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Proposition 5.1.

F1:=c0 x^2+c2 y^2+d0 x^3+d1 x^2 y+d2 x y^2+d3 y^3+e0 x^4+e1 x^3 y+e2 x^2 y^2+e3 x y^3+e4 y^4

F2:=g11 x^2+g22 y^2+g33 z^2+2 g31 x z+2 g32 y z+2(x^2+y^2+z^2)(b1 x+b2 y+ b3 z)+a(x^2+y^2+z^2)^2

Coefficient[F2/.{z->F1},x^2]/.{x->0,y->0}

Coefficient[F2/.{z->F1},y^2]/.{x->0,y->0}

Coefficient[F2/.{z->F1},x^3]/.{x->0,y->0}

Coefficient[F2/.{z->F1},x^2 y]/.{x->0,y->0}

Coefficient[F2/.{z->F1},x y^2]/.{x->0,y->0}

Coefficient[F2/.{z->F1},y^3]/.{x->0,y->0}

Coefficient[F2/.{z->F1},x^4]/.{x->0,y->0}

Coefficient[F2/.{z->F1},x^3 y]/.{x->0,y->0}

Coefficient[F2/.{z->F1},x^2 y^2]/.{x->0,y->0}

Coefficient[F2/.{z->F1},x y^3]/.{x->0,y->0}

Coefficient[F2/.{z->F1},y^4]/.{x->0,y->0}

Solve[{d0==2 b1+2 c0 g31,d2==(1/2) (4 b1+4 c2 g31)},{g31,b1}]

Solve[{d1==(1/2) (4 b2+4 c0 g32),d3==2 b2+2 c2 g32},{g32,b2}]

e1 is

Simplify[(1/6) (12 d1 g31+12 d0 g32)/.{g31->-((-d0+d2)/(2 (c0-c2))),g32->-((-d1+d3)/(2 (c0-c2)))}]

e3 is

Simplify[(1/6) (12 d3 g31+12 d2 g32)/.{g31->-((-d0+d2)/(2 (c0-c2))),g32->-((-d1+d3)/(2 (c0-c2)))}]

Solve[{e0==a+2 b3 c0+2 d0 g31+(c0^2) g33,e2==(1/4) (8 a+8 b3 c0+8 b3 c2+8 d2 g31+8 d1 g32+8 c0 c2 g33),e4==a+2 b3 c2+2 d3 g32+(c2^2) g33}/.{g31->-((-d0+d2)/(2 (c0-c2))),g32->-((-d1+d3)/(2 (c0-c2)))},{g33,b3,a}]

Proposition 5.2.

P[T_]:=-(d0^2)+(-4 (c0^4)+4 (c0^3) c2-6 (d0^2)+(d1^2)+2 d0 d2+4 c0 e0-4 c2 e0) (T^2)+(-8 d0 d1+4 d1 d2+4 d0 d3+4 c0 e1-4 c2 e1) (T^3)+(-12 (c0^3) c2+12 (c0^2) (c2^2)-5 (d0^2)-2 (d1^2)-4 d0 d2+3 (d2^2)+6 d1 d3+4 c0 e0-4 c2 e0+4 c0 e2-4 c2 e2) (T^4)+(-8 d0 d1+8 d2 d3+4 c0 e1-4 c2 e1+4 c0 e3-4 c2 e3) (T^5)+(-12 (c0^2) (c2^2)+12 c0 (c2^3)-3 (d1^2)-6 d0 d2+2 (d2^2)+4 d1 d3+5 (d3^2)+4 c0 e2-4 c2 e2+4 c0 e4-4 c2 e4) (T^6)+(-4 d1 d2-4 d0 d3+8 d2 d3+4 c0 e3-4 c2 e3) (T^7)+(-4 c0 (c2^3)+4 (c2^4)-(d2^2)-2 d1 d3+6 (d3^2)+4 c0 e4-4 c2 e4) (T^8)+(d3^2) (T^10)

Collect[P[t]/.{e1->(2 d0 d1-d1 d2-d0 d3)/(c0-c2), e3->(2 d2 d3-d1 d2-d0 d3)/(c2-c0)},t, Simplify]

Example 5.3.

