(* ::Package:: *)

(* ::Subsubsection:: *)
(**)
(*Goal (Lemma 2.11): to verify that Z7(z2,t)=-1*)


(* ::Text:: *)
(**)
(*It is obvious(verified) that z2=-sqrt(z2s)\[Element](-1,1) *)
(*General remarks. *)
(*We will verify the equality in an equivalent form, namely that*)
(*(1+b0+b2z2^2+b4 z2^4+b6 2z^6)=-(b1z+b3z^3+b5z^5+b7 z^7).*)
(*We do not claim that this is the simplest way.*)
(**)
(*We give  a particular example first (assuming that z2s, b0,...\[Omega], etc are defined in terms of t)*)


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


(* ::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[1/q(q+{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.Table[(-Sqrt[z2s])^j,{j,6,0,-2}])/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}],RootReduce[1/q{-Sqrt[p71+p72 \[Omega]\[Omega]],Sqrt[p51+p52 \[Omega]\[Omega]],-Sqrt[p31+p32 \[Omega]\[Omega]],Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[z2s])^j,{j,7,1,-2}]/.\[Omega]\[Omega]->\[Omega]/.{t->4/5}]}*)


(* ::Text:: *)
(**)
(*Now we will show two canonical forms of the LHS and RHS*)
(*(a) with radicals*)
(*(b) as bivariate integral polynomial p(t,y) (labeled by qq45) whose zero set is a superset of the curve (t,LHS(t))=(t,RHS(t)) (t\[Element](t0,1)) *)


(* ::Input:: *)
(*qq45=-4096+148480 t-2750720 t^2+34749824 t^3-336526192 t^4+2659653476 t^5-17816570327 t^6+103669805970 t^7-532673407533 t^8+2444555432428 t^9-10101516992278 t^10+37808103046200 t^11-128736341031482 t^12+400097778377888 t^13-1137744898086151 t^14+2965536850326350 t^15-7093301577214869 t^16+15579473043083968 t^17-31424558991009676 t^18+58188795780620280 t^19-98832142780309764 t^20+153771297026767552 t^21-218772834488230896 t^22+283964518819782336 t^23-335334923634828768 t^24+359082661480751680 t^25-347305120815149760 t^26+302022806671910528 t^27-234884594325708288 t^28+162340823171777024 t^29-98977513449947392 t^30+52763299284114944 t^31-24330463229814016 t^32+9577133976019968 t^33-3164554413297664 t^34+858818914918400 t^35-185838351155200 t^36+30727222591488 t^37-3631831056384 t^38+272092889088 t^39-9663676416 t^40+(-512 t^3+36480 t^4-862368 t^5+10578472 t^6-71258236 t^7+129341869 t^8+2786256320 t^9-38165474712 t^10+295155126592 t^11-1688825202801 t^12+7745491543804 t^13-29539529842228 t^14+95567084428168 t^15-265211205189724 t^16+634568767617900 t^17-1308675104905300 t^18+2309673007928456 t^19-3422964145264132 t^20+4063930244975440 t^21-3328525537082832 t^22+362287730145472 t^23+5031272957570880 t^24-11916590513322944 t^25+18346637582050816 t^26-22202048974222720 t^27+22291811606586368 t^28-18958202256457728 t^29+13762753646345216 t^30-8537042285750272 t^31+4508930756182016 t^32-2012599688642560 t^33+750296622743552 t^34-229579662491648 t^35+56196285202432 t^36-10581868544000 t^37+1438010834944 t^38-125275471872 t^39+5234491392 t^40) y+(128 t^5+944 t^6-121024 t^7+2435295 t^8-25309618 t^9+150214572 t^10-338717930 t^11-2716061235 t^12+35586205432 t^13-228758901298 t^14+1038778158408 t^15-3641811626338 t^16+10209673867420 t^17-23229240646864 t^18+43067989979784 t^19-64999863846764 t^20+80441318627968 t^21-87334674313120 t^22+105688325071488 t^23-180692976848352 t^24+356158390490240 t^25-627743179812288 t^26+917571987183616 t^27-1106630331519232 t^28+1109792927015936 t^29-931896661092352 t^30+657693440925696 t^31-390390706241536 t^32+194478675722240 t^33-80920576049152 t^34+27904990773248 t^35-7880314781696 t^36+1787806613504 t^37-315265908736 t^38+40653291520 t^39-3388997632 t^40+134217728 t^41) y^2+(8 t^8-202 t^9-273 t^10+87928 t^11-1942927 t^12+24965964 t^13-227100424 t^14+1579091256 t^15-8743747008 t^16+39530964420 t^17-148289139060 t^18+466280733784 t^19-1235973648972 t^20+2765663780976 t^21-5205285494432 t^22+8148562651392 t^23-10336766132480 t^24+9948814128128 t^25-5692194596864 t^26-1905393430528 t^27+10212250810368 t^28-15862040682496 t^29+16889421414400 t^30-13848250941440 t^31+9035515887616 t^32-4720780378112 t^33+1959608975360 t^34-633610960896 t^35+154131496960 t^36-26576158720 t^37+2900361216 t^38-150994944 t^39) y^3+(-t^10+50 t^11-1203 t^12+18544 t^13-205715 t^14+1748874 t^15-11847485 t^16+65652532 t^17-303151596 t^18+1182149760 t^19-3931550832 t^20+11232064320 t^21-27706743616 t^22+59214199808 t^23-109855142400 t^24+177017808896 t^25-247564564480 t^26+299858362368 t^27-313427288064 t^28+281219727360 t^29-215000383488 t^30+138664345600 t^31-74425892864 t^32+32629850112 t^33-11380981760 t^34+3036676096 t^35-581959680 t^36+71303168 t^37-4194304 t^38) y^4;*)


