Proving conditions on absolute value of roots of a 5-degree polynomial

Question

**I just need to find conditions on Subscript[[CapitalPhi], [Pi]] such that the fifth root of the polynomial is greater than 1 in absolute value. **

I have a 5-degree characteristic polynomial, for which I have roots in the form of 'Root' object. For certain values of the coefficients, I need to test if the absolute value of a specific root, (or the eigenvalue is greater than 1). I tried working with ConditionalExpression, If, etc. but no luck.

The following is the characteristic polynomial in z: ([CapitalPhi]_[Pi] is just a parameter in the model)

z^5 - 0.00204575 Subscript[\[CapitalPhi], \[Pi]] - 
 0.0487214 z Subscript[\[CapitalPhi], \[Pi]] + 
 1.04752 z^2 Subscript[\[CapitalPhi], \[Pi]] - 
 0.996752 z^3 Subscript[\[CapitalPhi], \[Pi]]
These are the roots:

z == Root[-3.81007 #1^4 + #1^5 + #1^2 (-0.659423 - 
        1.04752 Subscript[\[CapitalPhi], \[Pi]]) + #1 (-0.0163848 + 
        0.0487214 Subscript[\[CapitalPhi], \[Pi]]) + #1^3 (3.49068 + 
        0.996752 Subscript[\[CapitalPhi], \[Pi]]) + 
     0.00204575 Subscript[\[CapitalPhi], \[Pi]] &, 1] || 
 z == Root[-3.81007 #1^4 + #1^5 + #1^2 (-0.659423 - 
        1.04752 Subscript[\[CapitalPhi], \[Pi]]) + #1 (-0.0163848 + 
        0.0487214 Subscript[\[CapitalPhi], \[Pi]]) + #1^3 (3.49068 + 
        0.996752 Subscript[\[CapitalPhi], \[Pi]]) + 
     0.00204575 Subscript[\[CapitalPhi], \[Pi]] &, 2] || 
 z == Root[-3.81007 #1^4 + #1^5 + #1^2 (-0.659423 - 
        1.04752 Subscript[\[CapitalPhi], \[Pi]]) + #1 (-0.0163848 + 
        0.0487214 Subscript[\[CapitalPhi], \[Pi]]) + #1^3 (3.49068 + 
        0.996752 Subscript[\[CapitalPhi], \[Pi]]) + 
     0.00204575 Subscript[\[CapitalPhi], \[Pi]] &, 3] || 
 z == Root[-3.81007 #1^4 + #1^5 + #1^2 (-0.659423 - 
        1.04752 Subscript[\[CapitalPhi], \[Pi]]) + #1 (-0.0163848 + 
        0.0487214 Subscript[\[CapitalPhi], \[Pi]]) + #1^3 (3.49068 + 
        0.996752 Subscript[\[CapitalPhi], \[Pi]]) + 
     0.00204575 Subscript[\[CapitalPhi], \[Pi]] &, 4] || 
 z == Root[-3.81007 #1^4 + #1^5 + #1^2 (-0.659423 - 
        1.04752 Subscript[\[CapitalPhi], \[Pi]]) + #1 (-0.0163848 + 
        0.0487214 Subscript[\[CapitalPhi], \[Pi]]) + #1^3 (3.49068 + 
        0.996752 Subscript[\[CapitalPhi], \[Pi]]) + 
     0.00204575 Subscript[\[CapitalPhi], \[Pi]] &, 5]

I saved the last root as Root100 and then took the absolute value: The last two roots are imaginary and I just want to check if their absolute value is greater than 1 for [CapitalPhi]_[Pi] greater than or less than 1. Starting with the fifth root, I did the following:

Root100 = 
 z == Root[-3.8100716752443358` #1^4 + #1^5 + #1^2 \
(-0.6594231916363339` - 
        1.0475194467246431` Subscript[\[CapitalPhi], \[Pi]]) + #1 \
(-0.016384816689513255` + 
        0.048721356558687944` Subscript[\[CapitalPhi], \[Pi]]) + #1^3 \
(3.4906781487894887` + 
        0.9967523435624097` Subscript[\[CapitalPhi], \[Pi]]) + 
     0.0020457466035455397` Subscript[\[CapitalPhi], \[Pi]] &, 5]

