Research Report, August 13, 2009 Analysis of Bertini Experiments on Unions of Lines Barry Dayton (bhdayton@neiu.edu) APPENDICES Appendix A: Double Six Example from [2,3]. Equations s3,s4 in these appencices will always refer to the following. Equations a2,b3,etc. will vary from appendix to appendix. s3 = -0.07854101667099082*w^3 + 0.00672799793795011*w^2*x + 0.2780246796876111*w*x^2 - 0.14165235195750658*x^3 + 0.22685723299180435*w^2*y - 0.2403499894387601*w*x*y + 0.0012291623108112155*x^2*y - 0.1438584059373855*w*y^2 + 0.0902219369918389*x*y^2 + 0.025574906573001755*y^3 - 0.0592598023651423*w^2*z - 0.4546810996934818*w*x*z + 0.4446720551610471*x^2*z + 0.32518657502558823*w*y*z - 0.06904769142960078*x*y*z - 0.11708568721122255*y^2*z + 0.1483753405118448*w*z^2 - 0.4267886154828161*x*z^2 + 0.085244919842413*y*z^2 + 0.11860231987193742*z^3 s4 = 0.011399336716779169*w^4 + 0.0730237512815985*w^3*x - 0.22583589596001*w^2*x^2 + 0.2411008635304777*w*x^3 - 0.08133236801887503*x^4 + 0.021076725454793777*w^3*y - 0.2708478130165189*w^2*x*y + 0.36084578098551395*w*x^2*y - 0.15538465319784464*x^3*y - 0.10298377530090473*w^2*y^2 + 0.2875764232358533*w*x*y^2 - 0.15276272385590237*x^2*y^2 + 0.07014779363108166*w*y^3 - 0.07456372405174631*x*y^3 - 0.012821764272206411*y^4 + 0.09215724744950174*w^3*z + 0.0239362342559908*w^2*x*z - 0.29821062304579565*w*x^2*z + 0.20216639766351885*x^3*z - 0.062265375631754986*w^2*y*z - 0.14668106196461192*w*x*y*z + 0.192809154275353*x^2*y*z - 0.13526962687319175*w*y^2*z + 0.17033051314869155*x*y^2*z + 0.05688385604800236*y^3*z + 0.23678715501597117*w^2*z^2 - 0.1961547713373862*w*x*z^2 - 0.06361680041337349*x^2*z^2 - 0.23078915451228385*w*y*z^2 + 0.11763445757973791*x*y*z^2 - 0.019153727957407652*y^2*z^2 + 0.24048681974356*w*z^3 - 0.14387097401778862*x*z^3 - 0.1405582756118241*y*z^3 + 0.0837669673183713*z^4 APPENDIX B: 3-3 and 3-4 Configurations Equations s3, s4 above and a3 = 0.13054565569140206*w^3 - 0.1601379925607383*w^2*x - 0.15721805854050172*w*x^2 + 0.11004784028442606*x^3 - 0.29717376330851275*w^2*y + 0.3729764846652774*w*x*y - 0.05429915420522553*x^2*y + 0.1764249918651528*w*y^2 - 0.11909231142259409*x*y^2 - 0.03052970440719539*y^3 + 0.18958669761202113*w^2*z + 0.2414575956266261*w*x*z - 0.3657279774876291*x^2*z - 0.44968571900861487*w*y*z + 0.16985233626982457*x*y*z + 0.1465659499593698*y^2*z - 0.05429250044401394*w*z^2 + 0.363313769005202*x*z^2 - 0.13516763981943217*y*z^2 - 0.10053530775156876*z^3; a4 = -0.09904902611134198*w^4 - 0.01601120456955225*w^3*x + 0.2397654658459088*w^2*x^2 - 0.1743681855424718*w*x^3 - 0.08425289058582923*x^4 + 0.18070640106653876*w^3*y + 0.26499881379756024*w^2*x*y - 0.06940243067769743*w*x^2*y - 0.21651646512579226*x^3*y - 0.2596299560076209*w^2*y^2 + 0.14583708986119293*w*x*y^2 - 0.147817767944064*x^2*y^2 + 0.08655449303713406*w*y^3 - 0.042800372378279467*x*y^3 - 0.005421206921316752*y^4 - 0.22726413322866917*w^3*z + 0.22449801321206922*w^2*x*z + 0.07020893354085221*w*x^2*z + 0.180534755956721*x^3*z - 0.0993097478753132*w^2*y*z + 0.11295391789150684*w*x*y*z + 0.43460894831554475*x^2*y*z - 0.027152662139480362*w*y^2*z + 0.1359703956027507*x*y^2*z - 0.003789680305709474*y^3*z - 0.15083609687054803*w^2*z^2 + 0.0004194626649479697*w*x*z^2 - 0.0045713646243985395*x^2*z^2 - 0.32185862351844097*w*y*z^2 - 0.0936208495818799*x*y*z^2 + 0.07999649644077701*y^2*z^2 + 0.12923608830990896*w*z^3 - 0.23047146559715814*x*z^3 - 0.14680171088088226*y*z^3 + 0.1314250132439309*z^4; b4 = 0.08068013605354234*w^4 + 0.046499506446596554*w^3*x - 0.12356654679882398*w^2*x^2 - 0.024525550855664748*w*x^3 - 0.37823477477864476*x^4 - 0.13245746087960378*w^3*y - 0.1366229960458313*w^2*x*y + 0.12091689651313081*w*x^2*y - 0.3569714014688565*x^3*y + 0.08448682903422058*w^2*y^2 + 0.05089026933608163*w*x*y^2 - 0.009063674624221153*x^2*y^2 - 0.03546343931400162*w*y^3 + 0.04248741774452265*x*y^3 + 0.00713399775311911*y^4 + 0.10585378530554657*w^3*z + 0.04976208810065258*w^2*x*z + 0.4040270860658223*w*x^2*z + 0.4086158870385463*x^3*z - 0.10828792061550477*w^2*y*z + 0.14255259859299796*w*x*y*z + 0.052156222324646694*x^2*y*z - 0.042898236868210246*w*y^2*z - 0.09091346126658593*x*y^2*z + 0.01651690623387541*y^3*z - 0.18543213189962024*w^2*z^2 + 0.022141351322456655*w*x*z^2 + 0.10865195422294187*x^2*z^2 + 0.11427742238453968*w*y*z^2 + 0.19562586253524686*x*y*z^2 - 0.10031769341097432*y^2*z^2 - 0.34626839101513435*w*z^3 + 0.027270267345296285*x*z^3 + 0.08591949222389196*y*z^3 - 0.1268064789053836*z^4; Settings used in Bertini Computations TRACKTOLBEFOREEG: 1e-8; TRACKTOLDURINGEG: 1e-7; FINALTOL: 1e-6; APPENDIX C: Example 1 a1 = 0.8423317652433764*w^2 - 0.6550822097359125*w*x + 0.10855788354597991*x^2 - 0.019276086400998697*w*y + 0.06563320025316842*x*y - 0.2850211381265898*y^2 + 0.2151697709358656*w*z - 0.27251790301621065*x*z + 0.6690188244496514*y*z - 0.3022382894417572*z^2; a2 = -0.545317434427467*w^2 - 0.02815997674307029*w*x - 0.4998009758764356*x^2 - 0.1772392543540178*w*y - 0.43803489437287246*x*y + 0.48887497568747224*y^2 + 0.17409771703561416*w*z + 0.3980428851281009*x*z - 0.5852016017413417*y*z + 0.2770875619103029*z^2; a3 = -0.0924686831944905*w^2 + 0.2802891441390395*w*x - 0.22417910120015191*x^2 - 0.3720740991695751*w*y + 0.11889053070507773*x*y + 0.23244986233645465*y^2 - 0.2032309845739078*w*z - 0.09582325632664643*x*z - 0.20233734312072474*y*z + 0.027215679432459283*z^2; b1 = -0.10990213746181605*w^2 - 0.05791308600017342*w*x + 0.36542641537702564*x^2 - 0.3015629470592789*w*y - 0.34726596397096543*x*y + 0.318489305264731*y^2 - 0.2889276119412282*w*z - 0.17523418998701848*x*z - 0.0013026540117735009*y*z - 0.13990024021211306*z^2; b2 = 0.14420745081123584*w^2 - 0.040714596033184614*w*x + 0.07901149697800953*x^2 - 0.5271287483872981*w*y + 0.12335572909858283*x*y + 0.26223662404493425*y^2 - 0.2637976012347384*w*z - 0.19709356911866188*x*z - 0.05434554240100202*y*z - 0.19275725507073474*z^2; b3 = 0.2019419687757345*w^2 + 0.7032154565428419*w*x - 0.2621840030416183*x^2 - 0.1966659494536842*w*y + 0.3270550348902404*x*y - 0.1239113362834538*y^2 - 0.33833896130646846*w*z - 0.49059174885708234*x*z + 0.25980892886261525*y*z + 0.038936705698299004*z^2; b4 = 0.16564324753799975*w^2 + 0.005396790150385626*w*x + 0.20601957975595103*x^2 - 0.3761570501778822*w*y - 0.427009300846504*x*y + 0.26178166938615843*y^2 - 0.316106861917485*w*z - 0.45212323253384445*x*z + 0.32165059714874333*y*z - 0.15507448379868063*z^2; c1 = -0.336047193788699*w^2 + 0.8012569811082089*w*x + 0.055241808856933916*x^2 + 0.03134016962970116*w*y + 0.12123506577663144*x*y - 0.020724809477107028*y^2 - 0.3921137240593352*w*z - 0.21675171310442556*x*z - 0.07990453763287103*y*z + 0.137036485391528*z^2; c2 = 0.47113767846298626*w^2 + 0.042104277876521*w*x - 0.17604124024590317*x^2 - 0.5444215610736005*w*y + 0.060685076578796904*x*y + 0.14239709284254065*y^2 - 0.2350646710573646*w*z - 0.4832081695826114*x*z + 0.3248764659875741*y*z - 0.17498942778131177*z^2; c3 = -0.3126265964249045*w^2 - 0.08563039750215286*w*x + 0.055494298844855405*x^2 - 0.23674647933892523*w*y - 0.7616813262884645*x*y + 0.4780460599780467*y^2 - 0.11666693859227205*w*z - 0.04545924135848303*x*z - 0.09367463080758118*y*z + 0.05161784215111746*z^2; c4 = -0.13655249299588545*w^2 - 0.10587672032509167*w*x - 0.08579507259600772*x^2 - 0.48240109190800246*w*y + 0.4619886779337418*x*y + 0.29978910719525026*y^2 - 0.08513217806576719*w*z + 0.2643673323226019*x*z - 0.5828505467636143*y*z - 0.09956177709290377*z^2; c5 = 0.0160312967710768*w^2 + 0.18538406303235752*w*x - 0.8780196489045738*x^2 + 0.0037364420517448935*w*y - 0.09730966559218107*x*y + 0.028410849100774373*y^2 + 0.28953960548138874*w*z + 0.06510862973777132*x*z - 0.03908699399693089*y*z + 0.30751719393081384*z^2; Random linear equation used to solve {a1,a2,a3,e1} to get points e1= -1 - 0.527621347683513*w - 0.9489767134559433*x - 0.19678424943258932*y + 0.6380505759759205*z Points [x,y,z,w] in P^3 claimed as solution of ideal but not or P1 = [-0.8646751921586296, 0.8357319153526305, 0.3770454395823783, -0.1958377844015793] P2 = [-0.34761698403498653, 0.5057209658233641, 1.1548930556163037, -0.062084899422913176]; Line contained in V() etc. <-0.8552832354815139*w + 0.6569077610203435*x + 0.4241581605007889*y - 0.7952457698114355*z, 0.292892299965422*w - 0.5940001795146785*x - 0.06265223139623144*y + 0.39741236712817773*z> APPENDIX D. The Double 4 not in cubic a4 = 0.009887943915829932*w^4 + 0.019299958701535773*w^3*x - 0.08904763182574421*w^2*x^2 + 0.045540668000223695*w*x^3 - 0.020878773504312748*x^4 - 0.08250306132935921*w^3*y + 0.14086595949441963*w^2*x*y + 0.06962673886583108*w*x^2*y + 0.07731848798164746*x^3*y + 0.0868538799226636*w^2*y^2 - 0.2852070189201125*w*x*y^2 - 0.22324160062799214*x^2*y^2 + 0.047047020859971735*w*y^3 + 0.3169662222406017*x*y^3 - 0.1116611597340631*y^4 - 0.0536961191107958*w^3*z + 0.1227460443587556*w^2*x*z + 0.02180748272692798*w*x^2*z + 0.0048725452252306916*x^3*z + 0.08303821724380181*w^2*y*z - 0.4429423635698887*w*x*y*z - 0.08531646757086449*x^2*y*z + 0.20009116495899834*w*y^2*z + 0.4282269660755917*x*y^2*z - 0.280685954612915*y^3*z + 0.021911809617904135*w^2*z^2 - 0.18659720761181953*w*x*z^2 + 0.07716046245302091*x^2*z^2 + 0.20330831755871653*w*y*z^2 + 0.11399596215632452*x*y*z^2 - 0.24294886675209307*y^2*z^2 + 0.058415417586010016*w*z^3 - 0.012315571379497334*x*z^3 - 0.09101398675625064*y*z^3 - 0.016354477470944897*z^4; b4 = 0.01827279847899918*w^4 - 0.05748963128218592*w^3*x + 0.06713358460495703*w^2*x^2 - 0.04483108136586145*w*x^3 - 0.005449424808353229*x^4 - 0.01406522303460962*w^3*y + 0.12880975812989445*w^2*x*y - 0.2789134484919151*w*x^2*y + 0.24650436643300186*x^3*y - 0.16069859628425184*w^2*y^2 + 0.4500794143566874*w*x*y^2 - 0.32922621208334013*x^2*y^2 - 0.023381697976944123*w*y^3 - 0.088364296072268*x*y^3 + 0.09766662050863362*y^4 + 0.05470290218179271*w^3*z - 0.22620740482068236*w^2*x*z + 0.28291053703970515*w*x^2*z - 0.030076330761200573*x^3*z + 0.032320253508516646*w^2*y*z + 0.10360120164917763*w*x*y*z - 0.40112198021797374*x^2*y*z - 0.17227102005925246*w*y^2*z + 0.3070693768616076*x*y^2*z + 0.07354702631475112*y^3*z + 0.04504132952684536*w^2*z^2 - 0.135776874949854*w*x*z^2 + 0.04499956637652671*x^2*z^2 + 0.0026308220126809354*w*y*z^2 + 0.13602840712237604*x*y*z^2 - 0.03840629991395299*y^2*z^2 + 0.00109631555347016*w*z^3 + 0.006449213350705143*x*z^3 - 0.008279259525815371*y*z^3 - 0.006528189338348878*z^4; c4 = 0.06658012759451086*w^4 - 0.1725387242354847*w^3*x + 0.061818779574354436*w^2*x^2 + 0.01907078974241421*w*x^3 + 0.03492442616792407*x^4 - 0.13927705980823218*w^3*y + 0.5474951796690766*w^2*x*y - 0.2791186806984801*w*x^2*y - 0.2012845275584901*x^3*y - 0.1512562649633206*w^2*y^2 - 0.17006642962757793*w*x*y^2 + 0.4831053892083966*x^2*y^2 + 0.18432670215249783*w*y^3 - 0.3427208207426017*x*y^3 + 0.0882192297839945*y^4 + 0.0393947721800592*w^3*z - 0.03572956041223892*w^2*x*z + 0.010968395528500405*w*x^2*z + 0.048005806525514304*x^3*z - 