On Valuations, the Characteristic Polynomial, and Complex Subspace Arrangements

On Valuations, the Characteristic Polynomial, and Complex Subspace Arrangements

We present a new combinatorial method to determine the characteristic polynomial of any subspace arrangement that is defined over an infinite field, generalizing the work of Blass and Sagan. Our methods stem from the theory of valuations and Groemer's integral theorem. As a corollary of our main theorem, we obtain a result of Zaslavsky about the number of chambers of a real hyperplane arrangeme...

On Cohomology Algebras of Complex Subspace Arrangements

The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work wit...

Characteristic Polynomials of Subspace Arrangements and Finite Fields

Let A be any subspace arrangement in R defined over the integers and let Fq denote the finite field with q elements. Let q be a large prime. We prove that the characteristic polynomial /(A, q) of A counts the number of points in Fq that do not lie in any of the subspaces of A, viewed as subsets of Fq . This observation, which generalizes a theorem of Blass and Sagan about subarrangements of the...

A Geometric Interpretation of the Characteristic Polynomial of Reflection Arrangements

We consider projections of points onto fundamental chambers of finite real reflection groups. Our main result shows that for groups of type An, Bn, and Dn, the coefficients of the characteristic polynomial of the reflection arrangement are proportional to the spherical volumes of the sets of points that are projected onto faces of a given dimension. We also provide strong evidence that the same...

Cohomology Algebras of Complex Subspace Arrangements 3