(* ::Package:: *)

(* ::Section:: *)
(**)
(*Z7(z1,t)=1. Again, we verify instead that*)
(**)
(*-1+b0(t)+b2 z1^2+b4 z1^4+b6 z1^6=-(b1z1+b3z1^3+b5z1^5+b7z1^7)*)
(**)
(*(in one stroke we also have Z7(z3,t)=Z7(z5,t)=1)*)
(**)


(* ::Input:: *)
(*Off[Root::nup]*)


(* ::Input:: *)
(*\[Omega]=Sqrt[(4-11 t+8 t^2)/t];*)


(* ::Input:: *)
(*iep3=z^3-(z^2 (-14+81 t-130 t^2-134 t^3+728 t^4-1020 t^5+680 t^6-216 t^7+24 t^8+(-1+2 t) (2+5 t-40 t^2+64 t^3-38 t^4+8 t^5) \[Omega]\[Omega]))/(8 (-1+t)^3 t^2 (-1+5 t-6 t^2+t^3))+(z (-2-17 t+245 t^2-812 t^3+98 t^4+5437 t^5-13045 t^6+7456 t^7+19604 t^8-46768 t^9+47296 t^10-26752 t^11+8608 t^12-1440 t^13+96 t^14+t (-1+2 t) (7-23 t-36 t^2+138 t^3+575 t^4-3055 t^5+5670 t^6-5392 t^7+2736 t^8-688 t^9+64 t^10) \[Omega]\[Omega]))/(32 (-1+t)^3 t^5 (-1+5 t-6 t^2+t^3)^2)-(-225+2930 t-16685 t^2+53762 t^3-104265 t^4+112187 t^5-23793 t^6-119612 t^7+201884 t^8-171608 t^9+89696 t^10-29616 t^11+5984 t^12-672 t^13+32 t^14+(-1+2 t) (25-90 t-629 t^2+5374 t^3-17553 t^4+32643 t^5-37939 t^6+28146 t^7-13104 t^8+3632 t^9-536 t^10+32 t^11) \[Omega]\[Omega])/(32 (-1+t)^3 t^2 (-1+5 t-6 t^2+t^3)^3);*)


