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@DanielLoughran Just out of curiosity, would considering rational solutions instead of integral ones make the problem much easier? Do you know of any results (or similar) in the case of rational solutions? thanks again.
Yes, I've tried looking at the literature a bit, but all seems geared towards forms instead of just polynomials. Do you know any references which might deal with a case similar to mine (i.e. with polys instead of forms)? Thanks!