(* ::Input:: *)
(*rp1=8+54 t-2385 t^2+21528 t^3-100271 t^4+284780 t^5-512936 t^6+546616 t^7-185728 t^8-398908 t^9+761100 t^10-684968 t^11+377396 t^12-129808 t^13+25824 t^14-2304 t^15;*)
(*rp2=(16-8 t-1275 t^2+9302 t^3-31095 t^4+53980 t^5-28974 t^6-80664 t^7+217764 t^8-264956 t^9+189108 t^10-76880 t^11+11764 t^12+3552 t^13-1888 t^14+256 t^15);*)
(*rq=4 (-1+t)^6 t^2 (-1+2 t)^6;*)
(*rp3=2(12960-292016 t+2854698 t^2-14792140 t^3+32588991 t^4+85143228 t^5-925203078 t^6+3299368184 t^7-4942483971 t^8-7668657248 t^9+62474910910 t^10-170963084656 t^11+246192050150 t^12-34672143568 t^13-789233922456 t^14+2311627705560 t^15-4090792828656 t^16+5282309403760 t^17-5223088300728 t^18+3967261586528 t^19-2228596974256 t^20+799166250432 t^21-35198764320 t^22-180928743808 t^23+144029257712 t^24-66339007104 t^25+20749249664 t^26-4414670336 t^27+584887296 t^28-34091008 t^29-1572864 t^30+262144 t^31);*)
(*rp4=2(6624 t-161376 t^2+1695878 t^3-9509785 t^4+24259520 t^5+42626172 t^6-628269302 t^7+2609963815 t^8-5418910112 t^9+384019718 t^10+38614387936 t^11-152399515228 t^12+362607645744 t^13-617651612344 t^14+779178270248 t^15-703631968400 t^16+373394610624 t^17+57362354376 t^18-372214821664 t^19+456997541776 t^20-358199885632 t^21+201881079680 t^22-82461667616 t^23+22845169104 t^24-3148220352 t^25-456967680 t^26+354630144 t^27-87122944 t^28+10960896 t^29-589824 t^30);*)
(*rqq=4 (-1+t)^6 t^2 (-1+2 t)^6;*)
(**)


(* ::Input:: *)
(*RootReduce[{Root[qq45/.t->4/5,3],(rp1+rp2 \[Omega]\[Omega]-Sqrt[(rp3+rp4 \[Omega]\[Omega])])/rqq/.\[Omega]\[Omega]->\[Omega]/.t->4/5}]*)


(* ::Text:: *)
(*the three expressions are the same (lhs, rhs, radical expression)*)


(* ::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[z2s])^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[z2s])^j,{j,7,1,-2}]/.\[Omega]\[Omega]->\[Omega],y==(rp1+rp2 \[Omega]-Sqrt[(rp3+rp4 \[Omega])])/rqq}],{t,Root[-1 + 12 # - 20 #^2 + 8 #^3& , 2, 0],.999},{y,0,1},WorkingPrecision->80,PlotPoints->80]*)


(* ::Text:: *)
(**)
(*radical expression and the superset as a zeroset of a bivariate polynomial*)


