The Number of Singular Vector Tuples of a general, [n, n, n, n], tensor .



                              By Shalosh B. Ekhad



Theorem: Let a(n) be the number of singular vector tuples of a, 4,

    -dimensional tensor whose dimensions are all, n



                   The first, 70, terms of that sequence are



[1, 24, 997, 51264, 2940841, 180296088, 11559133741, 765337680384,

    51921457661905, 3590122671128664, 252070718210663749, 17922684123178825536,

    1287832671004683373753, 93368940577497932331288, 6821632357294515590873917,

    501741975445243527381995520, 37121266623211130111114816929,

    2760712710223967190110979892824, 206267049696409355312012281872181,

    15475371710788303634337687336412224, 1165407938272751946314416465834086889,

    88061985234092428001001009986265842520,

    6674841972181830152646239523454436828797,

    507365393175531718935697292222219542209024,

    38665843085507634007413709865882990263469041,

    2953740030556494206297508806673140156327404248,

    226139608318389776420643582181314917081622957061,

    17348847768732971801911980996414434839471856232768,

    1333495999756767180730069621847180744764824179044825,

    102679492043788645453302773152678289049925015355480088,

    7919467782979625176350608079488125284094134882129213277,

    611762406707957516785060447056716898028654120149205549056,

    47326297794171750329395108432285966368588072502073467990849,

    3666210684055121798038265960682272734448162887732387739437656,

    284375989192008571736288450556047894658675111432554006369198101,

    22085002503311452557061359446412536526728124547277566177787892288,

    1717125381944152148262069150947173852066288121156424979958627271625,

    133653371349899710142461127502334621416505147105558621766272672907736,

    10413736715423917925250315399202563233072956051319130146752533719981053,

    812193134071724372030028233213580873621215660428760806777376397599783424,

    63404056487163136402118230809098571181603513366964484046606874882278962769,
    495404870295776115711352400557583854048523148264415431829750157916901736\

    3736, 3874100318148354357315358073965097674811439028231751173394317375796\

    21537848213, 303201910495363193592724593578518175292538326522697267166188\

    14117664053202386240, 237480383431377454502749370889146851756541219771240\

    0933613898364537188007266415065, 1861416594576426438350907503501931870952\

    13802254962408894820969634047160142280221656, 146004317099596470885615652\

    85289383892223776953914711697290780844366152779284725947757, 114599098890\
    084064149897435277243557481269148658099883288884044278976119295523929820\

    7744, 9000735680286626034068167574334329542335256741979641844973723539135\

    8299924670403492251105, 7073669380508352939748022113605592581392110691450\

    778592637430119392203264791502783256731864, 55625095315167919032619948449\

    2492324591698905227339332845928022594121900029904848181878033749, 4376691\
    962300787767887375973657362959160931004444537813623483415588718508818765\

    0501694445489216, 3445568690886615141423834635981332745335900510981797976\

    345680993602708821591585237841811565673065, 27139797839901901106029741500\

    6938046841879926951597708635047601874185952373678877541675978258404568, 2\
    138821834931500666706483254859097343585216990790743086036561160356374638\

    7794261792645171987486563229, 1686385703886231464111507816879306616123474\

    300541011700087381363447966207871114366608478451366840144384, 13302888122\
    049843048583783811778374771716769379195154923357133253194697246458059225\

    4702500946111712214577, 1049867536127299365636549722751643747785586228564\

    8134225237865420533609968529304268060706932598043796535256, 8289260757799\
    465145690753369951530725015915491679748172062102797059658787631621789274\

    41457954079168327435141, 654761827515441723680762749350961718833905613301\

    68239058377228182666146594166925063393822640826631362216652096, 517405461\
    096698574072548645580103138436702717158366563877816107402754755571057774\

    1512817639228972097875063728153, 4090278877495654738776431246449021753755\
    445680799901090751282626105638684605992342918691652237232042445965565803\

    76, 323477081411316703815112633244396816051373975539285755764374214440614\

    41018442893538538447435917662519366143191410589, 255916030760866495394527\
    698951579075886076340724451964065812248376001173136687814230899836681222\

    4871870032223390662656, 2025395240895160324669824819423853284655954325900\

    97915310662109613239469052923320596448435309068417769429529956502685313,
    160352438766309838626207434251742263871834256767821679966333363560195801\

    48925904327992960651893088013216529811030500811352, 126996055879892090550\
    012984024063981776695025754799704119954559825707666010351045306124517616\

    3841999507880978168128227321301, 1006119047236215049891970651026876005653\
    465553050036319143930926830585636462050949830520738795678147410970775185\

    52658278518336, 797349578873080205697751714231745229204003541172569345165\

    3750611331139698481136957917921419063990733623143125922769418501460361, 6\
    320979719089295536038281550236329777086192083175171402508021690987367731\

    80837166243210394531746401261452355825000537762507812568]

While we do know that the sequence is P-recursive, i.e. satisfies SOME homog\
    eneous linear recurrence equation with constant coefficients

the number of terms in the input, 70,

    does not suffice. If you are really interested, try to make it larger.



     nevertheless let's do some non-rigorous estimates of the asymptotics.

                                                        n
                                           C 80.98624909
  It seems that the sequence behaves like, --------------, for some constant C
                                             1.497184980
                                            n

                 These are close to our conjectured asymptotics

                                         n
                                     C 81
                                     -----
                                      3/2
                                     n



        estimating the constant we get that it is probably approximately



                                     0.021



              Hence a non-rigorous estimate of the asymptotics is



                                           n
                                   0.021 81
                                   ---------
                                      3/2
                                     n



         This ends this paper that took, 10.020, seconds  to generate.

                               Have a good day.