On the norm form inequality \(|F(x)|\leq h\) (Q2707220)

Let \(F(X)\in \mathbb{Z}[X_1,\dots , X_n]\) be a non-degenerate norm form of degree \(r\). For a positive real number \(h\) let \(Z_F(h)\) denote the number of solutions \(x\in \mathbb{Z}^n\) of the norm form inequality (*) \(|F(x)|\leq h\). By a result of \textit{W. M. Schmidt} [Math. Ann. 191, 1-20 (1971; Zbl 0207.35402)] \(Z_F(h)\) is finite for any \(h>0\). The main result of this paper is the inequality NEWLINE\[NEWLINEZ_F(h) \leq (16r)^{{1 \over 3}(n+11)^3}\cdot h^{(n + \sum_{m=2}^{n-1} {1 \over m})/r } \cdot (1 + \log h)^{n(n-1)/2}.NEWLINE\]NEWLINE The proof of this inequality is based on a sharp estimation of the number of proper linear subspaces of \(\mathbb{Q}^n\), whose union contains all solutions of (*).