(* ::Input:: *)
(*ContourPlot[Evaluate[{qq45==0,y==(rp1+rp2 \[Omega]-Sqrt[(rp3+rp4 \[Omega])])/rqq}],{t,Root[-1 + 12 # - 20 #^2 + 8 #^3& , 2, 0],.999},{y,0,1},WorkingPrecision->80,PlotPoints->80,ContourStyle->{Blue,Red}]*)


(* ::Text:: *)
(**)
(*Now the formal verification consist of showing that starting from the LHS/RHS we can derive the same radical expression/bivariate polynomial.*)
(*It seems that showing everything "on the polynomial" level turns out to be the standard way. If the LHS-RHS polynomials coincide, then one can show that zero set of qq45 consist of 4 separate continuous curves above (t0,1) and therefore testing one point (t=4/5) for identifying the radical expression (the 3rd curve part) is sufficient, QED.*)


(* ::Text:: *)
(**)
(*To derive from LHS the radical expression, we take z2s = (z2)^2 as a sum.  *)


(* ::Text:: *)
(**)
(*z2s as z2s=z2ms+z2mms  for the derivation (binomial theorem)*)


(* ::Input:: *)
(*z2ms=(-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+0))/(16 (-1+t)^3 t^2 (-1+5 t-6 t^2+t^3))*)


(* ::Input:: *)
(*z2mms=((-1+2 t) \[Omega]\[Omega] (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:: *)
(*Together[z2s-(z2ms+z2mms)]*)


(* ::Input:: *)
(*Together[(rp1+rp2 \[Omega]\[Omega])/rq-(1+1/q{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.{z2ms^3+3z2ms z2mms^2,z2ms^2+z2mms^2,z2ms,1})/.\[Omega]\[Omega]->\[Omega]]*)


(* ::Input:: *)
(*Together[(rp3+rp4 \[Omega]\[Omega])/rq^2-(1/q{p61+p62 \[Omega]\[Omega],p41+p42 \[Omega]\[Omega],p21+p22 \[Omega]\[Omega],p01+p02 \[Omega]\[Omega]}.{z2mms^2 z2mms+3z2ms^2 z2mms,2z2ms z2mms,z2mms,0})^2/.\[Omega]\[Omega]->\[Omega]]*)


(* ::Text:: *)
(**)
(*The above computation verifies that LHS and the canonical radical expression are the same. One can also derive from the canonical radical expression  the bivariate polynomial.*)


(* ::Text:: *)
(**)
(*Now e.g. to derive from RHS the bivariate integral polynomial is a lengthy, tedious, resultant computation (see below), but it works!*)


(* ::Section:: *)
(**)
(*Auxiliary goal: rhs has the same bivariate polynomial as lhs above (qq45) *)


