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A seminal theorem of Kurt Mahler, in his papers Zur Approximation algebraischer Zahlen. I-III., is the following:

Let $F(x,y) \in \mathbb{Z}[x,y]$ be a binary form of degree $d \geq 3$ and non-zero discriminant $\Delta(F)$. Put $A_F$ for the area of the region

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

and put $N_F(Z)$ for the number of points $(x,y) \in \mathbb{Z}^2$ such that $|F(x,y)| \leq Z$ and $F(x,y) \ne 0$. Then he proved that

$\displaystyle N_F(Z) = A_F Z^{\frac{2}{d}} + O_F \left(Z^{\frac{1}{d-1}}\right).$

Has this statement been generalized to higher dimensions? The most conservative generalization would be to decomposable forms, meaning homogeneous polynomials in $x_1, \cdots, x_n$ with integer coefficients which factor completely into linear factors over $\mathbb{C}$, say. In particular, the set-up is as follows: Let

$F(x_1, \cdots, x_n) = L_1(\mathbf{x}) \cdots L_d(\mathbf{x})$

be a decomposable form with integer coefficients, where $L_j(\mathbf{x})$ are linear forms with coefficients in $\overline{\mathbb{Q}}$, and $d > n$. Put $V_F$ for the Lebesgue measure of the set

$\displaystyle \{(x_1, \cdots, x_n) \in \mathbb{R}^n : |F(x_1, \cdots, x_n)| \leq 1\}$

and $N_F(Z)$ for the number of points $(x_1, \cdots, x_n) \in \mathbb{Z}^n$ such that $|F(x_1, \cdots, x_n)| \leq Z, F(x_1, \cdots, x_n) \ne 0$. Then has it been proven that

$\displaystyle N_F(Z) = V_F Z^{\frac{n}{d}} + o_F\left(Z^{\frac{n}{d}}\right)$

in general?

I did a search of papers which cites Mahler's Zur Approximation algebraischer Zahlen. I and did not find any of them that deals with this problem (at least from looking at their titles and abstracts).

A more optimistic and aggressive generalization would be to consider general homogeneous polynomials in $n$ variables. I am not sure what additional complications could arise.

Any help would be appreciated.

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  • $\begingroup$ Where Mahler's proof seems to fail in proving your generalization? Also, a small remark: You want to assume $d>n$ (to avoid, for instance, the case where $F$ is the norm form of some number field, which implies $N_F$ is infinite - think of units). $\endgroup$ – Ofir Gorodetsky Dec 30 '15 at 7:49
  • $\begingroup$ @OfirGorodetsky Mahler's papers are in German, a language which I am not fluent in, so it has been very hard to read his papers and I am sure my understanding is not quite right in any case. Further, you are right, I meant to phrase my question around incomplete decomposable forms $\endgroup$ – Stanley Yao Xiao Dec 30 '15 at 14:40
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The first situation, with decomposable forms, has been addressed by Ramachandra in his paper "A lattice-point problem for norm forms in several variables", Journal of Number Theory 1 (1969), 534-555. His theorem only covers norm forms (i.e., irreducible decomposable forms) and requires the degree to be very large compared to the number of variables. J.L. Thunder improved his result, and indeed achieved a full generalization of Mahler's theorem, in 2001 in the paper "Decomposable form inequalities", Annals of Mathematics, 153 (2001), 767-804.

However, as far as I know, the situation for general homogeneous $F$ in many variables has not been addressed.

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