(* ::Input:: *)
(*q=(2t-1)^10;*)
(**)
(*p61=16t^5(-10+27 t+288 t^2-2022 t^3+5825 t^4-9477 t^5+9336 t^6-5494 t^7+1712 t^8-122 t^9-80 t^10+16 t^11);*)
(*p62=(-16 t^6)(-25+206 t-726 t^2+1459 t^3-1927 t^4+1870 t^5-1394 t^6+724 t^7-210 t^8+24 t^9);*)
(*p41=(-4t^3)(50-687 t+4203 t^2-15695 t^3+41282 t^4-82204 t^5+125852 t^6-144150 t^7+117216 t^8-62520 t^9+18496 t^10-1080 t^11-960 t^12+192 t^13);*)
(*p42=4t^3(10-121 t+677 t^2-2521 t^3+7132 t^4-15412 t^5+24800 t^6-29742 t^7+27184 t^8-18840 t^9+9072 t^10-2520 t^11+288 t^12);*)
(*p21=(-2+40 t-360 t^2+2245 t^3-11098 t^4+42924 t^5-126314 t^6+282304 t^7-483968 t^8+638896 t^9-640864 t^10+470400 t^11-234144 t^12+65792 t^13-2784 t^14-3840 t^15+768 t^16);*)
(*p22=-t^3(65-774 t+4344 t^2-15694 t^3+41224 t^4-82096 t^5+124784 t^6-144768 t^7+127840 t^8-83808 t^9+37824 t^10-10080 t^11+1152 t^12);*)
(*p01=(1-t) (1-19 t+161 t^2-924 t^3+4066 t^4-13822 t^5+35840 t^6-70584 t^7+105400 t^8-118168 t^9+96704 t^10-54208 t^11+17760 t^12-1440 t^13-1024 t^14+256 t^15);*)
(*p02=t^3 (25-290 t+1636 t^2-6010 t^3+15992 t^4-32064 t^5+48928 t^6-56632 t^7+49024 t^8-30752 t^9+13120 t^10-3360 t^11+384 t^12);*)
(*p11=8t^3(t-1)(-125+4005 t-60169 t^2+570158 t^3-3862166 t^4+20045945 t^5-83223823 t^6+284217439 t^7-813613881 t^8+1978202656 t^9-4124892528 t^10+7432364016 t^11-11645707296 t^12+15960414880 t^13-19242695952 t^14+20530254512 t^15-19487643408 t^16+16504985216 t^17-12441863808 t^18+8263236224 t^19-4749495424 t^20+2309017088 t^21-926839040 t^22+300215040 t^23-76503296 t^24+14585856 t^25-1794048 t^26+73728 t^27+8192 t^28);*)
(*p12=8t^3(1-t)(-25+765 t-11171 t^2+105634 t^3-736638 t^4+4055851 t^5-18297419 t^6+69029409 t^7-220386161 t^8+599953820 t^9-1399432680 t^10+2804818368 t^11-4835265904 t^12+7165985152 t^13-9113381872 t^14+9916520688 t^15-9197207856 t^16+7238388416 t^17-4811313152 t^18+2688147584 t^19-1256283520 t^20+488377856 t^21-156990208 t^22+41689856 t^23-9309952 t^24+1799168 t^25-280576 t^26+24576 t^27);*)
(*p31=8 t^3(t-1)(-125+4005 t-59369 t^2+553998 t^3-3730470 t^4+19565545 t^5-83537119 t^6+297885767 t^7-901975729 t^8+2350136384 t^9-5333296784 t^10+10652922784 t^11-18880321184 t^12+29852879392 t^13-42255474560 t^14+53630116864 t^15-60971031168 t^16+61759492352 t^17-55128623872 t^18+42657332224 t^19-28037451008 t^20+15314942464 t^21-6803466496 t^22+2405327616 t^23-657957120 t^24+131469312 t^25-16441344 t^26+663552 t^27+73728 t^28);*)
(*p32=8 t^3(1-t)(-25+765 t-11171 t^2+107634 t^3-787518 t^4+4665067 t^5-22973819 t^6+95297577 t^7-336259225 t^8+1016942748 t^9-2647457192 t^10+5940364352 t^11-11479425312 t^12+19070848736 t^13-27171006848 t^14+33097766976 t^15-34341234112 t^16+30212895232 t^17-22415990528 t^18+13930544128 t^19-7190272256 t^20+3053491200 t^21-1060255488 t^22+303176448 t^23-73578240 t^24+15455232 t^25-2525184 t^26+221184 t^27);*)
(*p51=128t^8(t-1)(-1500+28960 t-257939 t^2+1404604 t^3-5211955 t^4+13795890 t^5-25855137 t^6+29614248 t^7+3063112 t^8-113012293 t^9+333031055 t^10-652968101 t^11+993241608 t^12-1221269400 t^13+1224164696 t^14-994694744 t^15+647101024 t^16-331790224 t^17+131533872 t^18-39147024 t^19+8229120 t^20-1046016 t^21+41472 t^22+4608 t^23);*)
(*p52=128 t^8(1-t)(-100+80 t+20039 t^2-260728 t^3+1786957 t^4-8269540 t^5+28531279 t^6-77414974 t^7+170435054 t^8-309998835 t^9+470441223 t^10-598385907 t^11+638090092 t^12-567729488 t^13+417123064 t^14-249429768 t^15+119783328 t^16-46124976 t^17+14587824 t^18-3960432 t^19+919872 t^20-157824 t^21+13824 t^22);*)
(*p71=2048 t^13(t-1)(-25+56 t+2520 t^2-25565 t^3+120032 t^4-330197 t^5+533239 t^6-339915 t^7-575358 t^8+1899107 t^9-2748394 t^10+2566194 t^11-1675233 t^12+781067 t^13-257689 t^14+57232 t^15-7392 t^16+288 t^17+32 t^18);*)
(*p72=2048 t^13(1-t)(-1-50 t+626 t^2-2319 t^3-1948 t^4+47711 t^5-201331 t^6+485563 t^7-779658 t^8+878711 t^9-712736 t^10+425072 t^11-193639 t^12+72003 t^13-23071 t^14+6068 t^15-1096 t^16+96 t^17);*)


