Problem:
let $P_1(x), P_2(x), Q_1(y), Q_2(y)$ be some polynomials of degree $d$ in $\mathbb{F}_p$. Let \begin{equation} A := \{ (x, y) \in \mathbb{F}_p^2 : P_1(x) = Q_1(y) \},\\ B := \{ (x, y) \in \mathbb{F}_p^2 : P_2(x) = Q_2(y) \}. \end{equation} Let us also assume that equations $P_1(x) = Q_1(y)$ and $P_2(x) = Q_2(y)$ are independent in certain sense: for example, polynomials $P_1(x) - Q_1(y)$ and $P_2(x) - Q_2(y)$ do not have common divisors. Can one imply that $$|A \cap B| \ll \max{(|A|, |B|)} ?$$
This is clear that each of the equations $P(x) = Q(y)$ may have at most $dp$ solutions, and this is nearly attained when, for example, $P(x) = x^d + 3, Q(y) = y^d + 3$, where $d | p - 1$.
There is the following variation of Bombieri's result by Chalk and Smith (see Theorem 2 in here, page 202):
Theorem. Let $(b_1, b_2) \in \mathbb{F}_p^2$ be nonzero and $f(x, y) ∈ \mathbb{F}_p[x, y]$ be a polynomial of degree $d \geqslant 1$ with the following property: there is no $c \in \mathbb{F}_p$ for which the polynomial $f(x, y)$ is divisible by $b_1x + b_2y + c$. Then $$ \bigg| \sum_{(x, y):f(x,y)=0} e^{2\pi i(b_1x+b_2y)/p} \bigg| \leqslant 2d^2p^{1/2}. $$
This is roughly saying, that as long as $P(x) - Q(y)$ does not have any linear divisors, then sets $A, B$ are equidistributed over $\mathbb{F_p} \times \mathbb{F_p}$.
This gives an intuition, that sets $A$ and $B$ should be somehow independent, and probably even an inequality $|A \cap B| \ll |A||B|/p^2 \ll d^2$ could have place, upon a certain condition of independence of polynomials $P_1, P_2, Q_1, Q_2$.
Are there any known results of this type?
