Periodicity of hyperplane arrangements with integral coefficients modulo positive integers
Hidehiko Kamiya1
, Akimichi Takemura2
and Hiroaki Terao3
1Okayama University Faculty of Economics Okayama Japan
2University of Tokyo Graduate School of Information Science and Technology Tokyo Japan
3Hokkaido University Department of Mathematics Hokkaido Japan
2University of Tokyo Graduate School of Information Science and Technology Tokyo Japan
3Hokkaido University Department of Mathematics Hokkaido Japan
DOI: 10.1007/s10801-007-0091-2
Abstract
We study central hyperplane arrangements with integral coefficients modulo positive integers q. We prove that the cardinality of the complement of the hyperplanes is a quasi-polynomial in two ways, first via the theory of elementary divisors and then via the theory of the Ehrhart quasi-polynomials. This result is useful for determining the characteristic polynomial of the corresponding real arrangement. With the former approach, we also prove that intersection lattices modulo q are periodic except for a finite number of q's.
Pages: 317–330
Keywords: keywords characteristic polynomial; Ehrhart quasi-polynomial; elementary divisor; hyperplane arrangement; intersection lattice
Full Text: PDF
References
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2. Athanasiadis, C.A.: Extended Linial hyperplane arrangements for root systems and a conjecture of Postnikov and Stanley. J. Algebr. Comb. 10, 207-225 (1999)
3. Athanasiadis, C.A.: Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes. Bull. Lond. Math. Soc. 36, 294-302 (2004)
4. Athanasiadis, C.A.: A combinatorial reciprocity theorem for hyperplane arrangements. arXiv:math.CO/0610482v1; Can. Math. Bull. (to appear)
5. Beck, M., Robins, S.: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Springer, Berlin (2007)
6. Beck, M., Zaslavsky, T.: Inside-out polytopes. Adv. Math. 205, 134-162 (2006)
7. Björner, A., Ekedahl, T.: Subspace arrangements over finite fields: cohomological and enumerative aspects. Adv. Math. 129, 159-187 (1997)
8. Blass, A., Sagan, B.: Characteristic and Ehrhart polynomials. J. Algebr. Comb. 7, 115-126 (1998)
9. Crapo, H., Rota, G.-C.: On the Foundations of Combinatorial Theory: Combinatorial Geometries.