mathContainer hidden">class="mathCode">$(m-1)(n-1)+2\le p\le mn$, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si3.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=0cab74e3453cfe274126454edc27bd01" title="Click to view the MathML source">u=mn−pclass="mathContainer hidden">class="mathCode">$u=mn-p$. The set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si11.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=7c03bd1d0076e15895afa623c0faba03" title="Click to view the MathML source">R^u×n×mclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^{u\times n\times m}$ of all real tensors with size class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si283.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=413c97fefcf0ea759d339d90a27cf985" title="Click to view the MathML source">u×n×mclass="mathContainer hidden">class="mathCode">$u\times n\times m$ is one to one corresponding to the set of bilinear maps class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si6.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=d36ad28e15d8e232b5865a0f7a135fb0" title="Click to view the MathML source">R^m×Rⁿ→R^uclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^m\times {\mathbb{R}}^n\to {\mathbb{R}}^u$. We show that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si30.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=c3261d2dd699ad93a6633c8461ad5f79" title="Click to view the MathML source">R^m×n×pclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^{m\times n\times p}$ has plural typical ranks p   and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si1230.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=caa6903fb2405f646dfd08f2612976d6" title="Click to view the MathML source">p+1class="mathContainer hidden">class="mathCode">$p+1$ if and only if there exists a nonsingular bilinear map class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si6.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=d36ad28e15d8e232b5865a0f7a135fb0" title="Click to view the MathML source">R^m×Rⁿ→R^uclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^m\times {\mathbb{R}}^n\to {\mathbb{R}}^u$. We show that there is a dense open subset class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si10.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=9b8b65e1f46241c66b6bc5a13e438fac" title="Click to view the MathML source">Oclass="mathContainer hidden">class="mathCode">$\mathcal{O}$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si11.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=7c03bd1d0076e15895afa623c0faba03" title="Click to view the MathML source">R^u×n×mclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^{u\times n\times m}$ such that for any class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si12.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=3a85f533432f7fba5b11b3f82d6a8d34" title="Click to view the MathML source">Y∈Oclass="mathContainer hidden">class="mathCode">$Y\in \mathcal{O}$, 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si13.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=e9e74c8a89fe4554e20028030ccd0b9b" title="Click to view the MathML source">Tclass="mathContainer hidden">class="mathCode">$\mathcal{T}$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si1206.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=8568b4002952b3d3668b884ca4e48167" title="Click to view the MathML source">R^n×p×mclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^{n\times p\times m}$ and continuous surjective open maps class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si15.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=3f5ef75c9e9a5b87345635826f14262b" title="Click to view the MathML source">ν:O→R^u×pclass="mathContainer hidden">class="mathCode">$\nu :\mathcal{O}\to {\mathbb{R}}^{u\times p}$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si16.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=481dfe1801b2fd7c31a116d748ca7bdb" title="Click to view the MathML source">σ:T→R^u×pclass="mathContainer hidden">class="mathCode">$\sigma :\mathcal{T}\to {\mathbb{R}}^{u\times p}$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si1165.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=08f50dda754edf540eeab2ff8d498a44" title="Click to view the MathML source">R^u×pclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^{u\times p}$ is the set of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si18.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=319b6376c10f8ef01bcdf6bedf99013c" title="Click to view the MathML source">u×pclass="mathContainer hidden">class="mathCode">$u\times p$ matrices with entries in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si19.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=990d9f7614754697d5bdc24a1b766c83" title="Click to view the MathML source">Rclass="mathContainer hidden">class="mathCode">$\mathbb{R}$, such that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si1221.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=7f0592c1e0674dca256103e587b25ca5" title="Click to view the MathML source">ν(Y)=σ(T)class="mathContainer hidden">class="mathCode">$\nu (Y)=\sigma (T)$, then class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303490&_mathId=si1210.gif&_user=111111111&_pii=S0021869316303490&_rdoc=1&_issn=00218693&md5=41a77e66c7d9980604628dcb00ee46b2">class="imgLazyJSB inlineImage" height="15" width="78" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316303490-si1210.gif">class="mathContainer hidden">class="mathCode"> if and only if the ideal of maximal minors of the matrix defined by Y is a real prime ideal.