# Generalization of a theorem of Mahler to higher dimensions

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.

• 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). – Ofir Gorodetsky Dec 30 '15 at 7:49
• @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 – Stanley Yao Xiao Dec 30 '15 at 14:40

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