Mathematica region of parameters with roots

Question

I am currently trying to obtain an analytical expression for a range of validity of an expression. I am doing this with reduce in mathematica but it keeps giving me the expression in terms of the Roots function. I tried to solve it and these roots have analytical solutions, is there any way to get a cleaner expression, preferably in terms of the parameters only and no root function... Thank you

Code:

 FullSimplify[Reduce[{-((6 \[Alpha]^4 + 21 \[Alpha]^3 \[Lambda] + 27 \[Alpha]^2 \[Lambda]^2 + 15 \[Alpha] \[Lambda]^3 + 3 \[Lambda]^4 + Sqrt[3] Sqrt[-(\[Alpha] + \[Lambda])^6 (-72 + 16 \[Alpha]^3 \[Lambda] + 21 \[Lambda]^2 + 4 \[Alpha] \[Lambda] (-9 + 4 \[Lambda]^2) + 4 \[Alpha]^2 (-15 + 8 \[Lambda]^2))])/( 4 (\[Alpha] + \[Lambda])^4)) < 0 && -((6 \[Alpha]^4 + 21 \[Alpha]^3 \[Lambda] + 27 \[Alpha]^2 \[Lambda]^2 + 15 \[Alpha] \[Lambda]^3 +  3 \[Lambda]^4 - Sqrt[3] Sqrt[-(\[Alpha] + \[Lambda])^6 (-72 + 16 \[Alpha]^3 \[Lambda] + 21 \[Lambda]^2 + 4 \[Alpha] \[Lambda] (-9 + 4 \[Lambda]^2) +4 \[Alpha]^2 (-15 + 8 \[Lambda]^2))])/( 4 (\[Alpha] + \[Lambda])^4)) < 0 && -((3 (-2 \[Beta] + \[Lambda] + \[Lambda] \[Mu]))/(\[Alpha] \
+ \[Lambda])) < 0 && \[Alpha] (\[Alpha] + \[Lambda]) > -3 && \[Lambda] \
(\[Lambda] + \[Alpha]) > 3}, {\[Alpha], \[Beta], \[Lambda], \[Mu]}, 
Reals, Cubics -> True, Quartics -> True]]

Answer 1

To convert low order Root objects use ToRadicals

sol1 = FullSimplify[
   Reduce[-((6 α^4 + 21 α^3 λ + 
           27 α^2 λ^2 + 15 α λ^3 + 
           3 λ^4 + 
           Sqrt[3] Sqrt[-(α + λ)^6 (-72 + 
                16 α^3 λ + 21 λ^2 + 
                4 α λ (-9 + 4 λ^2) + 
                4 α^2 (-15 + 
                   8 λ^2))])/(4 (α + λ)^4)) < 
      0 && -((6 α^4 + 21 α^3 λ + 
           27 α^2 λ^2 + 15 α λ^3 + 
           3 λ^4 - 
           Sqrt[3] Sqrt[-(α + λ)^6 (-72 + 
                16 α^3 λ + 21 λ^2 + 
                4 α λ (-9 + 4 λ^2) + 
                4 α^2 (-15 + 
                   8 λ^2))])/(4 (α + λ)^4)) < 
      0 && -((3 (-2 β + λ + λ μ))/(α + λ)) < 
      0 && α (α + λ) > -3 && λ (λ + α) > 3, {α, β, λ, μ}, Reals]] // ToRadicals

(* long output removed *)

Further simplification requires additional assumptions/constraints on the parameters, e.g., if all the parameters are positive

sol2 = Simplify[sol1, Thread[{α, β, λ, μ} > 0]]

(* 2 β < λ + λ μ && ((λ <= (1/(
      48 α))(-21 - 
        32 α^2 + (441 + 3072 α^2 + 256 α^4)/(-9261 + 
           152064 α^2 + 84096 α^4 + 4096 α^6 + 
           288 Sqrt[
            2] α Sqrt[(-3 + 2 α^2)^3 (1029 + 64 α^2)])^(
         1/3) + (-9261 + 152064 α^2 + 84096 α^4 + 
          4096 α^6 + 
          288 Sqrt[2] α Sqrt[(-3 + 2 α^2)^3 (1029 + 
              64 α^2)])^(1/3)) && (2 α < Sqrt[
        6] || (2 α > Sqrt[6] && 
         Sqrt[12 + α^2] < α + 2 λ))) || (Sqrt[
      12 + α^2] < α + 2 λ && 
     2 α <= Sqrt[
      6] && λ <= -(1/(
       96 α))(42 + 
         64 α^2 + ((1 - I Sqrt[3]) (441 + 3072 α^2 + 
              256 α^4))/(-9261 + 152064 α^2 + 
            84096 α^4 + 4096 α^6 + 
            288 Sqrt[
             2] α Sqrt[(-3 + 2 α^2)^3 (1029 + 64 α^2)])^(
          1/3) + (1 + I Sqrt[3]) (-9261 + 152064 α^2 + 
            84096 α^4 + 4096 α^6 + 
            288 Sqrt[
             2] α Sqrt[(-3 + 2 α^2)^3 (1029 + 64 α^2)])^(
          1/3)))) *)

Note that the expression contains complex subexpressions, e.g., (1+I*Sqrt[3]).