>> Sorry, this is not correct. It is in FIELD (for EXPR INT)

Excuse me, I was to quick again. Here is the (hopefully correct) anaylysis:

exquo(simplify((A-2^a)*(A+2^a))::UP(A,EXPR INT),(A-2^a)::UP(A,EXPR INT))

calls exquo$SUP(EXPR INT). This implements exact division of polynomials p1 by p2 as usual. After each subtraction - done via fmecg$SUP - the result is again stored in p1. exquo terminates when p1 is the empty list - note that SUPs are stored as lists of pairs (degree, coefficient) - or the degree of p2 is larger than p1. In the latter case, exquo fails.

Thus, in our case, at one point p1 is 4^a-2^(2*a), which is zero mathematically, but axiom does not know it. In particular, p1 is not the empty list, but rather a constant polynomial...

It would be interesting to see how MuPAD or Aldor handle this.

I should have added:

(1) -> exquo(normalize(simplify(((A-2^a)*(A+2^a)))::EXPR INT),normalize((A-2^a)::EXPR INT))


          a log(4)    2
        %e         - A
   (1)  ---------------
           a log(2)
         %e         - A
                                          Type: Union(Expression Integer,...)

(2) -> exquo(simplify((A-2^a)*(A+2^a))::UP(A,EXPR INT),(A-2^a)::UP(A,EXPR INT))

   (2)  "failed"
                                                    Type: Union("failed",...)

I'm afraid that this cannot be fixed easily, since there is no general mechanism to determine whether an expression is zero or not, which is needed in exquo. The instance of exquo involved is the one in SMP.