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Decomposable form inequalities. (English) Zbl 1058.11028
Let \(L_i({\mathbf x})\in\mathbb{C}[x_1,\dots x_n]\) be linear forms for \(1\leq i\leq d\), and let \(F({\mathbf x})=\prod^d_1 L_i({\mathbf x})\). Write \[ V(F)= \text{Vol}\{{\mathbf x}\in \mathbb{R}^n:| F({\mathbf x})|\leq 1\}. \] The first result states that if \(V(F)\) is finite then \(V(F)\ll_{n,d} 1\), providing that \(F\) does not vanish at any nonzero integer point. Let \(N_F(m)\) denote the number of integral solutions \({\mathbf x}\) of the inequality \(| F({\mathbf x})|\leq m\). To describe the behaviour of \(N_F(m)\) the natural condition on \(F\) is that it should be of ‘finite type’. By this it is meant that, if \(F\) is restricted to any rational subspace, then the volume analogous to \(V(F)\) is finite.
The second theorem then states that \(N_F(m)\) is finite for all \(m\) if and only if \(F\) is of finite type, and in this case one has \[ N_F(m)\ll_{n,d} m^{n/d}. \] Finally it is shown that \(N_F(m)\sim V(F)m^{n/d}\) as \(m\to\infty\), if \(F\) is of finite type. The error term in the asymptotic formula is given explicitly, and depends on the height of \(F\).
These results dramatically extend work of K. Mahler [Acta Math. 62, 91–166 (1934; Zbl 0008.19801; JFM 60.0159.04)] and W. M. Schmidt [Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics, 1467. Springer-Verlag, Berlin (1991; Zbl 0754.11020)] for the case \(n= 2\).

MSC:
11D75 Diophantine inequalities
11D45 Counting solutions of Diophantine equations
11D57 Multiplicative and norm form equations
11J25 Diophantine inequalities
11D72 Diophantine equations in many variables
11H46 Products of linear forms
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