(* ::Input:: *)
(*qq45=-4096+148480 t-2750720 t^2+34749824 t^3-336526192 t^4+2659653476 t^5-17816570327 t^6+103669805970 t^7-532673407533 t^8+2444555432428 t^9-10101516992278 t^10+37808103046200 t^11-128736341031482 t^12+400097778377888 t^13-1137744898086151 t^14+2965536850326350 t^15-7093301577214869 t^16+15579473043083968 t^17-31424558991009676 t^18+58188795780620280 t^19-98832142780309764 t^20+153771297026767552 t^21-218772834488230896 t^22+283964518819782336 t^23-335334923634828768 t^24+359082661480751680 t^25-347305120815149760 t^26+302022806671910528 t^27-234884594325708288 t^28+162340823171777024 t^29-98977513449947392 t^30+52763299284114944 t^31-24330463229814016 t^32+9577133976019968 t^33-3164554413297664 t^34+858818914918400 t^35-185838351155200 t^36+30727222591488 t^37-3631831056384 t^38+272092889088 t^39-9663676416 t^40+(-512 t^3+36480 t^4-862368 t^5+10578472 t^6-71258236 t^7+129341869 t^8+2786256320 t^9-38165474712 t^10+295155126592 t^11-1688825202801 t^12+7745491543804 t^13-29539529842228 t^14+95567084428168 t^15-265211205189724 t^16+634568767617900 t^17-1308675104905300 t^18+2309673007928456 t^19-3422964145264132 t^20+4063930244975440 t^21-3328525537082832 t^22+362287730145472 t^23+5031272957570880 t^24-11916590513322944 t^25+18346637582050816 t^26-22202048974222720 t^27+22291811606586368 t^28-18958202256457728 t^29+13762753646345216 t^30-8537042285750272 t^31+4508930756182016 t^32-2012599688642560 t^33+750296622743552 t^34-229579662491648 t^35+56196285202432 t^36-10581868544000 t^37+1438010834944 t^38-125275471872 t^39+5234491392 t^40) y+(128 t^5+944 t^6-121024 t^7+2435295 t^8-25309618 t^9+150214572 t^10-338717930 t^11-2716061235 t^12+35586205432 t^13-228758901298 t^14+1038778158408 t^15-3641811626338 t^16+10209673867420 t^17-23229240646864 t^18+43067989979784 t^19-64999863846764 t^20+80441318627968 t^21-87334674313120 t^22+105688325071488 t^23-180692976848352 t^24+356158390490240 t^25-627743179812288 t^26+917571987183616 t^27-1106630331519232 t^28+1109792927015936 t^29-931896661092352 t^30+657693440925696 t^31-390390706241536 t^32+194478675722240 t^33-80920576049152 t^34+27904990773248 t^35-7880314781696 t^36+1787806613504 t^37-315265908736 t^38+40653291520 t^39-3388997632 t^40+134217728 t^41) y^2+(8 t^8-202 t^9-273 t^10+87928 t^11-1942927 t^12+24965964 t^13-227100424 t^14+1579091256 t^15-8743747008 t^16+39530964420 t^17-148289139060 t^18+466280733784 t^19-1235973648972 t^20+2765663780976 t^21-5205285494432 t^22+8148562651392 t^23-10336766132480 t^24+9948814128128 t^25-5692194596864 t^26-1905393430528 t^27+10212250810368 t^28-15862040682496 t^29+16889421414400 t^30-13848250941440 t^31+9035515887616 t^32-4720780378112 t^33+1959608975360 t^34-633610960896 t^35+154131496960 t^36-26576158720 t^37+2900361216 t^38-150994944 t^39) y^3+(-t^10+50 t^11-1203 t^12+18544 t^13-205715 t^14+1748874 t^15-11847485 t^16+65652532 t^17-303151596 t^18+1182149760 t^19-3931550832 t^20+11232064320 t^21-27706743616 t^22+59214199808 t^23-109855142400 t^24+177017808896 t^25-247564564480 t^26+299858362368 t^27-313427288064 t^28+281219727360 t^29-215000383488 t^30+138664345600 t^31-74425892864 t^32+32629850112 t^33-11380981760 t^34+3036676096 t^35-581959680 t^36+71303168 t^37-4194304 t^38) y^4;*)


(* ::Text:: *)
(**)
(*particular check, then building up, step by step, the "radical/root"-tower*)


(* ::Input:: *)
(*Root[qq45/.t->4/5,3]*)


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


(* ::Input:: *)
(*RootReduce[1/q{0,0,0,Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[Z])^j,{j,7,1,-2}]/.Z->z2s/.\[Omega]\[Omega]->\[Omega]/.{{t->4/5}}]*)


(* ::Input:: *)
(*RootReduce[-Sqrt[z2s (p11+p12 \[Omega]\[Omega])]/q/.\[Omega]\[Omega]->\[Omega]/.t->4/5]*)


(* ::Input:: *)
(*rh1a=Factor[Resultant[(((16 (-1+t)^3 t^2 (-1+5 t-6 t^2+t^3)))z22-(-6+25 t+14 t^2-266 t^3+648 t^4-752 t^5+464 t^6-144 t^7+16 t^8)-\[Omega]\[Omega](-1+2t)(2-3 t-8 t^2+20 t^3-14 t^4+4 t^5))^2-\[Omega]\[Omega]^2(-1+2t)^2(2((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])))),t \[Omega]\[Omega]^2-(4-11 t+8 t^2),\[Omega]\[Omega]]][[5]]*)


(* ::Input:: *)
(*RootReduce[z2s/.\[Omega]\[Omega]->\[Omega]/.t->4/5]*)


(* ::Input:: *)
(*Solve[rh1a==0/.t->4/5]*)


(* ::Input:: *)
(*RootReduce[-Sqrt[z2s (p11+p12 \[Omega]\[Omega])]/q/.\[Omega]\[Omega]->\[Omega]/.t->4/5]*)


(* ::Input:: *)
(*rh1b=Factor[Resultant[(b1 q)^2-(z22(p11+p12 \[Omega]\[Omega])),t \[Omega]\[Omega]^2-(4-11 t+8 t^2),\[Omega]\[Omega]]][[3]]*)