Solve[(x^2+y^2+z^2)^2-2 a x^2-2 b y^2-2 c z^2+d^2==0,z]

Series[(Sqrt[1+2t]),{t,0,2}]

Series[(Sqrt[1+t]),{t,0,2}]

F3[x_,y_]:=s2√(c-x^2-y^2+s1 L√(1+1/L^2(2(a-c)x^2+2(b-c)y^2)))

c0 is

Simplify[D[F3[x,y],{x,2}]/2/.{x->0,y->0}]

c2 is

Simplify[D[F3[x,y],{y,2}]/2/.{x->0,y->0}]

c0-c2 is

Simplify[-(((L+(-a+c) s1) s2)/(2 L (Sqrt[c+L s1])))-(-(((L+(-b+c) s1) s2)/(2 L (Sqrt[c+L s1]))))]

e0 is

Simplify[D[F3[x,y],{x,4}]/24/.{x->0,y->0,s1->1}]

Simplify[D[F3[x,y],{x,4}]/24/.{x->0,y->0,s1->-1}]

e2 is

Simplify[D[D[F3[x,y],{x,2}]/4,{y,2}]/.{x->0,y->0,s1->1}]

Simplify[D[D[F3[x,y],{x,2}]/4,{y,2}]/.{x->0,y->0,s1->-1}]

Expand[-2(a-c)(b-c)(c+L)-L(a-c-L)(b-c-L)+(a (2 b c-2 (c^2)+3 b L-3 c L-(L^2))+(c+L) (2 (c^2)+c L+(L^2)-b (2 c+L)))]

e4 is

Simplify[D[F3[x,y],{y,4}]/24/.{x->0,y->0,s1->1}]

Simplify[D[F3[x,y],{y,4}]/24/.{x->0,y->0,s1->-1}]

Q[t_]:=(t^2+1)(e0+e2 t^2+e4 t^4)-(c0+c2 t^2)^3/.{c0->-(((L+(-a+c) s1) s2)/(2 L (Sqrt[c+L s1]))),c2->-(((L+(-b+c) s1) s2)/(2 L (Sqrt[c+L s1]))),e0->((-2 s1(((a-c))^2) (c+L s1)-L (((-a+c+L s1))^2)) s2)/(8 (L^3) (((c+L s1))^(3/2))),e2->(1/(4 (L^3) (((c+L s1))^(3/2))))(-2s1(a-c)(b-c)(c+s1 L)-L(a-c-L s1)(b-c-L s1)) s2,e4->((-2 s1 (((b-c))^2) (c+L s1)-L (((-b+c+L s1))^2)) s2)/(8 (L^3) (((c+L s1))^(3/2)))}

Factor[Q[t]/.{s1->1,s2->1,L->(Sqrt[c^2-d^2])}]

Factor[Q[t]/.{s1->-1,s2->1,L->(Sqrt[c^2-d^2])}]

Factor[Q[t]/.{s1->1,s2->-1,L->(Sqrt[c^2-d^2])}]

Factor[Q[t]/.{s1->-1,s2->-1,L->(Sqrt[c^2-d^2])}]

Proof of Theorem 2.10.

NT:=-5RT KT^2ET1(RT DT-2(b T+a)CT^2)+DT^3RT B1+2DT^2DT1 RT^2B2-DT^2RT^2KT ((3d3 T+d2)(5RT-(b^2+1)T^2)+(d1 T+3d0)(b^2+1))+DT^2B3+2(b T+a)DT DT1 RT CT B4+10(b T+a)DT DT2 RT^2KT CT^2+DT B5-4(b T+a)DT1 RT KT CT^3(5(b T+a) CT1+2b CT)+4(b T+a)CT^4KT(3d0 B6+d1 B7+d2 B8-3d3 T B9)+4CT^4B10

B1:=KT(42(b^2+1)RT-20(a^2+b^2+1))-4c2 RT(3a b RT-2(a^2+b^2+1-a^2b^2)T+2a b(a^2+1))+2(b^2+1)c1 RT(3RT+4a b T+4a^2+4)-4(b^2+1)RT RT1 c0

B2:=-(4(b^2+1)T+5a b)KT-((b^2+1)T^2+a^2+1)KT1+4(a^2+1)(b^2+1)CT-2((a^2+1)c2+(b^2+1)c0)RT

