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 thmlsrc">text 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≥3thContainer hidden">thCode">th altimg="si1.gif" overflow="scroll">m,n≥3th>, thmlsrc">text 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≤mnthContainer hidden">thCode">th altimg="si2.gif" overflow="scroll">tretchy="false">(m−1tretchy="false">)tretchy="false">(n−1tretchy="false">)+2≤p≤mnth>, and thmlsrc">text 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−pthContainer hidden">thCode">th altimg="si3.gif" overflow="scroll">u=mn−pth>. The set thmlsrc">text 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×m}thContainer hidden">thCode">th altimg="si11.gif" overflow="scroll">thvariant="double-struck">Ru×n×mth> of all real tensors with size thmlsrc">text 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×mthContainer hidden">thCode">th altimg="si283.gif" overflow="scroll">u×n×mth> is one to one corresponding to the set of bilinear maps thmlsrc">text 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^uthContainer hidden">thCode">th altimg="si6.gif" overflow="scroll">thvariant="double-struck">Rm×thvariant="double-struck">Rntretchy="false">→thvariant="double-struck">Ruth>. We show that thmlsrc">text 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×p}thContainer hidden">thCode">th altimg="si30.gif" overflow="scroll">thvariant="double-struck">Rm×n×pth> has plural typical ranks p   and thmlsrc">text 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+1thContainer hidden">thCode">th altimg="si1230.gif" overflow="scroll">p+1th> if and only if there exists a nonsingular bilinear map thmlsrc">text 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^uthContainer hidden">thCode">th altimg="si6.gif" overflow="scroll">thvariant="double-struck">Rm×thvariant="double-struck">Rntretchy="false">→thvariant="double-struck">Ruth>. We show that there is a dense open subset thmlsrc">text 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">OthContainer hidden">thCode">th altimg="si10.gif" overflow="scroll">thvariant="script">Oth> of thmlsrc">text 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×m}thContainer hidden">thCode">th altimg="si11.gif" overflow="scroll">thvariant="double-struck">Ru×n×mth> such that for any thmlsrc">text 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∈OthContainer hidden">thCode">th altimg="si12.gif" overflow="scroll">Y∈thvariant="script">Oth>, 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 thmlsrc">text 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">TthContainer hidden">thCode">th altimg="si13.gif" overflow="scroll">thvariant="script">Tth> of thmlsrc">text 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×m}thContainer hidden">thCode">th altimg="si1206.gif" overflow="scroll">thvariant="double-struck">Rn×p×mth> and continuous surjective open maps thmlsrc">text 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×pthContainer hidden">thCode">th altimg="si15.gif" overflow="scroll">ν:thvariant="script">Otretchy="false">→thvariant="double-struck">Ru×pth> and thmlsrc">text 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×pthContainer hidden">thCode">th altimg="si16.gif" overflow="scroll">σ:thvariant="script">Ttretchy="false">→thvariant="double-struck">Ru×pth>, where thmlsrc">text 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×pthContainer hidden">thCode">th altimg="si1165.gif" overflow="scroll">thvariant="double-struck">Ru×pth> is the set of thmlsrc">text 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×pthContainer hidden">thCode">th altimg="si18.gif" overflow="scroll">u×pth> matrices with entries in thmlsrc">text 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">RthContainer hidden">thCode">th altimg="si19.gif" overflow="scroll">thvariant="double-struck">Rth>, such that if thmlsrc">text 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)thContainer hidden">thCode">th altimg="si1221.gif" overflow="scroll">νtretchy="false">(Ytretchy="false">)=σtretchy="false">(Ttretchy="false">)th>, then thmlsrc">title="View the MathML source" 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">t="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">t>t="15" border="0" style="vertical-align:bottom" width="78" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869316303490-si1210.gif">t>thContainer hidden">thCode">th altimg="si1210.gif" overflow="scroll">thvariant="normal">rankace width="0.2em">ace>T=pth> if and only if the ideal of maximal minors of the matrix defined by Y is a real prime ideal.