0.05442209584244736*w^2*y*z - 0.056348281344752206*w*x*y*z - 0.10763655778828897*x^2*y*z + 0.07948760216640699*w*y^2*z + 0.10599862538553255*x*y^2*z - 0.03197992143837371*y^3*z - 0.04751424295672078*w^2*z^2 + 0.043234309577465144*w*x*z^2 - 0.02204891471578591*x^2*z^2 + 0.127289869800976*w*y*z^2 + 0.014934520690793797*x*y*z^2 - 0.12158863327068606*y^2*z^2 - 0.02007179468646983*w*z^3 - 0.002101834691106233*x*z^3 + 0.024669945268264284*y*z^3 + 0.004614416164199115*z^4; d4 = 0.023054256574348946*w^4 - 0.18445130682614386*w^3*x + 0.28273526013971084*w^2*x^2 - 0.06483409712735659*w*x^3 - 0.05075668802583251*x^4 + 0.05190315437367382*w^3*y + 0.16377332611840564*w^2*x*y - 0.4678043567500384*w*x^2*y + 0.28572043627587346*x^3*y - 0.13439548912682428*w^2*y^2 + 0.16673826092863833*w*x*y^2 - 0.2109185895902956*x^2*y^2 + 0.013271970602970404*w*y^3 + 0.16830443509564336*x*y^3 - 0.03193577979065122*y^4 - 0.02723808398270967*w^3*z + 0.16469850155272875*w^2*x*z - 0.11003943568197372*w*x^2*z - 0.09062336614890797*x^3*z - 0.14360267518400158*w^2*y*z - 0.1675692758827488*w*x*y*z + 0.44856188112289536*x^2*y*z + 0.21800209960081166*w*y^2*z - 0.1987947215274668*x*y^2*z + 0.017385213942885287*y^3*z - 0.049936434467905896*w^2*z^2 + 0.08473412373064244*w*x*z^2 - 0.03256107587464512*x^2*z^2 + 0.0929903569806654*w*y*z^2 - 0.14903285722792234*x*y*z^2 + 0.023516214690732822*y^2*z^2 + 0.01032721404192302*w*z^3 + 0.0079641515159334*x*z^3 - 0.032942451356231965*y*z^3 + 0.004173888860658973*z^4; e4 = 0.054332447333606454*w^4 - 0.17796384160527562*w^3*x + 0.3506312488791379*w^2*x^2 - 0.45313587947405237*w*x^3 + 0.17449648334164947*x^4 - 0.13809794322221308*w^3*y + 0.12114757635531184*w^2*x*y + 0.21223129302603946*w*x^2*y + 0.023620906443569713*x^3*y + 0.13333542371730256*w^2*y^2 - 0.17432976589971608*w*x*y^2 - 0.2338741222185726*x^2*y^2 + 0.02321100764943867*w*y^3 + 0.10130839467923736*x*y^3 - 0.05142407378589515*y^4 + 0.022503575468233016*w^3*z + 0.008813735433047715*w^2*x*z - 0.10362190088414996*w*x^2*z + 0.14253845046392252*x^3*z - 0.06286419383966253*w^2*y*z + 0.16841830158057874*w*x*y*z - 0.3042125371193019*x^2*y*z + 0.05604700718117499*w*y^2*z + 0.06899450248053053*x*y^2*z - 0.07219736259246018*y^3*z - 0.13998772015971886*w^2*z^2 + 0.3588887357515366*w*x*z^2 - 0.15474876234672075*x^2*z^2 - 0.0006816240866635015*w*y*z^2 - 0.16754977250968572*x*y*z^2 + 0.0998663514415403*y^2*z^2 - 0.08501932389219259*w*z^3 - 0.007467414083796706*x*z^3 + 0.1518034912780055*y*z^3 + 0.02565433165667173*z^4; f4 = 0.015951255259730106*w^4 - 0.17772902961593764*w^3*x + 0.17771171523099674*w^2*x^2 + 0.023542056979688385*w*x^3 - 0.015854736721711064*x^4 + 0.2573032508767276*w^3*y - 0.13794179999607173*w^2*x*y + 0.16338098222258815*w*x^2*y - 0.2709271758851762*x^3*y - 