The density of integers represented by a binary form

Question

Suppose that $F(x,y)$ is a binary form of degree $d \geq 3$ with integral coefficients, and non-zero discriminant. It is known (from a paper due to Erdős and Mahler from 1938) that the density of integers in $[1,B]$ which can be represented by $F$ is $\Theta(B^{2/d})$ as $B \rightarrow \infty$, provided that $F(1,0) > 0$.

It is not too much of a leap to guess that we can replace $\Theta(B^{2/d})$ with an asymptotic formula, i.e. there should exist a positive constant $C_F$ such that $$\displaystyle \#\{t \in \mathbb{N} \cap [1,B]: \exists (x,y) \in \mathbb{Z}^2 \text{ s.t. } F(x,y) = t \} \sim C_F B^{2/d}.$$

This fact has been confirmed by C. Hooley for the case $d = 3$ and $$F(x,y) = ax^3 + 3bx^2 y + 3cxy^2 + dy^3$$ irreducible in two separate papers, the first in 1967 and the second in 2000. In the latter, he obtains the constant

$$\displaystyle C_F = \left(1 - \frac{3}{2 \Delta'}\right) \frac{\sqrt[3]{3}\Gamma(1/3)^2}{2 \sqrt[3]{\Delta} \Gamma(2/3)}, $$

where $\Delta$ is an integer such that $D(F) = 81 \Delta^2$, where $D(F)$ is the discriminant of $F$, and $\Delta' = \Delta/G$, where $G = \gcd(b^2 - ac, bc - ad, c^2 - bd)$.

Hooley's work seems to indicate that $C_F$ should exist in general, but probably not very easy to compute. I have not seen any later works where $C_F$ (if it exists) is worked out explicitly.

Has there been further work on investigating what $C_F$ is in other cases, in particular cases with $d \geq 4$? If not, are there any heuristics as to what it should be in general?

Answer 1

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.

Answer 2

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.