(* ::Input:: *)
(*ContourPlot[Evaluate[{y==1/q(-q+{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.Table[(-Sqrt[Root[iep3,1]])^j,{j,6,0,-2}])/.\[Omega]\[Omega]->\[Omega],y==1/q{-Sqrt[p71+p72 \[Omega]\[Omega]],Sqrt[p51+p52 \[Omega]\[Omega]],-Sqrt[p31+p32 \[Omega]\[Omega]],Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[Root[iep3,1]])^j,{j,7,1,-2}]/.\[Omega]\[Omega]->\[Omega]}],{t,Root[-1 + 12 # - 20 #^2 + 8 #^3& , 2, 0],.999},{y,-5,5},WorkingPrecision->80,PlotPoints->80]*)


(* ::Input:: *)
(*ContourPlot[Evaluate[{y==1/q(-q+{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.Table[(Sqrt[Root[iep3,2]])^j,{j,6,0,-2}])/.\[Omega]\[Omega]->\[Omega],y==1/q{-Sqrt[p71+p72 \[Omega]\[Omega]],Sqrt[p51+p52 \[Omega]\[Omega]],-Sqrt[p31+p32 \[Omega]\[Omega]],Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(Sqrt[Root[iep3,2]])^j,{j,7,1,-2}]/.\[Omega]\[Omega]->\[Omega]}],{t,Root[-1 + 12 # - 20 #^2 + 8 #^3& , 2, 0],.999},{y,-5,5},WorkingPrecision->80,PlotPoints->80]*)


(* ::Input:: *)
(*ContourPlot[Evaluate[{y==1/q(-q+{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.Table[(-Sqrt[Root[iep3,3]])^j,{j,6,0,-2}])/.\[Omega]\[Omega]->\[Omega],y==1/q{-Sqrt[p71+p72 \[Omega]\[Omega]],Sqrt[p51+p52 \[Omega]\[Omega]],-Sqrt[p31+p32 \[Omega]\[Omega]],Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[Root[iep3,3]])^j,{j,7,1,-2}]/.\[Omega]\[Omega]->\[Omega]}],{t,Root[-1 + 12 # - 20 #^2 + 8 #^3& , 2, 0],.999},{y,-5,5},WorkingPrecision->80,PlotPoints->80]*)


(* ::Text:: *)
(**)
(*Canonical bivariate poly for both lhs and rhs*)