Root101 = Abs[Root100]

Abs[z == Root[-3.81007 #1^4 + #1^5 + #1^2 (-0.659423 - 
        1.04752 Subscript[\[CapitalPhi], \[Pi]]) + #1 (-0.0163848 + 
        0.0487214 Subscript[\[CapitalPhi], \[Pi]]) + #1^3 (3.49068 + 
        0.996752 Subscript[\[CapitalPhi], \[Pi]]) + 
     0.00204575 Subscript[\[CapitalPhi], \[Pi]] &, 5]]

I have tried the following :

ConditionalExpression[Root101 > 1, 
 Subscript[\[CapitalPhi], \[Pi]] > 1]

Refine[Root101, Assumptions -> Subscript[\[CapitalPhi], \[Pi]] > 1]

FullSimplify[Root100, Subscript[\[CapitalPhi], \[Pi]] > 1]

Root101 = If[Subscript[\[CapitalPhi], \[Pi]] > 1, z]

If you could please suggest a way to prove that the absolute value of the fifth root would be greater than 1 for Subscript[[CapitalPhi], [Pi]] greater than as well as less than 1. I would really appreciate help with this.

****To make it more clear: I just need to find conditions on Subscript[[CapitalPhi], [Pi]] to prove that the fifth root of the polynomial is greater than 1 in absolute value. ****

Thank you very much, @Akku14 for your reply, but I think wasn't able to explain my objective too well.

I'm new to mathematica and would really appreciate any help with this. Thank you.

Answer 1

$Version

(* "12.3.1 for Mac OS X x86 (64-bit) (June 19, 2021)" *)

Clear["Global`*"]

poly = z^5 - 0.00204575 Subscript[Φ, π] - 
    0.0487214 z Subscript[Φ, π] + 
    1.04752 z^2 Subscript[Φ, π] - 
    0.996752 z^3 Subscript[Φ, π] // Rationalize[#, 0] &;

roots = Values[Solve[poly == 0, z]];

The absolute value of the last root equals 1 when

pts = FindRoot[
    Abs[roots[[-1]]] == 1, {Subscript[Φ, π], #}] & /@ {-1, 2}

(* {{Subscript[Φ, π] -> -0.604281}, {Subscript[Φ, π] -> 1.6579}} *)

Plot[{Abs[roots[[-1]]], 1}, {Subscript[Φ, π], -2, 3},
 Epilog -> {Red, AbsolutePointSize[4],
   Tooltip[Point[{Subscript[Φ, π], 1}], 
     Subscript[Φ, π]] /. pts},
 Frame -> True,
 PlotRange -> All,
 PlotPoints -> 75,
 MaxRecursion -> 10]

The absolute value of the last root is less than 1 for

Less @@ Insert[Values[pts] // Flatten, Subscript[Φ, π], 2]

(* -0.604281 < Subscript[Φ, π] < 1.6579 *)

EDIT: For the revised polynomial given in a comment

Clear["Global`*"]

poly2 = 0.` + 0.016384816689513255` z + 0.6594231916363339` z^2 - 
    3.4906781487894887` z^3 + 3.8100716752443358` z^4 - z^5 - 
    0.0020457466035455397` Subscript[Φ, π] - 
    0.048721356558687944` z Subscript[Φ, π] + 
    1.0475194467246431` z^2 Subscript[Φ, π] - 
    0.9967523435624097` z^3 Subscript[Φ, π] // 
   Rationalize[#, 0] &;

The last root is

root5 = Solve[poly2 == 0, z][[-1, 1, -1]];

Plotting the absolute value of the last root

plt = Plot[Abs[root5], {Subscript[Φ, π], -3, 3},
  MeshFunctions -> {#2 &},
  Mesh -> {{1}},
  MeshStyle -> {Red, AbsolutePointSize[4]}, Frame -> True]

val = Cases[plt // Normal, Point[pt_] :> pt, Infinity][[All, 1]]

(* {-0.384239, -0.0794879} *)

The absolute value of the last root is greater than 1 when