B3:=KT^2(-4a RT CT1+12(b^2+1)(b T+a)T CT+8a b(b T+a)CT+12b RT CT+2b RT(2c0+c1 T))+KT KT1(4a b T(b T+a)CT-12(b T+a)RT CT+b T RT(2c0+c1 T)+a T RT CT1)+KT(-4a(b^2+6)RT CT^2-(18(b^2+1)T(b T+a)+16a b(b T+a)-4b(a^2+1))RT CT CT1+8(a^2+1)(b^2+1)(b T+a)CT^2-8(b T+a)((a^2+1)c2+(b^2+1)c0)RT CT+4a b T(b c1-2a c2)RT CT+4a(b^2+1)(2c0+c1 T)RT CT-2a c2(2c0+c1 T)RT^2-2b c0 CT1 RT^2-12(b^2+1)(b T+a)T RT1 CT^2-8a b(b T+a)RT1 CT^2-4(b T+a)RT RT1 CT CT1+4b RT RT1 CT^2-4(b T+a)RT^2CT1^2-8c2(b T+a)RT^2CT+4b RT^2CT CT1)-2(b T+a)KT1 CT(2a b T RT1 CT+5RT1 RT CT+6RT^2CT1)-4(b T+a)KT2 RT CT^2(RT+a b T)-8(a^2+1)(b^2+1)(b T+a)RT1 CT^3+8(b T+a)((a^2+1)c2+(b^2+1)c0)RT RT1 CT^2-8(a^2+1)(b^2+1)(b T+a)RT CT^2CT1

B4:=2(3(b^2+1)T+5a b)KT CT+5RT KT CT1+2((b^2+1)T^2+a^2+1)KT1 CT-8(a^2+1)(b^2+1)CT^2+4RT CT((a^2+1)c2+(b^2+1)c0)

B5:=9CT^2KT^3+KT^2CT^2(6RT1 CT+4RT CT1+8b(b T+a)CT-8(b T+a)^2CT1)-8(b T+a)^2KT KT1 CT^3+KT CT^2(-8(b T+a)(b c1-2a c2)RT CT-8b(b T+a)RT CT1 CT+8b^2RT CT^2-4b(b T+a)RT CT CT1-48(b T+a)^2CT((a^2-1)c2+(b^2+1)c0)+24a c1(b T+a)RT1 CT-48(b T+a)CT(2a(c2-c0)-b c1)+12(b T+a)^2RT CT1^2-8b(b T+a)RT1 CT^2+8(b T+a)^2RT1 CT CT1+16c2(b T+a)^2RT CT-8b(b T+a)RT CT CT1)+8(b T+a)^2KT1 CT^3(RT1 CT+3RT CT1)+8(b T+a)^2RT CT^4KT2-16b(b T+a)RT CT^4((a^2-1)c2+(b^2+1)c0)-32(b T+a)^2RT CT^3CT1((a^2-1)c2+(b^2+1)c0)+16a(b T+a)RT RT1 c1 CT^3CT1+4a b RT RT1 c1 CT^4+8a(b^2+1)(b T+a)RT c1 CT^4-8b RT CT^4(2a(c2-c0)-b c1)-32(b T+a)RT CT^3CT1(2a(c2-c0)-b c1)

B6:=-(b^2+1)(b T+a)+2a

B7:=-(b^2+1)T(b T+a)+a RT1+2(a T+b)

B8:=-(a^2-3)(b T+a)+a T RT1-4a-4(b T+a)RT

B9:=(a^2-1)(b T+a)+2a+4(b T+a)RT

B10:=KT^2(2b CT-4(b T+a)CT1)-2(b T+a)CT KT KT1+KT(a RT1 c1 CT-4(b T+a)CT((a^2-1)c2+(b^2+1)c0)-4a(c2-c0)CT+2b c1 CT-2a(b T+a)^2c2(2c0+c1 T)-2b(b T+a)^2c0 CT1+2a b(b T+a)c1 CT+a(b T+a)^2c1 CT1+(1/2)(b T+a)RT1 c1 (2c0+c1 T)-2(b T+a)(c2-c0)(2c0+c1 T)+(b T+a)c1 CT1)+8b(b T+a)^2((a^2-1)c2+(b^2+1)c0)CT^2+16(b T+a)^3CT CT1((a^2-1)c2+(b^2+1)c0)-8a(b T+a)^2RT1 c1 CT CT1-2a b(b T+a)RT1 c1 CT^2-4a(b^2+1)(b T+a)^2c1 CT^2+4b(b T+a)CT^2(2a(c2-c0)-b c1)+16(b T+a)^2CT CT1(2a(c2-c0)-b c1)

