User blog comment:Nayuta Ito/Calculating values of SIR function, by inventor himself./@comment-1605058-20171001092316

There is a slightly quicker argument that \(SID(1,n,d)\leq d\): we may assume that 0 is not a root, so the polynomial is \(a_mx^m+\dots+a_1x+a_0,a_m,a_0\neq 0\). It follows from the rational root theorem that any integer root divides \(a_0\), hence its absolute value is at most \(|a_0|\leq d\).

For two-variable polynomials, I agree things are getting supercalifragilisticexpialidociously difficult. Degree 2 case is not terribly difficult and can be done elementarily. Degree 3 case (which is basically equivalent to elliptic curves case) is computable using a result due to Baker. For degree 4 I think things also reduce to elliptic curves, but higher degree is nearly hopeless (we know there usually are only finitely many solutions, but don't know how to find them), but I think it is believed that the answers are still computable in this case.

And then there are higher numbers of variables where, outside some isolated cases, we know literally nothing...