Decomposable form inequalities

الأصل · EN

We consider Diophantine inequalities of the kind |f(x)| ≤ m, where F(X) ∈ Z[X] is a homogeneous polynomial which can be expressed as a product of d homogeneous linear forms in n variables with complex coefficients and m≥ 1. We say such a form is of finite type if the total volume of all real solutions to this inequality is finite and if, for every n'-dimensional subspace S Rⁿ defined over Q, the corresponding n'-dimensional volume for F restricted to S is also finite. We show that the number of integral solutions x ∈ Zⁿ to our inequality above is finite for all m if and only if the form F is of finite type. When F is of finite type, we show that the number of integral solutions is estimated asymptotically as m→ ∞ by the total number of integral solutions is estimated asymptotically as m→ ∞ by the total volume of all real solutions. This generalizes a previous result due to Mahler for the case n=2. Further, we prove a conjecture of W. M. Schmidt, showing that for F of finite type the number of integral solutions is bounded above by c(n,d)m(n/d), where c(n,d) is an effectively computable constant depending only on n and d.