AT1:={-(T^2+1)^2ET1+2T CT^2(2CT+c0-c2)}(4ET-T ET1)+4c0 CT^2{(T^2+1)ET1-4c2 T CT^2}

AT2:=ET1{-(T^2+1)^2ET1+2T CT^2(2CT+c0-c2)}+4c2 T CT^2{(T^2+1)ET1-4c2 T CT^2}

NTR:=20 (c0-c2)^2 T^2(AT1 x+AT2 y)

M:={{NT-NTR/.{KT->RT1 CT-RT CT1,KT1->2(b^2+1)CT-2c2 RT,KT2->2(b^2+1)CT1-2 c2 RT1}}/.{RT->(b^2+1)T^2+2 a b T+a^2+1,RT1->2(b^2+1)T+2 a b,RT2->2(b^2+1),CT->c0 +c1 T+ c2 T^2,CT1->c1+2 c2 T,CT2->2 c2,DT->d0+d1 T+d2 T^2+d3 T^3,DT1->d1+2 T d2+3 d3 T^2,DT2->2 d2 +6 d3 T,ET1->e1+2 e2 T+3 e3 T^2+4 e4 T^3,ET->e0+e1 T+e2 T^2+e3 T^3+e4 T^4}}/.{a->2 c0 x,b->2 c2 y, c1->0,d0->4 e0 x+e1 y,d1->3 e1 x+2 e2 y,d2->2 e2 x+3 e3 y,d3->e3 x+4 e4 y}

Simplify[M/.{x->0,y->0}]

Simplify[Coefficient[M,x,1]/.{y->0}]

Simplify[Coefficient[M,y,1]/.{x->0}]

For AT=AT1/(2(c0-c2))

q0:=e0-c0^3

q1:=e1

q2:=e2+e0-3 c0^2 c2

q3:=e3+e1

q4:=e4+e2-3 c0 c2^2

q5:=e3

q6:=e4-c2^3

ga01:=-q1/q0

ga11:=-q2/q0

ga21:=-q3/q0

ga31:=-q4/q0

ga41:=-q5/q0

ga51:=-q6/q0

ga02:=-(q1+q2 I+q3 I^2+q4 I^3+q5 I^4+q6 I^5)/(q0-q2+q4-q6+(q1-q3+q5)I)

ga12:=-((q2+q3 I+q4 I^2+q5 I^3+q6 I^4)/(q0-q2+q4-q6+(q1-q3+q5)I))

ga22:=-((q3+q4 I+q5 I^2+q6 I^3)/(q0-q2+q4-q6+(q1-q3+q5)I))

ga32:=-((q4+q5 I+q6 I^2)/(q0-q2+q4-q6+(q1-q3+q5)I))

ga42:=-((q5+q6 I)/(q0-q2+q4-q6+(q1-q3+q5)I))

ga52:=-(q6/(q0-q2+q4-q6+(q1-q3+q5)I))

ga03:=-(q1-q2 I+q3 I^2-q4 I^3+q5 I^4-q6 I^5)/(q0-q2+q4-q6-(q1-q3+q5)I)

ga13:=-((q2-q3 I+q4 I^2-q5 I^3+q6 I^4)/(q0-q2+q4-q6-(q1-q3+q5)I))

ga23:=-((q3-q4 I+q5 I^2-q6 I^3)/(q0-q2+q4-q6-(q1-q3+q5)I))

ga33:=-((q4-q5 I+q6 I^2)/(q0-q2+q4-q6-(q1-q3+q5)I))

ga43:=-((q5-q6 I)/(q0-q2+q4-q6-(q1-q3+q5)I))

ga53:=-(q6/(q0-q2+q4-q6-(q1-q3+q5)I))

A1:=(AT1/.{CT->c0 + c2 T^2,ET1->e1+2 e2 T+3 e3 T^2+4 e4 T^3,ET->e0+e1 T+e2 T^2+e3 T^3+e4 T^4})

