The density of integers represented by a binary form

This question was open in general for degree $d \geq 5$ at the time this question was posted and in addition to the irreducible cubic case Hooley also dealt with the special case of bi-quadratic quartic forms in a paper in 1986 (bi-quadratic as in forms of the shape $Ax^4 + Bx^2y^2 + Cy^4$); see http://www.degruyter.com/view/j/crll.1986.issue-366/crll.1986.366.32/crll.1986.366.32.xml?format=INT .

The determination of the existence of the constant $C_F$ in general is done in the following joint paper of myself and Cam Stewart: http://arxiv.org/abs/1605.03427 . We showed that $C_F$ is equal to $W_F A_F$, where $A_F$ is the area of the region

$$\displaystyle \{(x,y) \in \mathbb{R}^2 : |F(x,y)| \leq 1 \}$$

and $W_F$ is a positive rational number which depends on the $\text{GL}_2(\mathbb{Q})$-automorphism group $\text{Aut}(F)$ of $F$.

However, in this paper we did not generalize Hooley's work entirely since we were not able to determine the value of $W_F$ as an explicit function of the coefficients of $F$. This is likely impossible to do when $d \geq 5$, in a similar way that the general quintic (and beyond) is unsolvable from a Galois theory perspective. However, much like the fact that degree 3 and 4 polynomials are always solvable, it is possible to determine $\text{Aut}(F)$ and therefore $W_F$ explicitly when $F$ is a binary cubic or quartic form. This is contained in an upcoming paper of mine.

For forms of the form $Ax^d - By^d$ this seems to have been done by Bennett, Dummigan, and Wooley:

M. A. Bennett, N. P. Dummigan, and T. D. Wooley, The representation of integers by binary additive forms, Compositio Math. 111 (1998), no. 1, 15--33.