Isoperimetric Inequalities for Volumes Associated with Decomposable Forms

Abstract

We give sharp estimates for volumes in ℝn defined by decomposable forms. In particular, we show that if F(X1…, Xn) = Π i = 1 d (αi1X1 + … + αinXn) is a decomposable form with αij ∈ C, degree d > n, and discriminant DF ≠ 0, and if VF is the volume of the region {x∈ℝn:|F(x)| ⩽ 1}, then |DF|(d−n)!/d! VF ⩽ Cn, where Cn is the value of |DF|(d−n)!/d! VF when F(X1…, Xn) = X1… Xn(X1 +… + Xn); moreover, we show that the sequence {Cn} is asymptotic to (2/π)e1−γ(2n)n. These results generalize work of the first author on binary forms and will likely find application in the enumeration of solutions of decomposable form inequalities.