A2:=(AT2/.{CT->c0 + c2 T^2,ET1->e1+2 e2 T+3 e3 T^2+4 e4 T^3,ET->e0+e1 T+e2 T^2+e3 T^3+e4 T^4})

b11:=(A1/.{T->0})/(2(c0-c2))

b12:=-(A1/.{T->I})/(4(c0-c2))

b13:=-(A1/.{T->-I})/(4(c0-c2))

b21:=(A2/.{T->0})/(2(c0-c2))

b22:=-(A2/.{T->I})/(4(c0-c2))

b23:=-(A2/.{T->-I})/(4(c0-c2))

BT1:=(A1/.{T->t})/(2(c0-c2)t(t^2+1))-b11/t-b12/(t-I)-b13/(t+I)

BT2:=(A2/.{T->t})/(2(c0-c2)t(t^2+1))-b21/t-b22/(t-I)-b23/(t+I)

Cancel[Simplify[BT1]]

Rem1:=PolynomialRemainder[(1/(2 (c0-c2)))(-64 (c0^4) (c2^2)+96 (c0^3) (c2^3)-64 (c0^2) (c2^4)+16 c0 (c2^5)+64 (c0^2) c2 e0-72 c0 (c2^2) e0+24 (c2^3) e0-3 (e1^2)+20 (c0^3) e2-36 (c0^2) c2 e2+36 c0 (c2^2) e2-12 (c2^3) e2-8 e0 e2+56 (c0^2) c2 e1 t-54 c0 (c2^2) e1 t+18 (c2^3) e1 t-4 e0 e1 t-8 e1 e2 t+18 (c0^3) e3 t-18 (c0^2) c2 e3 t+18 c0 (c2^2) e3 t-6 (c2^3) e3 t-12 e0 e3 t-96 (c0^3) (c2^3) (t^2)+64 (c0^2) (c2^4) (t^2)-16 c0 (c2^5) (t^2)+56 c0 (c2^2) e0 (t^2)-24 (c2^3) e0 (t^2)-3 (e1^2) (t^2)+48 (c0^2) c2 e2 (t^2)-36 c0 (c2^2) e2 (t^2)+12 (c2^3) e2 (t^2)-8 e0 e2 (t^2)-4 (e2^2) (t^2)-10 e1 e3 (t^2)+16 (c0^3) e4 (t^2)-16 e0 e4 (t^2)+46 c0 (c2^2) e1 (t^3)-18 (c2^3) e1 (t^3)-8 e1 e2 (t^3)+40 (c0^2) c2 e3 (t^3)-18 c0 (c2^2) e3 (t^3)+6 (c2^3) e3 (t^3)-12 e0 e3 (t^3)-8 e2 e3 (t^3)-12 e1 e4 (t^3)-64 (c0^2) (c2^4) (t^4)+16 c0 (c2^5) (t^4)+16 (c2^3) e0 (t^4)+36 c0 (c2^2) e2 (t^4)-12 (c2^3) e2 (t^4)-4 (e2^2) (t^4)-10 e1 e3 (t^4)-3 (e3^2) (t^4)+32 (c0^2) c2 e4 (t^4)-16 e0 e4 (t^4)-8 e2 e4 (t^4)+12 (c2^3) e1 (t^5)+26 c0 (c2^2) e3 (t^5)-6 (c2^3) e3 (t^5)-8 e2 e3 (t^5)-12 e1 e4 (t^5)-4 e3 e4 (t^5)-16 c0 (c2^5) (t^6)+8 (c2^3) e2 (t^6)-3 (e3^2) (t^6)+16 c0 (c2^2) e4 (t^6)-8 e2 e4 (t^6)+4 (c2^3) e3 (t^7)-4 e3 e4 (t^7)),q0+q1 t+q2 t^2+q3 t^3+q4 t^4+q5 t^5+q6 t^6,t]

Therefore g0 is

Simplify[Coefficient[Rem1,t,0]+b11 ga01+b12 ga02+b13 ga03]

g0:={((c2^3) (24 (e0^2)+(e1^2)-12 e0 e2)+e0 (e3^2)-24 (c0^2) c2 e0 ((c2^3)-e4)-24 (e0^2) e4-(e1^2) e4+12 e0 e2 e4+(c0^3) (12 (c2^3) e2-(e3^2)-12 e2 e4))/(2 (c0-c2) ((c2^3)-e4))}

g1 is

Simplify[Coefficient[Rem1,t,1]+b11 ga11+b12 ga12+b13 ga13]

