I was originally going to ask this question on MO, but it's unfortunately a little vague and is not a thorough presentation of any of my own work.
Let $f(x,y)$ be a polynomial with integer coefficients. Under what conditions does $f(x,y)=0$ have no solutions $(a,b) \in \mathbb{Z} \times \mathbb{Z}$? I understand this question is deceivingly difficult and is essentially asking about rational points on projective varieties. I have made no progress further than the trivial cases; for instance, any horizontal shift or integer vertical shifts of: $$f(x,y)=x^2y+y^2x+3y^3-y-1$$ Is one such set of functions satisfying my condition, although does not provide nearly any insight into the general solution to this problem.
My question: while not a specific 'reference request,' I would like to know of any electronically published/readily available work done on this or a similar problem. Or, if anyone could provide insight toward approaching this problem, that would also be appreciated.