Isoperimetric Inequalities for Volumes Associated with Decomposable Forms

We give sharp estimates for volumes in ℝ n defined by decomposable forms. In particular, we show that if F ( X 1 …, X n ) = Π i = 1 d(α i1 X 1 + … + α in X n ) is a decomposable form with α ij ∈ C, degree d > n , and discriminant D F ≠ 0, and if V F is the volume of the region {x∈ℝ n :| F (x)| ⩽ 1}, then | D F | (d−n)!/d! V F ⩽ C n , where C n is the value of | D F | (d−n)!/d! V F when F ( X 1 …, X n ) = X 1 … X n ( X 1 +… + X n ); moreover, we show that the sequence { C n } is asymptotic to (2/π)e 1−γ (2 n ) 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.