g1:={(e1 (e3^2)+2 (c2^3) (21 e0 e1-4 e1 e2-7 e0 e3)-26 (c0^2) c2 e1 ((c2^3)-e4)+14 (c0^3) e3 ((c2^3)-e4)-42 e0 e1 e4+8 e1 e2 e4+14 e0 e3 e4)/(2 (c0-c2) ((c2^3)-e4))}

g2 is

Simplify[Coefficient[Rem1,t,2]+b11 ga21+b12 ga22+b13 ga23]

g2:={(e0 (e3^2)+e2 (e3^2)-32 c0 (c2^2) e0 ((c2^3)-e4)-19 (e1^2) e4-24 e0 e2 e4+8 (e2^2) e4+8 e1 e3 e4+16 (c0^3) ((c2^3)-e4) e4+16 e0 (e4^2)+(c2^3) (19 (e1^2)+24 e0 e2-8 (e2^2)-8 e1 e3-16 e0 e4)+(c0^2) c2 (8 (c2^3) e2-3 (e3^2)-8 e2 e4))/(2 (c0-c2) ((c2^3)-e4))}

g3 is

Simplify[Coefficient[Rem1,t,3]+b11 ga31+b12 ga32+b13 ga33]

g3:={(e1 (e3^2)+(e3^3)-36 c0 (c2^2) e1 ((c2^3)-e4)+12 (c0^2) c2 e3 ((c2^3)-e4)-22 e1 e2 e4-16 e0 e3 e4+14 e2 e3 e4+8 e1 (e4^2)+2 (c2^3) (11 e1 e2+8 e0 e3-7 e2 e3-4 e1 e4))/(2 (c0-c2) ((c2^3)-e4))}

g4 is

Simplify[Coefficient[Rem1,t,4]+b11 ga41+b12 ga42+b13 ga43]

g4:={(-8 (c2^6) e0-4 c0 (c2^5) e2+16 (c0^2) (c2^4) e4-4 (e2^2) e4-16 (c0^2) c2 (e4^2)-2 e4 (8 e1 e3-3 (e3^2)+4 e0 e4)+(c2^3) (4 (e2^2)+16 e1 e3-5 (e3^2)+16 e0 e4-12 e2 e4)+c0 (c2^2) (-3 (e3^2)+4 e2 e4)+e2 ((e3^2)+12 (e4^2)))/(2 (c0-c2) ((c2^3)-e4))}

g5 is

Simplify[Coefficient[Rem1,t,5]+b11 ga51+b12 ga52+b13 ga53]

g5:={(-10 (c2^6) e1-2 c0 (c2^5) e3+(e3^3)+2 c0 (c2^2) e3 e4-4 e2 e3 e4-10 e1 (e4^2)+6 e3 (e4^2)+(c2^3) (4 e2 e3+20 e1 e4-6 e3 e4))/(2 (c0-c2) ((c2^3)-e4))}

Cancel[Simplify[BT2]]

Rem2:=PolynomialRemainder[(1/(2 (c0-c2)))(20 (c0^2) c2 e1-18 c0 (c2^2) e1+6 (c2^3) e1-4 e1 e2+18 (c0^3) e3-54 (c0^2) c2 e3+54 c0 (c2^2) e3-18 (c2^3) e3-64 (c0^3) (c2^3) t+96 (c0^2) (c2^4) t-64 c0 (c2^5) t+16 (c2^6) t-(e1^2) t+40 (c0^2) c2 e2 t-36 c0 (c2^2) e2 t+12 (c2^3) e2 t-4 (e2^2) t-6 e1 e3 t+24 (c0^3) e4 t-72 (c0^2) c2 e4 t+72 c0 (c2^2) e4 t-24 (c2^3) e4 t+22 c0 (c2^2) e1 (t^2)-6 (c2^3) e1 (t^2)-4 e1 e2 (t^2)+60 (c0^2) c2 e3 (t^2)-54 c0 (c2^2) e3 (t^2)+18 (c2^3) e3 (t^2)-12 e2 e3 (t^2)-8 e1 e4 (t^2)-96 (c0^2) (c2^4) (t^3)+64 c0 (c2^5) (t^3)-16 (c2^6) (t^3)+44 c0 (c2^2) e2 (t^3)-12 (c2^3) e2 (t^3)-4 (e2^2) (t^3)-6 e1 e3 (t^3)-9 (e3^2) (t^3)+80 (c0^2) c2 e4 (t^3)-72 c0 (c2^2) e4 (t^3)+24 (c2^3) e4 (t^3)-16 e2 e4 (t^3)+8 (c2^3) e1 (t^4)+66 c0 (c2^2) e3 (t^4)-18 (c2^3) e3 (t^4)-12 e2 e3 (t^4)-8 e1 e4 (t^4)-24 e3 e4 (t^4)-64 c0 (c2^5) (t^5)+16 (c2^6) (t^5)+16 (c2^3) e2 (t^5)-9 (e3^2) (t^5)+88 c0 (c2^2) e4 (t^5)-24 (c2^3) e4 (t^5)-16 e2 e4 (t^5)-16 (e4^2) (t^5)+24 (c2^3) e3 (t^6)-24 e3 e4 (t^6)-16 (c2^6) (t^7)+32 (c2^3) e4 (t^7)-16 (e4^2) (t^7)),q0+q1 t+q2 t^2+q3 t^3+q4 t^4+q5 t^5+q6 t^6,t]

