In general, I prefer to put the parameters a_i as variables and then 
interpret the results.

Another approach you may try is to work in the field:

GF(2^d)['a_1,a_2,a_3'].fraction_field()['x_1,x_2,x_3']

but  then you may encounter specialiation problems with denominators, 
another problem is that your coefficients will probably be huge.

If you are interested in finite fields, you may also try to put the 
parameters a_i as variables and, moreover, add to your ideal the 
polynomials  a_i^(2^d)-a_j to denote that a_i is an arbitrary element of 
the ground field GF(2^d).

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