# Rationalizing expressions with Roots

I'm looking for a way to rationalize a very ugly expression given by the code I pasted in. What do I mean by this? The expression is a solution to the zero locus of some polynomial. In my previous question Solving system of equations with Root I had a very simple expression and I can't generalize this solution to the new case. What I want to do: I want to get rid of the Roots and get an expression for zero locus of some polynomial. Let's say that $y=\textrm{my code}$ so,$y-\textrm{my code}=0$. I want to rationalize this expression to get rid of the roots and have algebraic expression given by the radicals looking like $a_1 y^1 + a_2 y^2 +...a^n y^n =0$, where $a_i$ are some coefficients dependent on x and $y^i$ is $i$-th power of y. It would be nice to have a solution that works for wider classes of those equations, not only the one I pasted in. How can one do that? Is there a function in Mathematica that can do such things? Long story short: I want to do the same thing as in my previous question but with much more complicated expression.

Root[x^9 +
x^7 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 + 20 x^7 +
35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 + 3 x^5 +
12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 - 12 x^2 +
6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 + 8 x^7) #1^6 + (-1 +
x + 7 x^2 - 7 x^3 - 8 x^4 + 45 x^5 + 45 x^6) #1^7 &, 3] -
3 x^7 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 - 12 x^2 +
6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 + 8 x^7) #1^6 + (-1 +
x + 7 x^2 - 7 x^3 - 8 x^4 + 45 x^5 + 45 x^6) #1^7 &,
3]^2 - 2 x^5 Root[-x^10 + (2 x^8 + 19 x^9 +
26 x^10) #1 + (x^6 + 20 x^7 + 35 x^8 - 40 x^9 -
64 x^10) #1^2 + (-2 x^4 + 3 x^5 + 12 x^6 - 85 x^7 -
129 x^8 + 32 x^9 + 32 x^10) #1^3 + (-x^2 + 10 x^4 -
43 x^5 - 80 x^6 + 123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 +
x^3 - 22 x^4 + 113 x^5 + 169 x^6 - 75 x^7 -
52 x^8) #1^5 + (1 - 12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 -
146 x^6 + 8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^3 +
3 x^5 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 - 12 x^2 +
6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 + 8 x^7) #1^6 + (-1 +
x + 7 x^2 - 7 x^3 - 8 x^4 + 45 x^5 + 45 x^6) #1^7 &, 3]^4 +
x^3 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 + 20 x^7 +
35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 + 3 x^5 +
12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 - 12 x^2 +
6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 + 8 x^7) #1^6 + (-1 +
x + 7 x^2 - 7 x^3 - 8 x^4 + 45 x^5 + 45 x^6) #1^7 &, 3]^5 -
x^3 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 + 20 x^7 +
35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 + 3 x^5 +
12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 - 12 x^2 +
6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 + 8 x^7) #1^6 + (-1 +
x + 7 x^2 - 7 x^3 - 8 x^4 + 45 x^5 + 45 x^6) #1^7 &,
3]^6 + (-6 x^8 Root[-x^10 + (2 x^8 + 19 x^9 +
26 x^10) #1 + (x^6 + 20 x^7 + 35 x^8 - 40 x^9 -
64 x^10) #1^2 + (-2 x^4 + 3 x^5 + 12 x^6 - 85 x^7 -
129 x^8 + 32 x^9 + 32 x^10) #1^3 + (-x^2 + 10 x^4 -
43 x^5 - 80 x^6 + 123 x^7 + 146 x^8) #1^4 + (-x +
6 x^2 + x^3 - 22 x^4 + 113 x^5 + 169 x^6 - 75 x^7 -
52 x^8) #1^5 + (1 - 12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 -
146 x^6 + 8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 -
8 x^4 + 45 x^5 + 45 x^6) #1^7 &, 3] +
x^5 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^2 -
3 x^6 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^2 +
x^3 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^3 +
2 x^4 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^3 +
12 x^6 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^3 -
x^3 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^4 -
3 x^2 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^5 -
6 x^4 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^5 +
3 x^2 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &,
3]^6) #1 + (15 x^7 Root[-x^10 + (2 x^8 + 19 x^9 +
26 x^10) #1 + (x^6 + 20 x^7 + 35 x^8 - 40 x^9 -
64 x^10) #1^2 + (-2 x^4 + 3 x^5 + 12 x^6 - 85 x^7 -
129 x^8 + 32 x^9 + 32 x^10) #1^3 + (-x^2 + 10 x^4 -
43 x^5 - 80 x^6 + 123 x^7 + 146 x^8) #1^4 + (-x +
6 x^2 + x^3 - 22 x^4 + 113 x^5 + 169 x^6 - 75 x^7 -
52 x^8) #1^5 + (1 - 12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 -
