Abstract
We define the level of a subset X of Euclidean space to be the dimension of the smallest subspace such that the distance between each element of X and the subspace is bounded. We prove that the number of faces in the n-dimensional extended Shi arrangement Aˆn r having codimension k and level m is given by m⋅(nk)⋅Δr kΔm−1xn−1|x=rn−1, where Δ is the difference operator and Δr is the difference operator of step r, that is, Δrp(x)=p(x)−p(x−r). This generalizes a result of Athanasiadis which counts the number of faces of different dimensions in the extended Shi arrangement Aˆn r. The proof relies on a multi-variated Abel identity due to Françon.
Original language | English |
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Pages (from-to) | 55-64 |
Number of pages | 10 |
Journal | Advances in Applied Mathematics |
Volume | 109 |
DOIs | |
State | Published - Aug 2019 |
Bibliographical note
Publisher Copyright:© 2019 Elsevier Inc.
ASJC Scopus subject areas
- Applied Mathematics