(* ::Input:: *)
(*qqq9=-(-1+2 t)^33 (40960000 t^5-1943552000 t^6+45285478400 t^7-690435860480 t^8+7743927232256 t^9-68108919866432 t^10+488951367704432 t^11-2944750063979824 t^12+15175970060479888 t^13-67929601432109680 t^14+267153306116315968 t^15-931564200668344928 t^16+2901234873070618208 t^17-8117718281903826272 t^18+20504594864656612848 t^19-46938766854057281136 t^20+97689962025293254736 t^21-185313014207268621360 t^22+321040887724083234272 t^23-508698685285134280192 t^24+738004829056194882752 t^25-980903075179142614464 t^26+1194692400632446871424 t^27-1333127378402104336384 t^28+1362138681532595858944 t^29-1273134239034149536256 t^30+1086950187535630619648 t^31-846062083308585042944 t^32+598964136457371922432 t^33-384503711839106015232 t^34+223000519703744294912 t^35-116327416959585554432 t^36+54286377662854893568 t^37-22516675294486294528 t^38+8235452639907504128 t^39-2630483836098461696 t^40+725003938985705472 t^41-169841565680795648 t^42+33166940610494464 t^43-5261745685790720 t^44+654409453273088 t^45-60564139147264 t^46+3837553278976 t^47-142807662592 t^48+2147483648 t^49)-(-1+2 t)^33 (-2048000 t^3+98897920 t^4-2301060096 t^5+34284058112 t^6-366100346048 t^7+2964161793936 t^8-18691860011004 t^9+91956453666440 t^10-339027331297076 t^11+773368922964576 t^12+485970715647896 t^13-16256778555818192 t^14+103527969481922696 t^15-460126867212055792 t^16+1652565858521336564 t^17-5048648935221121816 t^18+13450699414800590588 t^19-31703027459360917632 t^20+66713310049532305456 t^21-126113618703213864096 t^22+215088122456108451248 t^23-331958151583050360640 t^24+464579532249156713728 t^25-590376046514544427072 t^26+681745680377918777344 t^27-715612717391552582400 t^28+682789851101715750656 t^29-592042736709950430720 t^30+466409561118141895680 t^31-333820399437898950656 t^32+217171088645076603904 t^33-128601163772556646400 t^34+69502708363299216384 t^35-34419756121710096384 t^36+15691832007097171968 t^37-6607933276682190848 t^38+2568139456612139008 t^39-913878080313819136 t^40+292975978453532672 t^41-82638299907751936 t^42+19910855360512000 t^43-3955441272881152 t^44+620075182718976 t^45-72266448633856 t^46+5709353713664 t^47-257698037760 t^48+4294967296 t^49) y-(-1+2 t)^33 (-4096+226304 t-6104320 t^2+106074496 t^3-1324324336 t^4+12568448724 t^5-93608240131 t^6+556354788196 t^7-2649553376841 t^8+10004029859306 t^9-28951916123562 t^10+59415590450564 t^11-77505287576674 t^12+176745320100456 t^13-1858385126219215 t^14+14082876445044376 t^15-73553759124965153 t^16+297320808330105826 t^17-990278962499888772 t^18+2814306694136055344 t^19-6972020462350010132 t^20+15272227070460526008 t^21-29882305705517548848 t^22+52630072431602463600 t^23-83949745932641885408 t^24+121888599331638854656 t^25-161772260127566524672 t^26+196952719105805828224 t^27-220553400610110662912 t^28+227577950035522799104 t^29-216519086804267177216 t^30+189819896193008328704 t^31-153040708719458708736 t^32+113097452158906555904 t^33-76262504269633038336 t^34+46661088994199511040 t^35-25737575460889006080 t^36+12705578545489403904 t^37-5568947040026361856 t^38+2148594644742111232 t^39-722991853128122368 t^40+210141777697439744 t^41-52239057422385152 t^42+10994821218959360 t^43-1936636476653568 t^44+280320846856192 t^45-32093069377536 t^46+2666906255360 t^47-132070244352 t^48+2147483648 t^49) y^2-(-1+2 t)^33 (-9728 t^3+319616 t^4-4262560 t^5+21231272 t^6+174017144 t^7-4358709897 t^8+46221876998 t^9-346041528932 t^10+2115496257784 t^11-11537827945399 t^12+59243824444198 t^13-289966450760380 t^14+1328416837044532 t^15-5559254885446588 t^16+20912000519681028 t^17-70248423450837684 t^18+210661040704675080 t^19-565417341215564252 t^20+1363135099498485384 t^21-2962035232396486768 t^22+5817500497339138992 t^23-10346745192826359104 t^24+16679998567994197376 t^25-24372090387510151232 t^26+32244610031834865792 t^27-38550755854067812864 t^28+41525171375735604480 t^29-40127474227170270208 t^30+34581308273774562304 t^31-26352234840173588480 t^32+17530083121667678208 t^33-9964650758102171648 t^34+4644220093518381056 t^35-1598290785896235008 t^36+240060898049327104 t^37+164502136755585024 t^38-177224368158670848 t^39+100248003193864192 t^40-41072046192459776 t^41+12939930401505280 t^42-3147949555056640 t^43+576997851922432 t^44-75406338162688 t^45+6282463412224 t^46-251255586816 t^47) y^3-(-1+2 t)^33 (128 t^5-6672 t^6+187232 t^7-3746757 t^8+58668800 t^9-745763428 t^10+7849951402 t^11-69467928703 t^12+523633324918 t^13-3399714007506 t^14+19188108219416 t^15-94847507984346 t^16+413060552367952 t^17-1592556998134424 t^18+5457724439694136 t^19-16682213545947196 t^20+45618158313068376 t^21-111908238505973808 t^22+246912778434890576 t^23-491178722971778912 t^24+882986026840374848 t^25-1437557480495007040 t^26+2123705498248846976 t^27-2851318733313996032 t^28+3482789737286860288 t^29-3871253606239122432 t^30+3912883543987812352 t^31-3589394351929233408 t^32+2978555229037125632 t^33-2225509571674685440 t^34+1488208430438318080 t^35-884019654558154752 t^36+462290661732909056 t^37-210542393341771776 t^38+82424345549864960 t^39-27289732007329792 t^40+7482714041090048 t^41-1651598350614528 t^42+281679574859776 t^43-34798294794240 t^44+2767837986816 t^45-106300440576 t^46) y^4-(-1+2 t)^33 (152 t^8-7398 t^9+165929 t^10-2211278 t^11+18132711 t^12-68726802 t^13-413256616 t^14+9534928924 t^15-93166858152 t^16+642663579152 t^17-3486379980012 t^18+15547377177468 t^19-58329088678740 t^20+186582400399240 t^21-512787829217984 t^22+1215033657475072 t^23-2480957459206208 t^24+4342539884621440 t^25-6432194348111360 t^26+7834870970099712 t^27-7297962682238976 t^28+3901717429653504 t^29+2127573466480640 t^30-9119737862225920 t^31+14631841735737344 t^32-16781407917899776 t^33+15285877524660224 t^34-11431014044008448 t^35+7090339839737856 t^36-3645066435887104 t^37+1540208050831360 t^38-526658570289152 t^39+142198158917632 t^40-29182012686336 t^41+4276210368512 t^42-398358216704 t^43+17716740096 t^44) y^5-(-1+2 t)^33 (-t^10+60 t^11-1743 t^12+32654 t^13-443355 t^14+4648056 t^15-39146185 t^16+272061558 t^17-1590536844 t^18+7934058600 t^19-34135077600 t^20+127717012800 t^21-418214409600 t^22+1204361107200 t^23-3061184889600 t^24+6885051486720 t^25-13725049881600 t^26+24268379535360 t^27-38059283988480 t^28+52891781529600 t^29-65022693212160 t^30+70522375372800 t^31-67228313518080 t^32+56050744688640 t^33-40606924800000 t^34+25349271846912 t^35-13487906488320 t^36+6029853065216 t^37-2221563445248 t^38+656576348160 t^39-149602435072 t^40+24662507520 t^41-2617245696 t^42+134217728 t^43) y^6;*)