Therefore h0 is

Simplify[Coefficient[Rem2,t,0]+b21 ga01+b22 ga02+b23 ga03]

h0:={(2 (c0^5) c2 e1-2 (c0^2) c2 e0 e1-(e1^3)+4 e0 e1 e2+10 (c0^6) e3+(e0^2) (-6 e1+10 e3)+(c0^3) (6 e0 e1-4 e1 e2-20 e0 e3))/(2 (c0-c2) ((c0^3)-e0))}

h1 is

Simplify[Coefficient[Rem2,t,1]+b21 ga11+b22 ga12+b23 ga13]

h1:={(-16 (c0^4) (c2^2) e0+16 c0 (c2^2) (e0^2)+4 (c0^5) c2 e2-(e1^2) e2+(c0^2) c2 (3 (e1^2)-4 e0 e2)+e0 (-6 (e1^2)+4 (e2^2)+16 e1 e3)+8 (c0^6) e4+(e0^2) (-12 e2+8 e4)+(c0^3) (5 (e1^2)+12 e0 e2-4 (e2^2)-16 e1 e3-16 e0 e4))/(2 (c0-c2) ((c0^3)-e0))}

h2 is

Simplify[Coefficient[Rem2,t,2]+b21 ga21+b22 ga22+b23 ga23]

h2:={-(12 (c0^4) (c2^2) e1-12 c0 (c2^2) e0 e1+(e1^3)-36 (c0^5) c2 e3+36 (c0^2) c2 e0 e3+(e1^2) e3+2 e0 (4 e0-11 e2) e3+2 e0 e1 (7 e2-8 e4)-2 (c0^3) (7 e1 e2+4 e0 e3-11 e2 e3-8 e1 e4))/(2 (c0-c2) ((c0^3)-e0))}

h3 is

Simplify[Coefficient[Rem2,t,3]+b21 ga31+b22 ga32+b23 ga33]

h3:={(16 (c2^3) (e0^2)-8 (c0^4) (c2^2) e2-(e1^2) e2-8 e0 (e2^2)+c0 (c2^2) (3 (e1^2)+8 e0 e2)-8 e0 e1 e3+19 e0 (e3^2)+32 (c0^5) c2 e4-32 (c0^2) c2 e0 e4-16 (e0^2) e4-(e1^2) e4+24 e0 e2 e4+(c0^3) (-16 (c2^3) e0+8 (e2^2)+8 e1 e3-19 (e3^2)+16 e0 e4-24 e2 e4))/(2 (c0-c2) ((c0^3)-e0))}

h4 is

Simplify[Coefficient[Rem2,t,4]+b21 ga41+b22 ga42+b23 ga43]

h4:={-(-14 (c2^3) e0 e1-26 (c0^4) (c2^2) e3+26 c0 (c2^2) e0 e3+(e1^2) e3+8 e0 e2 e3+14 e0 e1 e4-42 e0 e3 e4+2 (c0^3) (7 (c2^3) e1-4 e2 e3-7 e1 e4+21 e3 e4))/(2 (c0-c2) ((c0^3)-e0))}

h5 is

Simplify[Coefficient[Rem2,t,5]+b21 ga51+b22 ga52+b23 ga53]

