# 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.

• The Root part is a root to a 5th degree polynomial. In general, they cannot be expressed in terms of radicals. Whether or not your particular solution can, for all x, is not a question I can figure out in my head. Probably not, though. One can probably solve it with this, which I believe solves it in terms of theta functions. But Root is probably easier to work with. Dec 11, 2016 at 19:45
• Root[] is like any other numeric function such as Sin[]. There are algorithms to compute their values to any number of decimals, there are algorithms for simplifying and combining them. Just treat like you would f[x]. -- Your 2nd question confuses me: Do you want to work with Root or without it? If without, see my first comment. Dec 11, 2016 at 20:08
• What both Michael and the answer you reference are saying is that it is not possible to express the roots of a general 5th order polynomial in terms of radicals. Root represents a number and you can manipulate it symbolically like any other number. Dec 11, 2016 at 20:31
• As I said, see my first comment. Perhaps it can be done "in terms of theta functions" (link above). -- Related perhaps: (109578), not a solution, just relevant to learning to live with Root as a closed-form expression. Dec 11, 2016 at 20:32
• I can't do what I want without explicit form in terms of x. As I said: I want to have algebraic curve defined by zero locus of some polynomial y^n-solution = 0. So any power of y would work, as long as It's a closed expression even with square root or something (If I would have expression with square root then I could manipulate it to get what I want). Unfortunately a method from the last link on Roots (the one on how to live with them) doesn't give me a nice expression like it gives there. Dec 11, 2016 at 20:51

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.

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.

• That's exactly what I want, I'm really sorry for not being more clear. Thank you very much, you saved the day! Dec 11, 2016 at 21:20
• By the way, there's no extra problem that you mentioned, everything is fine. Dec 11, 2016 at 21:25
• @Caims No problem & you're welcome. I'll leave the warning in case someone else tries the same thing on their problem. It's like squaring both sides of an equation: sometimes you get extra solutions, sometimes not. Dec 11, 2016 at 21:30

I'm not sure this is exactly what you are looking for, but one approach is to define y (and your others) as functions of x and then plot to see what it looks like. For example:

y[x_] := 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];
Plot[y[x], {x, -5, 5}] • Unfortunately, I need the expression for zero locus of the polynomial that I will get from the solution for y. Dec 11, 2016 at 19:10