(* ::Input:: *)
(*rh1c=Factor[Resultant[rh1a,rh1b,z22]][[5]]*)


(* ::Input:: *)
(*Solve[rh1c==0/.t->4/5]*)


(* ::Input:: *)
(*rh2b=Factor[Resultant[(b3 q)^2-(z22^3(p31+p32 \[Omega]\[Omega])),t \[Omega]\[Omega]^2-(4-11 t+8 t^2),\[Omega]\[Omega]]][[3]]*)


(* ::Input:: *)
(*rh2c=Factor[Resultant[rh1a,rh2b,z22]][[6]]*)


(* ::Input:: *)
(*Solve[rh2c==0/.t->4/5]*)


(* ::Input:: *)
(*RootReduce[Sqrt[z2s^3 (p31+p32 \[Omega]\[Omega])]/q/.\[Omega]\[Omega]->\[Omega]/.t->4/5]*)


(* ::Input:: *)
(*RootReduce[1/q{0,0,-Sqrt[p31+p32 \[Omega]\[Omega]],Sqrt[p11+p12 \[Omega]\[Omega]]}.Table[(-Sqrt[Z])^j,{j,7,1,-2}]/.Z->z2s/.\[Omega]\[Omega]->\[Omega]/.{{t->4/5}}]*)


(* ::Input:: *)
(*rh2d=Factor[Resultant[b1+b3-b13,rh1c,b1]];*)


(* ::Input:: *)
(*rh2e=Factor[Resultant[rh2d,rh2c,b3]][[2]]*)


(* ::Input:: *)
(*Solve[rh2e==0/.t->4/5]*)


(* ::Input:: *)
(*rh3b=Factor[Resultant[(b5 q)^2-(z22^5(p51+p52 \[Omega]\[Omega])),t \[Omega]\[Omega]^2-(4-11 t+8 t^2),\[Omega]\[Omega]]][[3]];*)


(* ::Input:: *)
(*rh3c=Factor[Resultant[rh1a,rh3b,z22]][[5]]*)


(* ::Input:: *)
(*Solve[rh3c==0/.t->4/5]*)


(* ::Input:: *)
(*RootReduce[-Sqrt[z2s^5 (p51+p52 \[Omega]\[Omega])]/q/.\[Omega]\[Omega]->\[Omega]/.t->4/5]*)


(* ::Input:: *)
(*rh4b=Factor[Resultant[(b7 q)^2-(z22^7(p71+p72 \[Omega]\[Omega])),t \[Omega]\[Omega]^2-(4-11 t+8 t^2),\[Omega]\[Omega]]][[3]];*)


(* ::Input:: *)
(*rh4c=Factor[Resultant[rh1a,rh4b,z22]][[6]]*)


(* ::Input:: *)
(*Solve[rh4c==0/.t->4/5]*)


(* ::Input:: *)
(*RootReduce[Sqrt[z2s^7 (p71+p72 \[Omega]\[Omega])]/q/.\[Omega]\[Omega]->\[Omega]/.t->4/5]*)


(* ::Input:: *)
(*rh4d=Factor[Resultant[b5+b7-b57,rh3c,b5]];*)


(* ::Input:: *)
(*rh4e=Factor[Resultant[rh4d,rh4c,b7]]*)


(* ::Input:: *)
(*Length[rh4e]*)


(* ::Input:: *)
(*rh4e[[5]]*)


(* ::Input:: *)
(*Solve[rh4e[[5]]==0/.t->4/5]*)


(* ::Input:: *)
(*RootReduce[1/q{-Sqrt[p71+p72 \[Omega]\[Omega]],Sqrt[p51+p52 \[Omega]\[Omega]],0,0}.Table[(-Sqrt[Z])^j,{j,7,1,-2}]/.Z->z2s/.\[Omega]\[Omega]->\[Omega]/.{{t->4/5}}]*)


(* ::Input:: *)
(*rh5a=Factor[Resultant[b13+b57-y,rh2e,b13]];*)


(* ::Input:: *)
(*rh5b=Factor[Resultant[rh5a,rh4e[[5]],b57]]*)


(* ::Input:: *)
(*rh5b[[5]]*)


(* ::Input:: *)
(*Solve[rh5b[[5]]==0/.t->4/5]*)


(* ::Input:: *)
(*Expand[rh5b[[5]]-qq45]*)
