As the title says, over the rationals, are there unconditional results for boundedness by the degree of $f(x,y)$ of the number of finitely many rational points on $f(x,y)=n$ for all rational $n$?

$f(x,y)$ must depend on both $x,y$.

Searching the web didn't work for me.

Major rewrite due to comments.

Let $f(x,y) \in \mathbb{Q}[x,y]$ and $f$ depends on both $x,y$.

Q1 Is it possible the number of rational solutions to $f(x,y)=n$ to be uniformly bounded for all rational $n$?

Looking for unconditional answer.

It is conjectured that polynomial injection $\mathbb{Q}^2 \to \mathbb{Q}$ exists and proving this will give bound $1$ (there are suspected injections).

As the title says, over the rationals, are there unconditional results for boundedness by the degree of $f(x,y)$ of the number of rational points on $f(x,y)=n$ for all rational $n$?

$f(x,y)$ must depend on both $x,y$.

Searching the web didn't work for me.

As the title says, over the rationals, are there unconditional results for boundedness by the degree of $f(x,y)$ of the number of finitely many rational points on $f(x,y)=n$ for all rational $n$?

$f(x,y)$ must depend on both $x,y$.

Searching the web didn't work for me.