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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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