IsEmpty( sys, R )  is a new signature of IsEmpty introduced in Maple 16.

It assumes that sys defines a semi-algebraic system, that means you are

interested in *real solutoins* of sys. It is possible that a system (which is your

case) has no real solutions (as semi-algebraic system), but has complex solutions

(as algebraic system).

To answer you question: both calling sequence are allowed, but they have different

means.

You can use "trace" to see the step-by-step execution of the procedure. Is this what you need?

Another way is use "showstat" to view the code and statements in that procedure.

Rong

Firstly,  y'=-2*x*y/(1+x^2) is not a valid maple expression.

Secondly, 2*x*y*dx+(1+x^2)*dy=0 is a valid maple expression but

not an maple ODE.

In maple, an ODE is typed as

diff(y(x),x)=-2*x*y(x)/(1+x^2).

For that ODE, you will get a pretty printed form similar to 

2*x*ydx+(1+x^2)dy=0

See dsolve for more details. Hope this helps.

Have you activated your Maple?

In general, the problem you described is very hard (10 variables, degree 20) for Maple and  the power

of personal computers today. However, if you just have one such problem to solve, there are still chances to

work it out by trying different methods. In that case, why not just paste the problem? 

If I understood well your need, the follow loop should work:

#####################

n:=5; # set  n here

S:=table();

for i from 0 to 3^n-1 do

    summands:=NULL:

    k:=i:

    for j to n do       

         summands:=summands, (k mod 3) +1:

          k:= floor(k/3):

    end do:

   S[summands]:=convert([summands],`+`):

end do: