# Simplifying polynomial Roots with assumptions

I have acquired the following root from a minimization problem:

Root[1 + 2/(1 - 2 p)^4 - 3/(1 - 2 p)^2 - 4 p + 4 p^2 -
16 Sqrt[-(((-1 + p)^3 p^3)/(-1 + 2 p)^8)] + (12 p -
12 p^2) #1 + (-12 p + 12 p^2) #1^2 + (-1 + 4 p - 4 p^2) #1^3 &,
2]


For numerics this is an unnecessary nasty form and often generates small imaginary parts on the range where the value is actually real.

I'm particularly interested of the value when $$0 \leq p \leq 1$$. In this case I can use ToRadicals to ease FullSimplifying the Root:

FullSimplify[ToRadicals@%, 0 <= p <= 1]


$$\frac{2 \left(2 (p-1) p+\sqrt{-(p-1) p}\right)}{(1-2 p)^2}$$

Now, this is much more understandable and numerically neat. This feels to me a bit of a hack, though; I would be interested of hearing of a more generic method for this kind of root simplification with assumptions; clearly I can't expect ToRadicals to work for all polynomials. Also, I found out that FullSimplify with assumptions does nothing to the original Root object...

• Did you include the constraint 0<=p<=1 in the minimization process? – Bob Hanlon Jan 14 '19 at 0:14
• @BobHanlon Yep, doesn't make a difference. – kirma Jan 14 '19 at 18:50

Mathematica often considers Root objects as simpler forms since they can have a much lower LeafCount. For example,

sol = Solve[x^3 - 7 x + 2 == 0, x]

(* {{x -> (-9 + 2 I Sqrt[237])^(1/3)/3^(2/3) + 7/(3 (-9 + 2 I Sqrt[237]))^(
1/3)}, {x -> -(((1 + I Sqrt[3]) (-9 + 2 I Sqrt[237])^(1/3))/(
2 3^(2/3))) - (7 (1 - I Sqrt[3]))/(
2 (3 (-9 + 2 I Sqrt[237]))^(
1/3))}, {x -> -(((1 - I Sqrt[3]) (-9 + 2 I Sqrt[237])^(1/3))/(
2 3^(2/3))) - (7 (1 + I Sqrt[3]))/(2 (3 (-9 + 2 I Sqrt[237]))^(1/3))}} *)

sol2 = sol // FullSimplify

(* {{x -> Root[2 - 7 #1 + #1^3 &, 3]}, {x -> Root[2 - 7 #1 + #1^3 &, 2]}, {x ->
Root[2 - 7 #1 + #1^3 &, 1]}} *)

LeafCount /@ {sol, sol2}

(* {187, 52} *)


To return to the radical form you would need to either use ToRadicals or a custom ComplexityFunction

In your case, until the constraint on p is known, the Root object is simpler

expr = Root[
1 + 2/(1 - 2 p)^4 - 3/(1 - 2 p)^2 - 4 p + 4 p^2 -
16 Sqrt[-(((-1 + p)^3 p^3)/(-1 + 2 p)^8)] + (12 p -
12 p^2) #1 + (-12 p + 12 p^2) #1^2 + (-1 + 4 p - 4 p^2) #1^3 &, 2];

expr2 = expr // ToRadicals // FullSimplify

(* (p + I Sqrt[3]
p + (1 - I Sqrt[3]) ((1 - 2 p)^4 Sqrt[-(((-1 + p)^3 p^3)/(1 - 2 p)^8)])^(
2/3) - 4 p (Sqrt[-(((-1 + p)^3 p^3)/(-1 + 2 p)^8)] (-1 + 2 p)^4)^(1/3) +
p^2 (-1 - I Sqrt[3] +
4 (Sqrt[-(((-1 + p)^3 p^3)/(-1 + 2 p)^8)] (-1 + 2 p)^4)^(1/3)))/((1 -
2 p)^2 ((1 - 2 p)^4 Sqrt[-(((-1 + p)^3 p^3)/(1 - 2 p)^8)])^(1/3)) *)

expr3 = expr // ToRadicals // FullSimplify[#, 0 <= p <= 1] &

(* (2 (2 (-1 + p) p + Sqrt[-(-1 + p) p]))/(1 - 2 p)^2 *)

LeafCount /@ {expr, expr2, expr3}

(* {95, 184, 26} *)

• The idea of changing ComplexityFunction is a good idea, but curiously enough even plain LeafCount makes simplification of the Root object take effectively forever, thus making this approach impractical when trying to avoid ToRadicals. I think this is pretty curious... – kirma Jan 14 '19 at 19:10

It looks like you are basically looking for the region in p where the expression is real-valued. You can find this using Reduce. With exp equal to your expression,

reg = Reduce[exp>0]
0 < p < 1/2 || 1/2 < p < 1

(2 (2 (-1 + p) p + Sqrt[-(-1 + p) p]))/(1 - 2 p)^2


• I'm wishing to avoid using ToRadicals - although you can certainly ease finding the region with Reduce, or, say, FunctionDomain. – kirma Jan 13 '19 at 8:27
Unlike radical expressions, Root objects with exact coefficients resist the phenomenon of parasitic imaginary parts. Thus,
 Root[1 + 2/(1 - 2 p)^4 - 3/(1 - 2 p)^2 - 4 p + 4 p^2 -