(* ::Package:: *)

(* ::Subsubsection:: *)
(**)
(*Goal: to verify that Z7'(z2,t)=0*)
(*instead we will show that z2*Z7(z2,t)=0*)


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


(* ::Input:: *)
(*q0=(16 (-1+t)^3 t^2 (-1+5 t-6 t^2+t^3));*)
(*pp1=(-6+25 t+14 t^2-266 t^3+648 t^4-752 t^5+464 t^6-144 t^7+16 t^8);*)
(*pp2=(2t-1)(2-3 t-8 t^2+20 t^3-14 t^4+4 t^5);*)
(*pp3=2(2t-1)^2t(10-41 t-48 t^2+590 t^3-1482 t^4+1870 t^5-1328 t^6+526 t^7-104 t^8+8 t^9);*)
(*pp4=2 (2t-1)^2 t^2((19-124 t+322 t^2-422 t^3+292 t^4-100 t^5+12 t^6));*)
(**)


(* ::Input:: *)
(*z2s=(-6+25 t+14 t^2-266 t^3+648 t^4-752 t^5+464 t^6-144 t^7+16 t^8+(-1+2 t) \[Omega]\[Omega] (2-3 t-8 t^2+20 t^3-14 t^4+4 t^5+Sqrt[2] \[Sqrt](t (10-41 t-48 t^2+590 t^3-1482 t^4+1870 t^5-1328 t^6+526 t^7-104 t^8+8 t^9+t (19-124 t+322 t^2-422 t^3+292 t^4-100 t^5+12 t^6) \[Omega]\[Omega]))))/(16 (-1+t)^3 t^2 (-1+5 t-6 t^2+t^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:: *)
(*RootReduce[1/q{Sqrt[p71+p72 \[Omega]\[Omega]],p61+p62 \[Omega]\[Omega],-Sqrt[p51+p52 \[Omega]\[Omega]],p41+p42 \[Omega]\[Omega],Sqrt[p31+p32 \[Omega]\[Omega]],p21+p22 \[Omega]\[Omega],-Sqrt[p11+p12 \[Omega]\[Omega]],p01+p02 \[Omega]\[Omega]}.Table[(-Sqrt[z2s])^j,{j,7,0,-1}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]*)


(* ::Input:: *)
(*RootReduce[{7Sqrt[p71+p72 \[Omega]\[Omega]],6(p61+p62 \[Omega]\[Omega]),-5Sqrt[p51+p52 \[Omega]\[Omega]],4(p41+p42 \[Omega]\[Omega]),3Sqrt[p31+p32 \[Omega]\[Omega]],2(p21+p22 \[Omega]\[Omega]),-Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,6,0,-1}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]*)


(* ::Input:: *)
(*RootReduce[{7Sqrt[p71+p72 \[Omega]\[Omega]],6(p61+p62 \[Omega]\[Omega]),-5Sqrt[p51+p52 \[Omega]\[Omega]],4(p41+p42 \[Omega]\[Omega]),3Sqrt[p31+p32 \[Omega]\[Omega]],2(p21+p22 \[Omega]\[Omega]),-Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,7,1,-1}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]*)


(* ::Input:: *)
(*RootReduce[{7Sqrt[p71+p72 \[Omega]\[Omega]],6(p61+p62 \[Omega]\[Omega]),-5Sqrt[p51+p52 \[Omega]\[Omega]],4(p41+p42 \[Omega]\[Omega]),3Sqrt[p31+p32 \[Omega]\[Omega]],2(p21+p22 \[Omega]\[Omega]),-Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,6,0,-1}]/.\[Omega]\[Omega]->\[Omega]/.{t->1/(11-4Sqrt[6])}]*)


(* ::Text:: *)
(**)
(*The first idea is again consider z2 Z7'(z2)*)


(* ::Input:: *)
(*RootReduce[6(p61+p62 \[Omega]\[Omega])z2s^3+4(p41+p42 \[Omega]\[Omega])z2s^2+2(p21+p22 \[Omega]\[Omega])z2s/.\[Omega]\[Omega]->\[Omega]/.t->4/5]*)


(* ::Input:: *)
(*RootReduce[{-7Sqrt[p71+p72 \[Omega]\[Omega]],+5Sqrt[p51+p52 \[Omega]\[Omega]],-3Sqrt[p31+p32 \[Omega]\[Omega]],Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,7,1,-2}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]*)


(* ::Input:: *)
(*RootReduce[{-7Sqrt[p71+p72 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,7,7,-2}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]*)


(* ::Input:: *)
(*qq1=Factor[Resultant[(q0 zs-pp1-pp2 \[Omega]\[Omega])^2-\[Omega]\[Omega]^2(pp3+pp4 \[Omega]\[Omega]),t \[Omega]\[Omega]^2-(4-11 t+8 t^2),\[Omega]\[Omega]]][[-1]];*)


(* ::Input:: *)
(*qq2=Factor[Resultant[y-2(p21+p22 \[Omega]\[Omega])zs-4(p41+p42 \[Omega]\[Omega])zs^2-6(p61+p62 \[Omega]\[Omega])zs^3,t \[Omega]\[Omega]^2-(4-11 t+8 t^2),\[Omega]\[Omega]]][[-1]];*)


(* ::Input:: *)
(*qq3=Collect[Factor[Resultant[qq1,qq2,zs]][[5]],y]*)


