@dharr great, thanks. It's good to see that the positive Lambda_1 and Lamnda_2 are unique. By the way, how did you explicitly work out the nondims and corresponding Eqq2 and Eqq1 (output (3) in nondimGamma1Gamma2.mw) using Hubert's theory? Is it an adaptation of Case_with_radicals.mw (but without the additional equation)? I can't get to the same Eqq2 and Eqq1 using that...

@dharr thank you for getting back to me on this.

In your worksheet you mention "Lambda__1 is the largest root of a degree 10 polynomial" and while answering to my point 1. in the comment above you mention "If the RootOf for Lambda__1 is consistently interpreted as the most positive real root, and this value is used to calculate Lambda__2 then both Lambda__1 and Lambda__2 are consistently positive".

What's the meaning of "most positive" in this context? and above all:

• Given strictly positive Gamma_1 and Gamma_2, is the positive Lambda_1 (and thus Lambda_2) we use unique? In other words, can I prove that the remaining 9 roots of the RootOf(_Z^10-polynomial) are either negative or not consistently positive for strictly positive Gamma_1 and Gamma_2?

@dharr great catch for spotting the mismatching notation! 

@dharr I am back to this thread. Your conclusion here was that numerical analysis showed that Lambda_1 is actually strictly positive (for Sigma = 0.01 .. 5) and Lambda_2 as well (for Sigma = 0.03 .. 5). For both Lambda_1 and Lambda_2 the plots are not just strictly positive but, importantly, they also show no weird cliffs as in the plots for the analytical/symbolic-form analysis.

You tried two different non-dimensionalizations and concluded that in these two cases analytical/symbolic-form analysis leads to a Lambda_1 with some weird cliffs and a Lambda_2 that is NOT strictly positive, i.e., the surface has a boundary. In other words, you couldn't find a symbolic form for Lambda_1 and Lambda_2 that is consistent with the numerical solutions (at least for the first root of the degree-10 polynomial and for T and Sigma both not too close to 0).

Questions :

1. (Assuming that it's tough to guess a non-dimensionalization Lambda_1 and Lambda_2 leading to two symbolic-forms whose plots are consistent with the well-behaved numerical solutions) is it possible to reverse-engineer a (even approximative) symbolic form from the numerical solutions?
2. Even if we managed to reverse-engineer such symbolic form, would I encounter issues if I wanted to take partial derivatives of Lambda_1 and Lambda_2 wrt to sigma_v1 and sigma_v2 (since T and Sigma "entangles" both sigma_v1 and sigma_v2)? Specifically, you transitioned from a Gamma_1=sigma_v1*sigma_d*gamma and Gamma_2=sigma_v2*sigma_d*gamma non-dimensionalization to a T=sigma_v2/sigma_v1 and Sigma=gamma*sigma_d*sqrt(sigma_v1^2+sigma_v2^2) non-dimensionalization just to get rid of the square roots, but wouldn't be more "intuitive" to work with Gamma_1 and Gamma_2? (it's easier to interpret a 3D plot with Gamma_1 and Gamma_2 as x- and y- axes)
3. Perhaps a stupid question: worst case scenario, are there ways to calculate partial derivatives when no analytical expressions are available? 

@mmcdara I am not sure I know how to make use of the previous answer to reply this question, which is very different. It would take me some time to go through all the details but, if I understood correctly, you prove that for Gamma>0 and -1<rho<+1 there is only one real and positive root. (Basically you provided a very detailed and formal analysis, which I really appreciate, of the existence of the unique positive root shown simply with plot3d for the ranges of Gamma and rho above, or am I wrong? I need to read more about Sturm's method).

In this question I am just interested in the sign of the partial derivatives of such unique positive root.

@mmcdara deserves best answer for the impressive analysis. It'll take me some time to go through it and digest it. Thanks a lot!

@acer thanks. I think I also figured out query 1). I think I just made a bad choice about my values for Gamma and rho when I tested the zeros of my function.

Since I am only interested in Gamma>0, s2 is not relevant to me and I can just play around with Gamma values > or < than the Gamma__0 given by s3, for which there will always be just one positive root. This is then consistent with the unique positive zero given by implicitplot3d with Explore().  

