@vv Thank you for the answer. I am not sure if the answer will help test millions of equations but it surely helps me to understand some of the possibilities (without using assuming).  

@Carl Love Many thanks.  I will give a try on simplify(..., symbolic) and also check assuming real. 

Acer's answer helps me with another problem - part b) of my original post.  

@acer Thanks.  The methods are not top-secret at all: 1st Method - solve the general equation with unknown symbolic coefficients for the monomials (beta, for instance).  Then, by using the relation between beta coefficients and alpha coefficients (such as the ones you see in my first post), find the solution.  2nd Method -solve the system with original alpha coefficients (they have an intricate relationship that depends on other calculations).  
My expectation was to reduce the number of systems to be solved using the general equation.  Of course, I have no choice but to use Method 2 in some cases (again due to the relationships in the alpha coefficients). 

I hope this sheds some light on the problem.  

@acer Many thanks.   I will avoid evalb in future calculations.  Since I am away from my notebook, I wonder whether assuming is the key to all alternatives.  If so, that will make the comparison almost impossible since I have millions of such a system to be tested and I can't see how I will write a specific assuming for each one of them. 

@Carl Love Many thanks again.  I will test it over the weekend and let you know. Could you explain what rcurry does and why it is faster?

@Carl Love Many thanks.  I will use the wrapper from now on.   

@Carl Love Many thanks. I don't know how I could miss making M local (Sorry for wasting your time).   Once M was made local, Threads:-Map returned the same number of elements.  Some more tests are needed though.   

Normally order is important for me but not in this case. I guess I am used to putting the monomials in lists and don't think much about it.  

@Carl Love, First of all, thank you for the comment. I came across an example that seems to show what you mentioned in your comment.   If I use Grid-Seq instead, Maple returns the same result as in map. But when I try to use it with a much larger input (list or set),  the calculations with the nodes went rather okay (2 hours), but when collating the results the memory jumped from 50Gb to 200 GB, and it is still running (more than 12 hours).   

@ecterrab Many thanks.   I wasn't aware of hastype.  I should have checked "See also".  Sorry for wasting your time.   

@acer Many thanks.   I could not find the slider in the first option.  I can see the controls but not the slider. 

The second option, there is no slider (as you said).  

The third option has everything I need.  Many thanks.  

@acer Many thanks. Great alternatives that I can easliy follow. 

 As I said that was the first time I tried to use plots in Maple.   Although, after the alternatives, a plot is created I could not see a slider (or something like that) that I can use to change the values of a.  Am I missing something here?   

@Carl Love Many thanks. 

@Carl Love Many thanks.    

• The way to delay the evaluation is to use single quotes, right?
• I agree, it is definitely awkward.  The use of a function is way more elegant.
• I could not figure out how to change a (that is, there is no animation).

My wish has to do with Grid and Threads packages:

1) Efficient usage of memory -   Once a grid command finishes the memory allocated should return to Maple (I am not sure whether this is possible or not). There is a post in the list showing that, in some calculations using Grid, Maple hangs for no apparent reason.

2) A clear explanation in the help files (examples) on which variables should be Grid:-Set and which don't.  (Although I have received help from tops users on the list and from maple support, the answers are not the same in some cases).

3) CodeTools:-ThreadSafetyCheck - Some of the functions that are threadsafe checked by this command causes Maple to hang.  Is the list of threadsafe functions updated?  

@Kitonum Many thanks.   I can't believe I have missed allsolutions.