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


(* ::Input:: *)
(*RootReduce[{-7Sqrt[p71+p72 \[Omega]\[Omega]],+5Sqrt[p51+p52 \[Omega]\[Omega]],-3Sqrt[p31+p32 \[Omega]\[Omega]],Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,7,1,-2}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]*)


(* ::Input:: *)
(*qq4=Factor[Resultant[y1^2-1(p11+p12 \[Omega]\[Omega])zs,t \[Omega]\[Omega]^2-(4-11 t+8 t^2),\[Omega]\[Omega]]][[-1]];*)


(* ::Input:: *)
(*qq4b=Factor[Resultant[qq4,qq1,zs]][[5]]*)


(* ::Input:: *)
(*RootReduce[Solve[%==0/.t->4/5]]*)


(* ::Input:: *)
(*qq5=Factor[Resultant[y3^2-9(p31+p32 \[Omega]\[Omega])zs^3,t \[Omega]\[Omega]^2-(4-11 t+8 t^2),\[Omega]\[Omega]]][[-1]];*)


(* ::Input:: *)
(*qq5b=Factor[Resultant[qq5,qq1,zs]][[6]]*)


(* ::Input:: *)
(*RootReduce[Solve[%==0/.t->4/5]]*)


(* ::Input:: *)
(*qq45a=Factor[Resultant[y13-(y1+y3),qq4b,y1]];*)


(* ::Input:: *)
(*qq45b=Factor[Resultant[qq45a,qq5b,y3]][[2]]*)


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


(* ::Input:: *)
(*RootReduce[{-3Sqrt[p31+p32 \[Omega]\[Omega]],Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,3,1,-2}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]*)


(* ::Input:: *)
(*RootReduce[{-3Sqrt[p31+p32 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,3,3,-2}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]*)


(* ::Input:: *)
(*RootReduce[{Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,1,1,-2}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]*)


(* ::Input:: *)
(*RootReduce[zs/.Solve[qq4==0/.{t->4/5,y1->Root[-46587212248392566935943088757112056944 - 9053298431907805277210427978515625000000 # + 564493410196292603574693202972412109375000 #^2 + 53487501361360045848414301872253417968750000 #^3 + 542101086242752217003726400434970855712890625 #^4& , 2, 0]}]]*)


(* ::Input:: *)
(*RootReduce[zs/.Solve[qq5==0/.{t->4/5,y3->Root[-698829018774823681853138957878220267979748100919 - 423234698843559591441610683213065660156250000000000 # - 15132057181764819782021162223815917968750000000000000 #^2 + 109409727737680077552795410156250000000000000000000000 #^3 + 9094947017729282379150390625000000000000000000000000 #^4& , 4, 0]}]]*)


(* ::Input:: *)
(*Solve[qq45b[[1]]==0/.{t->4/5,zs->Root[11176506961 + 101644115456 # - 2975270928384 #^2 + 7145894445056 #^3 + 451240001536 #^4& , 4, 0]},Reals]*)


(* ::Input:: *)
(**)


(* ::Input:: *)
(*qq6=Factor[Resultant[y5^2-25(p51+p52 \[Omega]\[Omega])zs^5,t \[Omega]\[Omega]^2-(4-11 t+8 t^2),\[Omega]\[Omega]]][[-1]];*)


(* ::Input:: *)
(*qq6b=Factor[Resultant[qq6,qq1,zs]][[5]]*)


(* ::Input:: *)
(*RootReduce[Solve[%==0/.t->4/5]]*)


(* ::Input:: *)
(*qq7=Factor[Resultant[y7^2-49(p71+p72 \[Omega]\[Omega])zs^7,t \[Omega]\[Omega]^2-(4-11 t+8 t^2),\[Omega]\[Omega]]][[-1]];*)


(* ::Input:: *)
(*qq7b=Factor[Resultant[qq7,qq1,zs]][[6]]*)


(* ::Input:: *)
(*RootReduce[Solve[%==0/.t->4/5]]*)


(* ::Input:: *)
(*RootReduce[{5Sqrt[p51+p52 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,5,5,-2}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]*)


(* ::Input:: *)
(*RootReduce[{-7Sqrt[p71+p72 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,7,7,-2}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]*)


(* ::Input:: *)
(*RootReduce[Root[-2325056565274010875774964383918770540755079 - 19723395805168432074062820751669324218750000000000 # - 43314256243503330925614455986022949218750000000000000 #^2 + 970855382586643099784851074218750000000000000000000000 #^3 + 9094947017729282379150390625000000000000000000000000 #^4& , 4, 0]+Root[-20042318286305529576256608102417097094636439 - 221467518400310951414854459514378906250000000000 # + 33668193874905802730861375808715820312500000000000 #^2 + 203116586949676275253295898437500000000000000000000 #^3 + 909494701772928237915039062500000000000000000000 #^4& , 2, 0]]*)


(* ::Input:: *)
(*qq67a=Factor[Resultant[y57-(y5+y7),qq6b,y5]];*)


(* ::Input:: *)
(*qq67b=Factor[Resultant[qq67a,qq7b,y7]][[4]]*)


(* ::Input:: *)
(*RootReduce[Solve[%==0/.t->4/5]]*)


(* ::Input:: *)
(*qq8=Resultant[qq67b,qq45b/.{y13->y-y57},y57]*)


(* ::Input:: *)
(*qq8b=Factor[qq8]*)


(* ::Input:: *)
(*Solve[qq8b[[4]]==0/.t->4/5,Reals]*)


(* ::Input:: *)
(*qq8c=qq8b[[4]]*)


(* ::Input:: *)
(*Expand[qq3-qq8c]*)
