THE CHARACTERISTIC POLYNOMIAL OF A MULTIARRANGEMENT

By Takuro Abe, Hiroaki Terao, Max Wakefield. 2006

54 citations recorded by the source corpus.

Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arragnement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates ‘global’ data to ‘local’ data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.

Subfields: Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/4ba84c0c760b34a40cc94f454987c7df43f23e38

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In Electrical and Electronic Engineering

1. Takuro Abe
2. Masahiko Yoshinaga

Citations in the corpus

• Divisionally free arrangements of hyperplanes (2015, methodology): “10 (Local-global formula, [7], Theorem 3.”
• Freeness of hyperplane arrangements and related topics (2012, background): “There is a notion of the characteristic polynomial of a multiarrangement (A,m) [3].”