146 x^6 + 8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 -
8 x^4 + 45 x^5 + 45 x^6) #1^7 &, 3]^2 -
3 x^4 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^3 +
2 x^5 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^3 -
2 x^2 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^4 -
3 x^3 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^4 -
18 x^5 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^4 +
3 x Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^5 +
2 x^2 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^5 +
6 x^3 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^5 -
3 x Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^6 +
3 x^3 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &,
3]^6) #1^2 + (-20 x^6 Root[-x^10 + (2 x^8 + 19 x^9 +
26 x^10) #1 + (x^6 + 20 x^7 + 35 x^8 - 40 x^9 -
64 x^10) #1^2 + (-2 x^4 + 3 x^5 + 12 x^6 - 85 x^7 -
129 x^8 + 32 x^9 + 32 x^10) #1^3 + (-x^2 + 10 x^4 -
43 x^5 - 80 x^6 + 123 x^7 + 146 x^8) #1^4 + (-x +
6 x^2 + x^3 - 22 x^4 + 113 x^5 + 169 x^6 - 75 x^7 -
52 x^8) #1^5 + (1 - 12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 -
146 x^6 + 8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 -
8 x^4 + 45 x^5 + 45 x^6) #1^7 &, 3]^3 +
3 x^3 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^4 +
2 x^4 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^4 -
Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 + 20 x^7 +
35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 + 3 x^5 +
12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 + 45 x^5 +
45 x^6) #1^7 &, 3]^5 +
x Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 + 20 x^7 +
35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 + 3 x^5 +
12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^5 +
12 x^4 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^5 +
Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 + 20 x^7 +
35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 + 3 x^5 +
12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 + 45 x^5 +
45 x^6) #1^7 &, 3]^6 -
x Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 + 20 x^7 +
35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 + 3 x^5 +
12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^6 -
4 x^2 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &,
3]^6) #1^3 + (15 x^5 Root[-x^10 + (2 x^8 + 19 x^9 +
26 x^10) #1 + (x^6 + 20 x^7 + 35 x^8 - 40 x^9 -
64 x^10) #1^2 + (-2 x^4 + 3 x^5 + 12 x^6 - 85 x^7 -
129 x^8 + 32 x^9 + 32 x^10) #1^3 + (-x^2 + 10 x^4 -
43 x^5 - 80 x^6 + 123 x^7 + 146 x^8) #1^4 + (-x +
6 x^2 + x^3 - 22 x^4 + 113 x^5 + 169 x^6 - 75 x^7 -
52 x^8) #1^5 + (1 - 12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 -
146 x^6 + 8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 -
8 x^4 + 45 x^5 + 45 x^6) #1^7 &, 3]^4 -
x^2 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^5 -
3 x^3 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^5 +
x Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 + 20 x^7 +
35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 + 3 x^5 +
12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^6 -
3 x^3 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &,
3]^6) #1^4 + (-6 x^4 Root[-x^10 + (2 x^8 + 19 x^9 +
26 x^10) #1 + (x^6 + 20 x^7 + 35 x^8 - 40 x^9 -
64 x^10) #1^2 + (-2 x^4 + 3 x^5 + 12 x^6 - 85 x^7 -
129 x^8 + 32 x^9 + 32 x^10) #1^3 + (-x^2 + 10 x^4 -
43 x^5 - 80 x^6 + 123 x^7 + 146 x^8) #1^4 + (-x +
6 x^2 + x^3 - 22 x^4 + 113 x^5 + 169 x^6 - 75 x^7 -
52 x^8) #1^5 + (1 - 12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 -
146 x^6 + 8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 -
8 x^4 + 45 x^5 + 45 x^6) #1^7 &, 3]^5 +
x^2 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 +
20 x^7 + 35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 +
3 x^5 + 12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 -
12 x^2 + 6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 +
8 x^7) #1^6 + (-1 + x + 7 x^2 - 7 x^3 - 8 x^4 +
45 x^5 + 45 x^6) #1^7 &, 3]^6) #1^5 +
x^3 Root[-x^10 + (2 x^8 + 19 x^9 + 26 x^10) #1 + (x^6 + 20 x^7 +
35 x^8 - 40 x^9 - 64 x^10) #1^2 + (-2 x^4 + 3 x^5 +
12 x^6 - 85 x^7 - 129 x^8 + 32 x^9 +
32 x^10) #1^3 + (-x^2 + 10 x^4 - 43 x^5 - 80 x^6 +
123 x^7 + 146 x^8) #1^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) #1^5 + (1 - 12 x^2 +
6 x^3 + 21 x^4 - 115 x^5 - 146 x^6 + 8 x^7) #1^6 + (-1 +
x + 7 x^2 - 7 x^3 - 8 x^4 + 45 x^5 + 45 x^6) #1^7 &,
3]^6 #1^6 &, 1];


