This question is a bit vague by design. Let $F$ be a field. I'm mostly interested in finite fields, but would also be interested in $R$ or $C$.
Let $S \subset F^d$ be an algebraic set and let $A = A_1 \times A_2 \times \ldots \times A_d$ be a product set in $F^d$. I'd like to know of any results that give upper bounds on the size of $A \cap S$.
I would expect any theorem here will have dependencies on the degree and complexity of the algebraic set and one probably has to exclude some degenerate cases.
One can do a bit better than the completely trivial estimate, just from the factor theorem. If S has a fixed degree, by fixing all but one coordinate, it isn't hard to see that $|A \cap S | \lesssim \frac{\prod_i |A_i|}{\max_i |A_i|}$, but I'd imagine one can do much better than this in many situations.
