"Given this set of three polynoinal equations in three real unknowns x, y, z ~
and a"

"constant VAL in the range of 0 to 1..."

x-y+z=v

4*x^3-3*x-(4*y^3-3*y)+4*z^3-3*z=0

16*x^5-20*x^3+5*x-(16*y^5-20*y^3+5*y)+16*z^5-20*z^3+5*z=0

"Am I correct in my premise that there is ONE and only one real solution for ~
a given"

"VAL?"

"Is this provable?"

"Does a closed form non-iterative expression exist to exactly determine x,y,z~
 given"

"VAL?"

"What is x,y,z for VAL of, say 0.534?"

;Simp(Solve(#3,z))
z=-x+y+v

[4*x^3-3*x-(4*y^3-3*y)+4*z^3-3*z=0,16*x^5-20*x^3+5*x-(16*y^5-20*y^3+5*y)+16*z~
^5-20*z^3+5*z=0]

;Simp(Sub(#13))
[12*x^2*(y+v)-12*x*(y^2+2*v*y+v^2)+12*v*y^2+12*v^2*y+v*(4*v^2-3)=0,80*x^4*(y+~
v)-160*x^3*(y^2+2*v*y+v^2)+20*x^2*(y+v)*(8*y^2+16*v*y+8*v^2-3)-20*x*(4*y^4+16~
*v*y^3+3*y^2*(8*v^2-1)+2*v*y*(8*v^2-3)+4*v^4-3*v^2)+v*(80*y^4+160*v*y^3+20*y^~
2*(8*v^2-3)+20*v*y*(4*v^2-3)+16*v^4-20*v^2+5)=0]

;Simp(Solve(#14',y))
y=SQRT(3)*(SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))-SQRT(3)*(x^2-2*v*x~
+v^2))/(6*(v-x)) OR y=SQRT(3)*(SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3)~
)+SQRT(3)*(x^2-2*v*x+v^2))/(6*(x-v))

;Fctr(User)
[0=0,-v*(60*x^4*(4*v^2-3)+120*v*x^3*(3-4*v^2)+6*x^2*(56*v^4-60*v^2+15)-v*x*(1~
12*v^4-240*v^2+135)+v^2*(16*v^4-60*v^2+45))/(9*(x-v)^2)=0]

;Expd(User)
-5*v^2*(16*v^4-24*v^2+9)/(9*(x-v))+20*v*x^2*(3-4*v^2)/3-2*v*(16*v^4-30*v^2+15~
)/3=0

;Simp(User)
60*x^3*(3-4*v^2)+60*v*x^2*(4*v^2-3)-6*x*(16*v^4-30*v^2+15)+v*(16*v^4-60*v^2+4~
5)=0

[x-y+z=v,4*x^3-3*x-(4*y^3-3*y)+4*z^3-3*z=0,16*x^5-20*x^3+5*x-(16*y^5-20*y^3+5~
*y)+16*z^5-20*z^3+5*z=0]

;Solve(User,[x,y,z])
SOLVE([x-y+z=0.534,4*x^3-3*x-(4*y^3-3*y)+4*z^3-3*z=0,16*x^5-20*x^3+5*x-(16*y^~
5-20*y^3+5*y)+16*z^5-20*z^3+5*z=0],[x,y,z])

SOLVE([x-y+z=v,4*x^3-3*x-(4*y^3-3*y)+4*z^3-3*z=0,16*x^5-20*x^3+5*x-(16*y^5-20~
*y^3+5*y)+16*z^5-20*z^3+5*z=0],[x,y,z])

;Expd(User')
[0=0,80*SQRT(3)*x^4*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(3*(v-x))~
+40*SQRT(3)*x^4*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(3*(x-v))+80*~
SQRT(3)*v*x^3*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(3*(v-x))+80*SQ~
RT(3)*x^5*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(3*(x-v)^2)-40*SQRT~
(3)*x^5*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(3*(x-v)^2)+40*SQRT(3~
)*v*x^4*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(3*(x-v)^2)-80*SQRT(3~
)*v^2*x^3*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(3*(x-v)^2)+20*SQRT~
(3)*v*x^2*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(3*(x-v)^2)+10*SQRT~
(3)*v*x^2*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(3*(x-v)^2)-10*SQRT~
(3)*v*x^2*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(x-v)^2-40*x^6/(x-v~
)-80*v*x^5/(x-v)+40*v^2*x^4/(x-v)+80*v^3*x^3/(3*(x-v))+40*v*x^3/(x-v)+40*x^7/~
(x-v)^2+40*v*x^6/(x-v)^2-120*v^2*x^5/(x-v)^2-40*v^3*x^4/(3*(x-v)^2)-20*v*x^4/~
(x-v)^2+80*v^4*x^3/(x-v)^2-112*v^5*x^2/(3*(x-v)^2)+40*v^3*x^2/(x-v)^2-10*v*x^~
2/(x-v)^2+112*v^6*x/(9*(x-v)^2)-80*v^4*x/(3*(x-v)^2)+15*v^2*x/(x-v)^2-16*v^7/~
(9*(x-v)^2)+20*v^5/(3*(x-v)^2)-5*v^3/(x-v)^2=0]

;Expd(#22')
[0=0,-10*SQRT(3)*v*x^2*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(x-v)^~
2+10*SQRT(3)*v^3*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(x-v)^2+20*S~
QRT(3)*v^2*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(x-v)+10*SQRT(3)*v~
*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))+20*v*x^4*(3-4*v^2)/(3*(x-v)^~
2)+40*v^2*x^3*(4*v^2-3)/(3*(x-v)^2)-2*v*x^2*(56*v^4-60*v^2+15)/(3*(x-v)^2)+v^~
2*x*(112*v^4-240*v^2+135)/(9*(x-v)^2)-v^3*(16*v^4-60*v^2+45)/(9*(x-v)^2)=0]

-10*SQRT(3)*v*x^2*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(x-v)^2+10*~
SQRT(3)*v^3*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(x-v)^2+20*SQRT(3~
)*v^2*SQRT(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))/(x-v)+10*SQRT(3)*v*SQRT~
(3*x^4-6*v^2*x^2+v*x*(4*v^2-3)-v^2*(v^2-3))+20*v*x^4*(3-4*v^2)/(3*(x-v)^2)+40~
*v^2*x^3*(4*v^2-3)/(3*(x-v)^2)-2*v*x^2*(56*v^4-60*v^2+15)/(3*(x-v)^2)+v^2*x*(~
112*v^4-240*v^2+135)/(9*(x-v)^2)-v^3*(16*v^4-60*v^2+45)/(9*(x-v)^2)=0

;Simp(User)
60*x^3*(4*v^2-3)+60*v*x^2*(3-4*v^2)+6*x*(16*v^4-30*v^2+15)-v*(16*v^4-60*v^2+4~
5)=0