In the final step in Coppersmith technique we have $n$ polynomials (possibly non-homogeneous) in $\mathbb Z[x_1,\dots,x_m]$ where $m\leq n$ and using elimination theory we extract the common integer roots of these $n$ polynomials.

Can we use resultants here? As I understand resultants are for algebraically closed fields and $\mathbb Z$ is not even a field. How does Coppersmith's technique work to obtain integer roots? Is there some variation here?

Is there any other technique applicable here? What are some good comprehensive explicit example friendly references to know more on this?

finitely manysolutions. Is it the case? Otherwise, your question amounts to find the integral points on some algebraic variety, which is hard even in the case of curves. So unless you provide more details, I cannot answer your question. $\endgroup$ – François Brunault Jan 8 at 19:07