@rwjewett Thank you for the problems, but to understand the scope of the project, we need to see the solutions. In particular, we need to know the extent to which you want Maple's computational abilities to be used. Here are pedagogical models at two extremes:

1. Minimal computational use of Maple: Maple is only being used as a mathematical typesetting system for a full-English-text exposition of a "by hand" solution as would be found in a textbook's companion solutions manual. If this is the case, it would be easier to do the project with actual mathematical typesetting software.
2. Minimal textual use of Maple: Maple's computational abilities are used to whatever extent possible with minimal exposition. This would be fairly easy for someone familiar with Maple, especially the Student packages.

There are many levels of gradation between these extremes.

@minhthien2016 As I said in a Reply to the main Question, that's not a solution to the original problem. Regarding my dimensionality argument, allowing a 7th parameter with 6 equations means that the solution space likely has topological dimension 7 - 6 = 1 in C^7, so the probability of an integer solution is no longer infinitesimal.

The solution that you found in your addendum is not a solution to the original equation! The coefficient of x^2 is not a parameter in the original. Allowing that coefficient to be negative changes VV's convexity argument. Allowing it to be a 7th parameter changes my dimensionality argument.

@Carl Love The header has changed, and no longer says Maple 2016.

@Earl I just noticed that the Question header says Maple 2016. I don't know why that is, because the OP has said previously that they have Maple 2019. The code I gave works in 2019. My 3rd example also doesn't work in 2016.

Can you give a meaningful example of legend entries that you would want on an inequal plot? (I wouldn't consider a one-entry legend meaningful; there must be at least two entries.)

We've been telling you for years now: Stop using evalm! Stop using linalg! But you never learn. You ask essentially the same handful of Questions over and over, year after year. 

@Carl Love Three addenda to my Answer:

1. As Acer pointed out, the tabs bar that I speak of doesn't appear if only one worksheet is open. Since the chance that I have only one open at the time of a kernel crash is near nil, this isn't an issue for me.

2. I think that the stop button on the tool bar will only interrupt Maple-level code, not compiled code (I'm not absolutely sure about this). If you press it while compiled code is running, the interruption doesn't occur until control comes back to Maple-level. But you can always kill a kernel the from the OS, such as with WIndows Task Manager.

3. If there's a long queue of screen output to display after the kernel has finished with your code, the stop button unfortunately does nothing. In that case, you're out of luck and need to kill the GUI (from, say, Windows Task Manager).

I remember old user interfaces / OSs (pre-Windows, pre-Maple) for which there was a control key (such as Ctrl-O) to discard the rest of the output queue. Something like that is needed.

@acer You wrote:

• I ... save the worksheet by closing the GUI using the top corner icon (window decoration button from the OS, which often looks like an X as Carl points out). 

I was referring to closing the worksheet alone---not the whole GUI---by clicking the X that appears to the right of the worksheet's name on the tabs bar, which is the bar below the main toolbar and above the context toolbar in the Standard GUI (in Windows, at least). Not closing the GUI is a major timesaver, especially if you have several worksheets open.

• When I do this the GUI still offers me the choice to save the worksheet, even though the kernel has crashed.

And the same things happens for me when I click my X. 

@Stretto If you put the for-loop in parentheses, then it's an expression, not a statement, and printlevel truly does nothing. It's akin to the distinction between if and `if`(...). 

I'm not claiming that printlevel will completely solve your problem. But it will let you see the results of statements at all levels of nesting, including within Maple's own procedures. 

Except for top-level statements, there's no difference between separating statements with colons or semicolons. Indeed, if you view the code (for example, with showstat if it's a procedure), you'll see that all statement separators are semicolons and redundant ones are eliminated.

@Christian Wolinski I don't know what you mean. For me, your code produces simple algebraic number solutions for the 4 variables. What do you mean by "incomplete"?

Could you provide an example worksheet showing what you're looking for for at least one of your problems? You can attach a worksheet to a Reply by using the green uparrow on the toolbar.

@ddaigle321 Yes, I think that using sets of two-element lists is the better option. And yes, they would be immutable, thus identical when mathematically equal. And because of that, there'll be significant savings when those recursive structures are saved as .m files rather than text files. 

@acer I was rushing to post my Reply "Not reliable" before my battery died, and then I went to sleep. I forgot to mention that saving MultiSets to text files doesn't work either, and that's where a ModuleDeconstruct could help. I was willing to accept that .m would never work. 

@ddaigle321 

It's looking like MultiSet is not, as you said, "reliable". In particular, it doesn't have a ModuleDeconstruct, which would make saving to a file easier. Your saved-file-reading problem is indeed a flaw of MultiSet. A primitive deconstructor would be

MSD:= MS-> subsindets(MS, 'MultiSet', e-> %MultiSet({convert(e, list)[]}));

The % creates an inert form and the value command can then be used for reconstruction. I don't think that that'll automatically deconstruct the embedded levels of a recursive MultiSet; if you need that, I can work on it.

That deconstructor has the side benefit of converting the MultiSet into an immutable structure, so mathematically equal copies will be automatically identified (to the same address).