I was wondering if the following problem has been studied and if so where I could find work on it:
Given a set of integer solutions S. Is there a polynomial with integer coefficients that has exactly the set of solutions S?
Obviously there are a lot of other related questions. As in, is the given set the exact set or not (i.e. can the polynomial have solutions that are not in the set etc.). However, I was not able to find any body of work about this topic which makes me think it might be trivially decidable. In this case it would be nice if someone could point me to some work.
Actually, this was solved at exactly the same time as Hilbert's 10th problem. See for instance https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's_theorem, which states that Diophantine sets are exactly the same as recursively enumerable sets (sets which can be emitted in their entirety, not necessarily in ascending order, by a Turing machine if allowed to run forever).
You could quibble about the use of Diophantine sets for encoding "the set of solutions $S$", but it's a reasonable interpretation.
There is a body of work on trying to find specific polynomials that encode for "interesting" sets $S$. For instance, you could try http://www.math.umd.edu/~laskow/Pubs/713/Diorepofprimes.pdf, or just search on the keyword "Diophantine representation"