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?



Polynomials with few prime factors
1) The polynomial in two variables $f(x,y)=2(x^2-2y^2)$ will take the prime
value $p=2$ infinitely often (Pell's equation).


2) Nontrivial results in the direction of your question
are due to Gihan Marasingha:


a) Almost primes represented by binary forms. J. Lond. Math. Soc. (2) 82 (2010), no. 2, 295–316.


b) On the representation of almost primes by pairs of quadratic forms.
Acta Arith. 124 (2006), no. 4, 327–355.