Edit: Example of an expression with the new problem: pastebin.com/uZLmJ6SF (I'm posting a link since the code as you can see is way too long) - I can't somehow resolve the issue of trivial solution. Why is it coming out this way? Function res from the answer gives value 1.

• It would be indeed be nice to be able to do what you ask for, but wouldn't it invalidate some well-established results of algebra? Jan 12, 2017 at 1:40
• What do you mean precisely? What results? If you mean Abel-Ruffini theorem then no, it's OK. Jan 12, 2017 at 1:53

Because the enormous expression in the question contains two levels of Root functions, it can be represented as the solution of two polynomials in two variables, y and an auxiliary variable, say, z. With the expression designated s for convenience, the two equations are readily derived, as follows. To obtain the first polynomial, extract the First argument of the outer Root, for instance by s // First, replace the inner Root by z, and apply the expression to y.

eq1 = ((s // First) /. Root[__] -> z)[y] == 0
(* x^9 + x^7 z - 3 x^7 z^2 - 2 x^5 z^3 + 3 x^5 z^4 + x^3 z^5 - x^3 z^6 +
x^3 y^6 z^6 + y^3 (-20 x^6 z^3 + 3 x^3 z^4 + 2 x^4 z^4 - z^5 + x z^5 +
12 x^4 z^5 + z^6 - x z^6 - 4 x^2 z^6) + y^5 (-6 x^4 z^5 + x^2 z^6) +
y (-6 x^8 z + x^5 z^2 - 3 x^6 z^2 + x^3 z^3 + 2 x^4 z^3 +
12 x^6 z^3 - x^3 z^4 - 3 x^2 z^5 - 6 x^4 z^5 + 3 x^2 z^6) +
y^4 (15 x^5 z^4 - x^2 z^5 - 3 x^3 z^5 + x z^6 - 3 x^3 z^6) +
y^2 (15 x^7 z^2 - 3 x^4 z^3 + 2 x^5 z^3 - 2 x^2 z^4 - 3 x^3 z^4 -
18 x^5 z^4 + 3 x z^5 + 2 x^2 z^5 + 6 x^3 z^5 - 3 x z^6 + 3 x^3 z^6) == 0 *)


To obtain the second polynomial, extract the inner Root from s, here with Cases, and apply the result to z.

eq2 = (Union@Cases[s, _Root, {4}])[[1, 1]][z] == 0
(* -x^10 + (2 x^8 + 19 x^9 + 26 x^10) z + (x^6 + 20 x^7 + 35 x^8 -
40 x^9 - 64 x^10) z^2 + (-2 x^4 + 3 x^5 + 12 x^6 - 85 x^7 -
129 x^8 + 32 x^9 + 32 x^10) z^3 + (-x^2 + 10 x^4 - 43 x^5 -
80 x^6 + 123 x^7 + 146 x^8) z^4 + (-x + 6 x^2 + x^3 - 22 x^4 +
113 x^5 + 169 x^6 - 75 x^7 - 52 x^8) z^5 + (1 - 12 x^2 + 6 x^3 +
21 x^4 - 115 x^5 - 146 x^6 + 8 x^7) z^6 + (-1 + x + 7 x^2 -
7 x^3 - 8 x^4 + 45 x^5 + 45 x^6) z^7 == 0 *)