0.4834288522291382*w^2*y^2 + 0.08199762511930593*w*x*y^2 + 0.07441735027255428*x^2*y^2 + 0.13289489644413127*w*y^3 + 0.23983037362910703*x*y^3 + 0.04642237849598549*y^4 - 0.03446349111041086*w^3*z + 0.2630287369590149*w^2*x*z - 0.21327521656563714*w*x^2*z + 0.027684753074725038*x^3*z - 0.3264769696664536*w^2*y*z + 0.14966071992259422*w*x*y*z - 0.1065571385054174*x^2*y*z + 0.09724258040317985*w*y^2*z + 0.25327293576681237*x*y^2*z + 0.1542020425475669*y^3*z + 0.03966840188021281*w^2*z^2 - 0.16960986724510202*w*x*z^2 + 0.040815353058085306*x^2*z^2 + 0.004720133727696767*w*y*z^2 + 0.15227793248739124*x*y*z^2 + 0.08573620913873294*y^2*z^2 + 0.07389230367007574*w*z^3 - 0.042815010836218595*x*z^3 - 0.049812326965637595*y*z^3 - 0.021313447460603166*z^4; g4 = 0.026733378690264212*w^4 - 0.04798338904676954*w^3*x + 0.15988664920291001*w^2*x^2 - 0.18949789396334882*w*x^3 + 0.06862688854256789*x^4 - 0.11070468993039834*w^3*y - 0.06841810612608777*w^2*x*y - 0.0268038097714973*w*x^2*y + 0.04199245997367752*x^3*y + 0.1986218304903635*w^2*y^2 + 0.21015481317832685*w*x*y^2 + 0.02908096791220281*x^2*y^2 - 0.10955893710278265*w*y^3 - 0.26265913588289785*x*y^3 + 0.03749005499774472*y^4 - 0.15478470939973585*w^3*z + 0.34124896585601977*w^2*x*z - 0.4334009461539378*w*x^2*z + 0.2221860315082342*x^3*z + 0.23838228349627413*w^2*y*z + 0.04691298596835171*w*x*y*z - 0.11655679852102532*x^2*y*z - 0.11578342516566897*w*y^2*z - 0.13521847547078053*x*y^2*z - 0.032730713498944075*y^3*z + 0.07482571480800826*w^2*z^2 - 0.2777202685720575*w*x*z^2 + 0.17013126253484598*x^2*z^2 + 0.1639627794925884*w*y*z^2 + 0.037979636701887566*x*y*z^2 - 0.2210667147914072*y^2*z^2 + 0.16729018826650685*w*z^3 - 0.07318035394881653*x*z^3 - 0.19343412631871693*y*z^3 - 0.04369372655070115*z^4; The following 8 points are zeros of a4,b4 and c4 but not d4,...,g4. For uniqueness of representation these points are given in affine coordinates, w = 1. p1 = {0.6772190792059358, 0.17053337674067143, 0.5776253955753501}; p2 = {3.1153096422400237, 2.082182969156931, -0.009612623338229551}; p3 = {0.4529408753972345, 0.5650816997757243, -0.86045901722832}; p4 = {0.880092135385136, 0.5065231920380551, 0.46131938413622364}; p5 = {-0.2714527483923413 - 4.487563071130615*I, -0.49739901667968867 - 1.0141191534828344*I, 0.11740186089367005 - 1.7987574133902928*I}; p6 = {-0.27145274839234157 + 4.487563071130617*I, -0.4973990166796887 + 1.0141191534828344*I, 0.11740186089367009 + 1.798757413390293*I}; p7 = {0.6367997158806823 - 0.02593310153590579*I, 0.010938009276501466 + 0.12509264935117267*I, 0.29246383349425215 - 0.21517043444251127*I}; p8 = {0.6367997158806823 + 0.02593310153590579*I, 0.010938009276501466 - 0.12509264935117267*I, 0.29246383349425215 + 0.21517043444251127*I};