Volumes and Diophantine inequalities associated with decomposable forms.
The author considers Diophantine inequalities of the shape (*) \(| F({\mathbf x})| \leq m\) in \({\mathbf x}\in{\mathbb Z}^n\) where \(F\in{\mathbb Z}[X_1,\ldots ,X_n]\) is a decomposable form, that is, \(F\) factors over the complex numbers into linear forms. Suppose \(F\) has degree \(d\). Denote by \(N(F,m)\) the number of solutions of (*) and by \(V(F)\) the \(n\)-dimensional volume of the set of \({\mathbf x}\in{\mathbb R}^n\) with \(| F({\mathbf x}| \leq 1\). In a previous paper [Ann. Math. (2) 153, 767--804 (2001; Zbl 1058.11028)] the author proved that if \(F\) is `of finite type' then \(| N(F,m)| -m^{d/n}V(F)| \leq C(F)m^{a(n)}\) where \(C(F)\) is some explicit constant depending on \(F\) and \(a(n)<n/d\) and moreover that \(N(F,m)\leq c(n,d)m^{n/d}\) where \(c(n,d)\) depends only on \(n\), \(d\). In the present paper the author introduces some other quantities related to \(F\), and compares these with \(V(F)\). Further, he derives upper bounds for \(N(F,m)\) in terms of these other quantities. In his arguments he uses Schmidt's Subspace Theorem, as well as techniques from the geometry of numbers. For instance, the quantity \(Q(F)\) is defined as follows. Consider all factorizations of \(F\) in the form \(F({\mathbf X})=L_1({\mathbf X})\cdots L_d({\mathbf X})\) where the first \(r\) factors \(L_1,\ldots, L_r\) are linear forms with real coefficients, the other \(2s=d-r\) factors are linear forms with non-real complex coefficients and \(L_{i+s}\) is the complex conjugate of \(L_i\) for \(i=r+1,\ldots ,r+s\). Define \(Q(F)\) as the length \(| | L_1\wedge\cdots \wedge L_d | | \) of the exterior product of the coefficient vectors of \(L_1,\ldots , L_d\), where the factorization \(L_1,\ldots ,L_d\) has been chosen such that this length is minimal. Further, let \({\mathcal H}(F)\) be the minimum of the heights of all decomposable forms equivalent to \(F\), i.e. the minimum of the heights of the forms \(F(A{\mathbf X})\) with \(A\in\text{ GL}_n({\mathbf Z})\). Among other things the author shows that if \(F\) satisfies some suitable non-degeneracy criterion, that \(Q(F)\gg {\mathcal H}(F)^{1/d}\) and \[ {\mathcal H}(F)^{-n/d}\ll V(F)\ll {\mathcal H}(F)^{-1/d}(1+(\log {\mathcal H}(F))^{n-1}, \] where constants implied by \(\ll\), \(\gg\) depend only on \(n,d\). Further he shows that for all \(m\geq 1\), \[ N(F,m)\ll m^{n/d}{\mathcal H}(F)^{-1/d} (1+(\log {\mathcal H}(F))^{n-1}+ m^{(n-1)/d}(1+(\log m)^{n-1}+ (\log {\mathcal H}(F))^{n-1}). \]
scientific article; zbMATH DE number 1961837