h5:={((c2^3) ((e1^2)+12 (-(c0^3)+e0) e2)+e0 (e3^2)+24 c0 (c2^2) ((c0^3)-e0) e4-(e1^2) e4-12 e0 e2 e4+24 e0 (e4^2)-(c0^3) ((e3^2)-12 e2 e4+24 (e4^2)))/(2 (c0-c2) ((c0^3)-e0))}

f1:=2(c0-c2)(c0^3-e0)(c2^3-e4)(5h0-g1)

f2:=2(c0-c2)(c0^3-e0)(c2^3-e4)(2h1-g2)

f3:=2(c0-c2)(c0^3-e0)(c2^3-e4)(h2-g3)

f4:=2(c0-c2)(c0^3-e0)(c2^3-e4)(h3-2g4)

f5:=2(c0-c2)(c0^3-e0)(c2^3-e4)(h4-5g5)

Collect[f1,{e1,e3},Simplify]

Collect[f2,{e1,e3},Simplify]

Collect[f3,{e1,e3},Simplify]

Collect[f4,{e1,e3},Simplify]

Collect[f5,{e1,e3},Simplify]

Collect[f2-f4,{e1,e3},Simplify]

Simplify[Cancel[(((c0^3)-e0)(3 (c0^2) c2-6 c0 (c2^2)+9 (c2^3)-e0+e2-7 e4))(((c0^3)-e0) (3 (c0^2) c2-e0-e2) (e3^2)-(e1^2) (9 (c0^3)-6 (c0^2) c2-7 e0+2 e2) ((c2^3)-e4))^2-(24 ((c0^3)-e0) e1 e3 ((c2^3)-e4))^2/.{e1->1,e3->√(((c2^3)-e4) (9 (c0^3)-6 (c0^2) c2+3 c0 (c2^2)-7 e0+e2-e4)/(((c0^3)-e0)(3 (c0^2) c2-6 c0 (c2^2)+9 (c2^3)-e0+e2-7 e4)))}]]

Collect[Expand[(-64 ((c2^3)-e4) (9 (c0^3)-6 (c0^2) c2+3 c0 (c2^2)-7 e0+e2-e4)+(((18 (c0^4) (c2^2)+3 (c2^3) (7 e0-2 e2)+5 e0 e2-(e2^2)+3 c0 (c2^2) (-5 e0+e2)-3 (c0^3) (12 (c2^3)+2 e2-7 e4)+3 (c0^2) c2 (6 (c2^3)+e2-5 e4)-16 e0 e4+5 e2 e4))^2))],e0, Simplify]

Simplify[Coefficient[(-64 ((c2^3)-e4) (9 (c0^3)-6 (c0^2) c2+3 c0 (c2^2)-7 e0+e2-e4)+(((18 (c0^4) (c2^2)+3 (c2^3) (7 e0-2 e2)+5 e0 e2-(e2^2)+3 c0 (c2^2) (-5 e0+e2)-3 (c0^3) (12 (c2^3)+2 e2-7 e4)+3 (c0^2) c2 (6 (c2^3)+e2-5 e4)-16 e0 e4+5 e2 e4))^2)),e0,2]]

Simplify[Coefficient[(-64 ((c2^3)-e4) (9 (c0^3)-6 (c0^2) c2+3 c0 (c2^2)-7 e0+e2-e4)+(((18 (c0^4) (c2^2)+3 (c2^3) (7 e0-2 e2)+5 e0 e2-(e2^2)+3 c0 (c2^2) (-5 e0+e2)-3 (c0^3) (12 (c2^3)+2 e2-7 e4)+3 (c0^2) c2 (6 (c2^3)+e2-5 e4)-16 e0 e4+5 e2 e4))^2)),e0,1]]

Simplify[Coefficient[(-64 ((c2^3)-e4) (9 (c0^3)-6 (c0^2) c2+3 c0 (c2^2)-7 e0+e2-e4)+(((18 (c0^4) (c2^2)+3 (c2^3) (7 e0-2 e2)+5 e0 e2-(e2^2)+3 c0 (c2^2) (-5 e0+e2)-3 (c0^3) (12 (c2^3)+2 e2-7 e4)+3 (c0^2) c2 (6 (c2^3)+e2-5 e4)-16 e0 e4+5 e2 e4))^2)),e0,0]]