When all the input variables $a_i$ are restricted to $\{0,1\}$, how does one compute the number of solutions to equations like
$ a_1a_4a_2 + a_1a_5a_3 + a_4a_6a_3 = c $
where $c$ is a non-negative integer? The terms are not limited to trinary products, they could be greater. It is trivial to compute the above example by hand (or by enumerating with a computer), but insight to the solution process would be helpful.
Edit: In response to the comments that suggest this problem could be NP-hard, I'll accept the bounds on a solution as well.
${\bf \text{Edit}}_2 :$ Originally the problem stated that the question was over GF(2), but from the suggestions in the comments, the title of "0-1 non-linear integer program" is more accurate.