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I'm trying to find the sign of some very long expressions, but I don't seem to make it work using Simplify[Sign[...], Assumptions ...]. Here is an example of my problem. I can easily find the sign of these two expressions:

exp1 = θA1^2 θAU (16 θB1^3 (1 - λ)^4 + 

 64 θB1^2 θBU (1 - λ)^3 (1 + λ) + 

 5 θBU^3 (1 - λ) (1 + λ)^3 + 

 54 θB1 θBU^2 (1 - λ^2)^2);

 Simplify[Sign[8 (exp1)] , 

   Assumptions -> {(1 - λ) > 0, θA1 > 0, θAU > 0, θB1 > 0, θBU > 0, λ > 0}]

This expression correctly calculates as 1. A second expression:

exp2 = θA1 θAU^2 (16 θB1^3 (1 - λ)^3 (1 + λ) +  44 θB1 θBU^2 (1 - λ) (1 + λ)^3 + 

 5 θBU^3 (1 + λ)^4 +  54 θB1^2 θBU (1 - λ^2)^2);

Simplify[Sign[8 (exp2)] ,  

  Assumptions -> {(1 - λ) > 0, θA1 > 0, θAU >     0, θB1 > 0, θBU > 0, λ > 0}]

Correctly calculates as 1 too. BUT when I try:

Simplify[Sign[8 (exp1 + exp2)] ,  

  Assumptions -> {(1 - λ) > 0, θA1 > 0, θAU >    0, θB1 > 0, θBU > 0, λ > 0}]

Then Mathematica cannot compute... Any ideas on what may be going on here?

You need to increase the value of the suboption "AssumptionsMaxNonlinearVariables" to a value higher than its default value (4):

SystemOptions["SimplificationOptions"]

{"SimplificationOptions" -> {"AssumptionsMaxExponent" -> 25,
"AssumptionsMaxNonlinearVariables" -> 4,
"AssumptionsMaxVariables" -> 21, "AutosimplifyTrigs" -> True,
"AutosimplifyTwoArgumentLog" -> True,
"ConvertTrigsToRadicals" -> False, "FiniteSumMaxTerms" -> 30,
"FunctionExpandMaxSteps" -> 15, "ListableFirst" -> True,
"RestartELProver" -> False, "SimplifyMaxExponents" -> 100,
"SimplifyToPiecewise" -> True}}

Set the value of the suboption "AssumptionsMaxNonlinearVariables" to a higher value (say, 10):

SetSystemOptions["SimplificationOptions" -> {"AssumptionsMaxNonlinearVariables" -> 10}];

Simplify[Sign[8 (exp1 + exp2)], 

   Assumptions -> {(1 - λ) > 0, θA1 > 0, θAU > 0, θB1 > 0, θBU > 0, λ > 0}]

1

Reset the suboption value to its default:

SetSystemOptions["SimplificationOptions" -> {"AssumptionsMaxNonlinearVariables" ->  4}];