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On a Jacobian Identity Associated with Real Hyperplane Arrangements

Published online by Cambridge University Press:  04 December 2007

Kazuhiko Aomoto  and
Peter J. Forrester
Kazuhiko Aomoto
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602.
Peter J. Forrester
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia.
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Abstract

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For $x\in (a_{j-1}, a_j)\ (j=1,\ldots, p+1;\ a_0\!:=-\infty, \ a_{p+1} \!:=\infty)$ the mapping $T_j\!: w=x-\sum ^p_{l=1}\lambda _l/(x-a_l)\ (\lambda _l$>$0, \ a_l\in$R) is onto R. It was shown by G. Boole in the 1850's that $\sum ^{p+1} _{j=1}[(\partial w/\partial x) ^{-1}] _{x=T^{-1}_j(w)}=1.$ We give an n-dimensional analogue of this result. The proof makes use of the Griffiths–Harris residue theorem from algebraic geometry.

Keywords

hyperplane arrangementsJacobiansGriffiths–Harris residue theorem
Type
Research Article
Information
Compositio Mathematica , Volume 121 , Issue 3 , May 2000 , pp. 263 - 295
Copyright
© 2000 Kluwer Academic Publishers