On a Jacobian Identity Associated with Real Hyperplane Arrangements

Abstract

For x ∈ (aj-1, aj) (j = 1, . . . ,p + 1; a0: = -∞, ap+1:= ∞) the mapping Tj: w = x - Σpl=1 λl/(x - al) (λ > 0, al ∈ R) is onto R It was shown by G. Boole in the 1850's that Σp+1j=1[(∂w/∂x)-1] x=T-1j(w) = 1. We give an n-dimensional analogue of this result. The proof makes use of the Griffiths-Harris residue theorem from algebraic geometry.