(* ::Input:: *)
(*Solve[qqq9==0/.t->4/5,Reals]*)


(* ::Input:: *)
(*ContourPlot[Evaluate[{qqq9==0}],{t,Root[-1 + 12 # - 20 #^2 + 8 #^3& , 2, 0],.999},{y,-1,1},WorkingPrecision->100,PlotPoints->120]*)


(* ::Input:: *)
(*ContourPlot[Evaluate[{qqq9==0,y==1/q(-q+{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.Table[(-Sqrt[Root[iep3,1]])^j,{j,6,0,-2}])/.\[Omega]\[Omega]->\[Omega]}],{t,Root[-1 + 12 # - 20 #^2 + 8 #^3& , 2, 0],.999},{y,-1,1},WorkingPrecision->100,PlotPoints->120]*)


(* ::Input:: *)
(*RootReduce[{Root[qqq9/.t->4/5,2],Root[qqq9/.t->4/5,3],Root[qqq9/.t->4/5,4],1/q(-q+{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.Table[(-Sqrt[Root[iep3,1]])^j,{j,6,0,-2}])/.\[Omega]\[Omega]->\[Omega]/.t->4/5,1/q(-q+{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.Table[(Sqrt[Root[iep3,2]])^j,{j,6,0,-2}])/.\[Omega]\[Omega]->\[Omega]/.t->4/5,1/q(-q+{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.Table[(-Sqrt[Root[iep3,3]])^j,{j,6,0,-2}])/.\[Omega]\[Omega]->\[Omega]/.t->4/5}]*)


(* ::Input:: *)
(*ContourPlot[Evaluate[{qqq9==0,y==1/q(-q+{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.Table[(Sqrt[Root[iep3,2]])^j,{j,6,0,-2}])/.\[Omega]\[Omega]->\[Omega]}],{t,Root[-1 + 12 # - 20 #^2 + 8 #^3& , 2, 0],.999},{y,-1,1},WorkingPrecision->100,PlotPoints->120]*)


(* ::Input:: *)
(*ContourPlot[Evaluate[{qqq9==0,y==1/q(-q+{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.Table[(-Sqrt[Root[iep3,3]])^j,{j,6,0,-2}])/.\[Omega]\[Omega]->\[Omega]}],{t,Root[-1 + 12 # - 20 #^2 + 8 #^3& , 2, 0],.999},{y,-1,1},WorkingPrecision->100,PlotPoints->120]*)
