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

It is obvious(verified) that z2=-sqrt(z2s)∈(-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,...ω, etc are defined in terms of t)

ω=(Sqrt[((4-11 t+8 (t^2))/t)]);

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) ωω (2-3 t-8 (t^2)+20 (t^3)-14 (t^4)+4 (t^5)+(Sqrt[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)) ωω))))/(16 (((-1+t))^3) (t^2) (-1+5 t-6 (t^2)+(t^3)))

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));

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

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

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∈(t0,1))

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);

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);

RootReduce[{Root[qq45/.t->4/5,3],(rp1+rp2 ωω-Sqrt[(rp3+rp4 ωω)])/rqq/.ωω->ω/.t->4/5}]

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

ContourPlot[Evaluate[{y==1/q(q+{p61+p62 ωω,p41+p42 ωω,p21+p22 ωω,p01+p02 ωω}.Table[(-Sqrt[z2s])^j,{j,6,0,-2}])/.ωω->ω,y==1/q{-Sqrt[p71+p72 ωω],Sqrt[p51+p52 ωω],-Sqrt[p31+p32 ωω],Sqrt[p11+p12 ωω]}.Table[(-Sqrt[z2s])^j,{j,7,1,-2}]/.ωω->ω,y==(rp1+rp2 ω-Sqrt[(rp3+rp4 ω)])/rqq}],{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),.999},{y,0,1},WorkingPrecision->80,PlotPoints->80]

radical expression and the superset as a zeroset of a bivariate polynomial

ContourPlot[Evaluate[{qq45==0,y==(rp1+rp2 ω-Sqrt[(rp3+rp4 ω)])/rqq}],{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),.999},{y,0,1},WorkingPrecision->80,PlotPoints->80,ContourStyle->{Blue,Red}]

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.

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

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

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) ωω (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)))

z2mms=((-1+2 t) ωω ((Sqrt[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)) ωω))))/(16 (((-1+t))^3) (t^2) (-1+5 t-6 (t^2)+(t^3)))

Together[z2s-(z2ms+z2mms)]

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

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

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.

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

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

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);

particular check, then building up, step by step, the “radical/root“-tower

Root[qq45/.t->4/5,3]

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

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

RootReduce[-Sqrt[z2s (p11+p12 ωω)]/q/.ωω->ω/.t->4/5]

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))-ωω(-1+2t)(2-3 t-8 (t^2)+20 (t^3)-14 (t^4)+4 (t^5)))^2-ωω^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)) ωω)))),t ωω^2-(4-11 t+8 (t^2)),ωω]][[5]]

RootReduce[z2s/.ωω->ω/.t->4/5]

Solve[rh1a==0/.t->4/5]

RootReduce[-Sqrt[z2s (p11+p12 ωω)]/q/.ωω->ω/.t->4/5]

rh1b=Factor[Resultant[(b1 q)^2-(z22(p11+p12 ωω)),t ωω^2-(4-11 t+8 (t^2)),ωω]][[3]]

rh1c=Factor[Resultant[rh1a,rh1b,z22]][[5]]

Solve[rh1c==0/.t->4/5]

rh2b=Factor[Resultant[(b3 q)^2-(z22^3(p31+p32 ωω)),t ωω^2-(4-11 t+8 (t^2)),ωω]][[3]]

rh2c=Factor[Resultant[rh1a,rh2b,z22]][[6]]

Solve[rh2c==0/.t->4/5]

RootReduce[Sqrt[z2s^3 (p31+p32 ωω)]/q/.ωω->ω/.t->4/5]

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

rh2d=Factor[Resultant[b1+b3-b13,rh1c,b1]];

rh2e=Factor[Resultant[rh2d,rh2c,b3]][[2]]

Solve[rh2e==0/.t->4/5]

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

rh3c=Factor[Resultant[rh1a,rh3b,z22]][[5]]

Solve[rh3c==0/.t->4/5]

RootReduce[-Sqrt[z2s^5 (p51+p52 ωω)]/q/.ωω->ω/.t->4/5]

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

rh4c=Factor[Resultant[rh1a,rh4b,z22]][[6]]

Solve[rh4c==0/.t->4/5]

RootReduce[Sqrt[z2s^7 (p71+p72 ωω)]/q/.ωω->ω/.t->4/5]

rh4d=Factor[Resultant[b5+b7-b57,rh3c,b5]];

rh4e=Factor[Resultant[rh4d,rh4c,b7]]

Length[rh4e]

rh4e[[5]]

Solve[rh4e[[5]]==0/.t->4/5]

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

rh5a=Factor[Resultant[b13+b57-y,rh2e,b13]];

rh5b=Factor[Resultant[rh5a,rh4e[[5]],b57]]

rh5b[[5]]

Solve[rh5b[[5]]==0/.t->4/5]

Expand[rh5b[[5]]-qq45]