@Markiyan Hirnyk I made a small adjustment to my plaintext code. I believe that now when you paste it into 2D input it'll work (although I have no interest in testing this myself). Give it another try. I do always expect/hope for my plaintext code to work in 2D input, although I personally wouldn't touch the 2D input with a ten-foot pole for programming. For static display of mathematical expressions, it's okay, not great.

@Duplax The link that you gave opens the file as an XML tree (in browser Chrome). But how did you download the file?

@rlopez Surely we all know that the actual matrix A that you use doesn't matter. Not being able to regenerate your A, your worksheet can be read as a static document, as if it were in a book. But I like to use Maple to generate "active" content, so that there's an extra dimension to it not present when reading a book. As a reader, I like to read a worksheet starting with its output removed and having the output computed line by line as I read.

My experience is that most (but certainly not all) good-reading worksheets have some randomly generated examples. I don't think that it's widely known that Maple's random results are reproducible by other users as long as you start from a restart point and don't use the command randomize() (with no arguments). Using randomize(n) for constants n is fine.

@taro I have very little experience with monomial orderings. I have more experience with rearranging the side relations to produce a desired outcome. All I can say about the monomial orderings is that the plex works better if you specify more variables.

@Carl Love Well, did that work for you? It disturbs me to not get any response to an Answer.

I also haven't been able to reproduce Dr Lopez's matrix A, rerunning the loop ten times, for a total of 50,000 candidate matrices. It isn't clear whether he used restart. If he had used restart, then he'd be able to say something like "From the matrices with determinant 1, I chose the 34th" and then that would be reproducible for anyone else running the code as long as they also started from restart.

Markiyan's point is that none of his randomly generated matrices are exactly the same matrix A as that used by Dr Lopez.

@Thomas Richard On Unix, you can just use Maple's system command to invoke Unix's mail command. No need for a shell script or command-line Maple.

@vv I appreciate what you're saying, and I was going to suggest using smaller random numbers, but I thought that I'd work with the OP's own function. But what do you mean by "will not work"? My code does produce an answer, and it's an answer that's different than the answer produced when R is constant; indeed, it's very similar to the answer produced by your code.

@taro What I like about my solution is that it uses one command and only the four equations given by the OP. The substitution equations need to be arranged so that the eliminated expressions are on the left and the substitued variable is on the right, but that is the only change that is needed.

Since the purpose of this post is to explain simplify with side relations, the need for this left-to-right reversal should be pointed out explicitly.

@AfshinK I figured it out anyway. Thanks.

Lest anyone think that the summation or the subscripts are the source of your error, consider the simpler case c=1, and remove the subscripts:

solve(m*exp(-(m-1)*t)*d^2-theta+theta*t-theta*t-m*eta*(1-exp(-t))^(m-1) = 0, t);

This produces the same error message, so the summation and the subscripts have nothing to do with it.

@Christian Wolinski By "remainder" I assume that you mean residual. I wouldn't expect to get any better residuals from factor than from fsolve. But the fsolve residuals aren't bad (in Maple 16):

Digits:= 500:
Sols500:= [fsolve(rho_poly, complex)]:
unapply(abs(rho_poly), rho)~(Sols500):
max(evalf[5]~(%));

Also consider the absolute errors:

Digits:= 600:
Sols600:= [fsolve(rho_poly, complex)]:
abs~(Sols600 -~ Sols500):
evalf[5](max(%));

And the relative errors:

abs~((Sols600 -~ Sols500) /~ Sols600):
evalf[5](max(%));

What does the above code return in your Maple?

@acer Your first two examples contradict each other: The input is the same, but the output differs. I think that you may have made a copy-and-paste error.

Why use factor? You can use fsolve(rho_poly, complex) to get all the roots.

@Markiyan Hirnyk I don't understand why one way works with IsPrime and another way with PrimeDecomposition. I suggest going with Axel's way.