Download always_just_one_root.mw

@dharr wow that's great! and it seems that apart from the region for small Sigma and T the surfaces are quite smooth for both lambda_1 and lambda_2...

@dharr thank you so much for all the effort!! I really appreciate your help. 

I am a bit confused though, perhaps I need to review your three worksheets again...Your conclusion is that we can't work out an expression for the positive portion of lambda_2 because we can't exactly identify its boundaries? 

@C_R thank you for the detailed answer and @mmcdara for your comments!

@C_R thank you for the detailed answer.

Isn't this equivalent to the solution I mentioned in my last comment to @mmcdara ??

That is, wouldn't it be simpler to just take the linear term of the polynomial (since the quadratic is surely negative), plug into it the condition to cancel the X and see when such expression is negative. Then, either:

1. lambda_1*delta_1+lambda_2*delta_2-lambda_3*delta_3>0 and theta<0     or
2. lambda_1*delta_1+lambda_2*delta_2-lambda_3*delta_3<0 and theta>0

In other words, it would suffice to show (which I can't as of now just with the equations I have) that lambda_1*delta_1+lambda_2*delta_2-lambda_3*delta_3 and theta having the same sign leads to a contradiction, thus proving that it can only be the case that the two terms have opposite signs or, equivalently, that not only the quadratic term but also the linear term is necessarily negative.  

@mmcdara yes I do agree that my strategy is flawed or at least missing some crucial elements, but I am a bit confused by your last paragraph in your script...would you mind elaborating please?

So, what you suddenly call Z would be the Non_Trivial_Theta_Root found just above, right? And what does HDT stands for? And then when you talk about the sign of -HDT do you simply mean positive (since all the lambdas are positive)? If I understood correctly the broader picture, you are basically saying that the polynomial in theta (diffcond) is not necessarily negative as I was hoping since it has a zero, i.e., it changes sign for theta=Non_Trivial_Theta_Root. In other words, it cannot be negative for any value of theta. But I am not sure I follow why it matters whether such zero is positive or negative...

Perhaps I am asking something really trivial which I can't see at the moment (might be too tired), but it would be helpful if you could rephrase that last paragraph... 

For example, wouldn't it be simpler to just take the linear term of the polynomial (since the quadratic is surely negative), plug into it the condition to cancel the X and see when such expression is negative. Then, either:

1. lambda_1*delta_1+lambda_2*delta_2-lambda_3*delta_3>0 and theta<0     or
2. lambda_1*delta_1+lambda_2*delta_2-lambda_3*delta_3<0 and theta>0

In other words, it would suffice to show (which I can't as of now just with the equations I have) that lambda_1*delta_1+lambda_2*delta_2-lambda_3*delta_3 and theta having the same sign leads to a contradiction, thus proving that it can only be the case that the two terms have opposite signs or, equivalently, that not only the quadratic term but also the linear term is necessarily negative.  

@mmcdara what if I add the following constraint? 

X__3*lambda__3-X__2*lambda__2-X__1*lambda__1=0

@mmcdara thanks, but I'd try to avoid numerical approaches as much as possible to show this. I agree with you I might be missing out some additional constraint...

Anyway here I further analyze the linear term (since the quadratic term in theta is trivially always negative). Perhaps output (8) is useful in some ways... 

EDIT: I am thinking deeply whether for my system the additional specification that is missing is that the linear term in theta is actually 0 (and thus the sum of both terms in theta, i.e., the linear and the quadratic, is negative overall...)

Download inequality_new.mw

@dharr I was trying to use the non-dimensionalized expressions obtained in Case_with_radicals.mw and Case_2.mw to express in compact form the solution to a system of 2 equations (in a similar way as what you helped me with previously) for further analysis: putting_all_together.mw. The script includes (1) the system whose solution I can't rewrite in compact form, (2) a working example for a simpler system, which I copied from a previous answer of yours.

My goal is the same as before: identify real and positive (ideally unique) solutions for lambda_1 and lambda_2, which in this case differ from each other. Thanks a lot for taking a look.