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Solving system of equations with Root


I have this system of equations. I want to express all variables in terms of x as you can see. x is a parameter.


Solve[{y (z - x) == x^6, x zg - z zg == x z^2 za zb - x z za zb zg, 
za zg - zb zg == z za zb - z za^2 zb, zb - zg == z za zb - z zb^2, 
za zb (z - zg) == zg (-zb + zg), za > 0, zb > 0, zg > 0, z > 0}, {y, za, zb, zg, z}, Reals]


However Mathematica doesn't know how to solve it. Solutions show me something called Root and I can't make sense out of any of the expressions I get. What I get is:


y -> ConditionalExpression[((
x^6)/(-x + 
Root[-x + 
2 x^2 + (1 - 3 x - x^2) #1 + (1 + 2 x - 4 x^2) #1^2 + (-1 + 
 5 x - x^2) #1^3 + (-1 + x - x^2) #1^4 + x^2 #1^5 &, 3]))

I also tried using reduce which gives me a set of solutions for y:


y == Root[

x^26 + (-x^18 + x^19 - x^20 + 5 x^21) #1 + (-x^12 + x^13 + 3 x^14 - 
   4 x^15 + 10 x^16) #1^2 + (x^6 - x^7 + 5 x^8 + 3 x^9 - 6 x^10 + 
   10 x^11) #1^3 + (1 - x + 3 x^3 + x^4 - 4 x^5 + 
   5 x^6) #1^4 + (-1 + x) #1^5 &, 1]

I've read this thread: How do I work with Root objects? but it doesn't solve my problem since I have this free parameter x that ruins everything. For example if I try ToRadicals or numerical value I don't get anything. How do I make sense of it? How can I proceed to get the full expression for y in terms of x without Root? I tried FindInstance but it doesn't work in my case since I have this free parameter x. If this helps I'm only interested in y, the rest of the parameters don't really matter. Any help is greatly appreciated.



Answer



From a comment, it seems maybe the OP would like to rationalize the expression:


Product[y - 
   Root[x^26 + (-x^18 + x^19 - x^20 + 5 x^21) #1 + (-x^12 + x^13 + 

         3 x^14 - 4 x^15 + 10 x^16) #1^2 + (x^6 - x^7 + 5 x^8 + 
         3 x^9 - 6 x^10 + 10 x^11) #1^3 + (1 - x + 3 x^3 + x^4 - 
         4 x^5 + 5 x^6) #1^4 + (-1 + x) #1^5 &, i], {i, 
   5}] // FullSimplify
(*
(1/(-1 + x))(x^26 + x^18 (-1 + x - x^2 + 5 x^3) y + 
  x^12 (-1 + x (1 + x (3 + 2 x (-2 + 5 x)))) y^2 + 
  x^6 (1 + x (-1 + x (5 + x (3 + 2 x (-3 + 5 x))))) y^3 + (1 - x + 
     3 x^3 + x^4 - 4 x^5 + 5 x^6) y^4 + (-1 + x) y^5)
*)


Set this equal to zero to represent the locus. Note that the product may or may not have introduced extraneous solutions.



Addendum


Silly me, this is equivalent to, with less work, the following:


First[
  Root[x^26 + (-x^18 + x^19 - x^20 + 5 x^21) #1 + (-x^12 + x^13 + 
        3 x^14 - 4 x^15 + 10 x^16) #1^2 + (x^6 - x^7 + 5 x^8 + 
        3 x^9 - 6 x^10 + 10 x^11) #1^3 + (1 - x + 3 x^3 + x^4 - 
        4 x^5 + 5 x^6) #1^4 + (-1 + x) #1^5 &, 1]

  ]@y
(*
x^26 + (-x^18 + x^19 - x^20 + 5 x^21) y + (-x^12 + x^13 + 3 x^14 - 
    4 x^15 + 10 x^16) y^2 + (x^6 - x^7 + 5 x^8 + 3 x^9 - 6 x^10 + 
    10 x^11) y^3 + (1 - x + 3 x^3 + x^4 - 4 x^5 + 5 x^6) y^4 + (-1 + 
    x) y^5
*)

There is still the general caveat about rationalizing leads potentially to extraneously solutions.


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