It is a long standing problem to investigate whether irreducible integral polynomials not divisible by a fixed square integer assumes square-free values infinitely often. The result is known conditioned on the $abc$-conjecture (due to Granville), and the best unconditional results are for $k$-free values of polynomials. That is, one can prove that an irreducible polynomial with no fixed $k$-th power divisor of degree $d$ assumes infinitely many $k$-free values if $k$ is sufficiently large with respect to $d$. For the single variable case, it is known that an irreducible polynomial $f(x)$ of degree $d$ with no fixed $k$-th power divisor assumes infinitely many $k$-free values as soon as $k > 3d/4$ (due to Heath-Brown and Browning). For the two variable case, in general we have the bound $k/d > 39/64$, due to Browning, and in the homogeneous case we have the estimate $k/d > 7/16$, also due to Browning.

A related problem is to investigate when an integral polynomial (say, of one or two variables) assumes values with very few prime factors (including multiplicity). This is a more stringent condition since many $k$-free values for example may have many distinct prime factors.

It appears to be known that in general an irreducible polynomial $f(x) \in \mathbb{Z}[x]$ assumes infinitely many values in $P_{d + 1}$, where $P_n$ denotes the set of numbers with at most $n$ prime factors (not necessarily distinct).

Is there a similar result for binary polynomials?