Of course, this process introduces many more solutions. For instance, at x == 2,

NSolve[{eq1, eq2} /. x -> 2, {y, z}] // Length
(* 42 *)


all but six of which are complex. The four smaller real solutions over the domain {x, 2, 10} are given by

t = Transpose@Table[y /. NSolve[{eq1, eq2}, {y, z}, Reals], {x, 1.25, 10, .25}];
pt = ListLinePlot[t, DataRange -> {1.25, 10}, PlotRange -> {0, 8}]


This can be compared with the single curve, s.

ps = Plot[s, {x, 1.25, 10}, PlotRange -> {{0, 10}, {0, 8}}]


which is identical to the lowest curve in the preceding plot.

Addendum: Reduction to a single seventh-order polynomial

As pointed out by Daniel Lichtblau in a comment below, z can be eliminated between the two polynomials (as requested by the OP).

res = Resultant[First@eq1, First@eq2, z]


to produce a single but very lengthy forty-second order polynomial in y. Plotting it

tr = Transpose@Table[y /. NSolve[res, y, Reals], {x, 1.25, 10, .25}];
ptr = ListLinePlot[tr, DataRange -> {1.25, 10}, PlotRange -> {0, 8}]


yields a plot identical to the first one above. Unexpectedly, I found that

fac = Factor[res];


splits res into three factors, the first being x^47, the third a thirty-fifth order polynomial, and the second

fac[[2]]
(* x^5 + 2 x^3 y + x y^2 - x^2 y^2 - 2 x^3 y^2 + 7 x^4 y^2 + 12 x^5 y^2 - y^3 - x y^3 +
6 x^2 y^3 + 5 x^3 y^3 - 14 x^4 y^3 - 4 x y^4 - 17 x^2 y^4 - 3 x^3 y^4 + 32 x^4 y^4 +
32 x^5 y^4 + 7 x y^5 + 7 x^2 y^5 - 44 x^3 y^5 - 64 x^4 y^5 + 14 x^2 y^6 + 26 x^3 y^6 -
x^2 y^7 *)


It is this factor that contains the solution associated with s, as can be seen from

fs = Solve[fac[[2]] == 0, y];
ptf = Plot[Evaluate[y /. fs], {x, 1.25, 10}, PlotRange -> {{0, 10}, {0, 8}}]


in which the lower curve is identical to the curve for s. Thus, f[[2]] is the desired polynomial and is only seventh-order in y.

• Thank you for your answer. Can I get rid of this auxilliary variable z somehow, without solving the equations? As I said, I'm looking for the zero locus of the polynomial in question, extra solutions are fine since I know what I can cancel in the end, but I don't know what to do with this extra variable. Jan 12, 2017 at 10:12
• This works: In[1173]:= AbsoluteTiming[res = Resultant[First@eq1, First@eq2, z];] Out[1173]= {29.670744, Null} Jan 12, 2017 at 16:32
• How do I generalize it to even harder expressions? I can't paste it because it's too long but after I use eq1 = ((s // First) /. Root[__] -> z)[y] == 0 on the new expression then I get x^7[y]==0 output. How can I fix it? eq2 works fine. Jan 12, 2017 at 21:07
• My new expression is multiplied by x^7 before all the roots by the way, it's pretty similar to the one in my first post but a lot longer. When I cancel this x^7 I get another trivial expression. I edited the OP with an expression with this problem, however canceling x^7 there leads to a good expression, I can't do that with my complicated one. How can I get rid of this problem? Jan 12, 2017 at 21:48
• @Caims I have added a few sentences of explanation which you may find helpful. Basically, extract the first argument of each of the Root functions and apply it to the appropriate dependent variable, here y for the outer Root and z for the inner Root. In addition, the first argument of the outer Root contains the inner Root, which must be replaced by z. I do not understand your question about x^7. If it multiplies everything, certainly it can be cancelled. Good luck. Jan 12, 2017 at 23:59