منابع مشابه
Primitive derivations, Shi arrangements and Bernoulli polynomials
LetW be a finite irreducible real reflection group, which is a Coxeter group. A primitive derivation D, introduced and studied by K. Saito (e.g., [4]), plays a crucial role in the theory of differential forms with logarithmic poles along the Coxeter arrangement. For example, we may describe the contact order filtration of the logarithmic derivation module using the primitive derivations ([10, 1...
متن کاملLattice point counts for the Shi arrangement and other affinographic hyperplane arrangements
Hyperplanes of the form xj = xi + c are called affinographic. For an affinographic hyperplane arrangement in Rn, such as the Shi arrangement, we study the function f(m) that counts integral points in [1,m]n that do not lie in any hyperplane of the arrangement. We show that f(m) is a piecewise polynomial function of positive integers m, composed of terms that appear gradually as m increases. Our...
متن کاملRoot Arrangements of Hyperbolic Polynomial-like Functions
A real polynomial P of degree n in one real variable is hyperbolic if its roots are all real. A real-valued function P is called a hyperbolic polynomial-like function (HPLF) of degree n if it has n real zeros and P (n) vanishes nowhere. Denote by x (i) k the roots of P , k = 1, . . . , n− i, i = 0, . . . , n− 1. Then in the absence of any equality of the form x (j) i = x (l) k (1) one has ∀i < ...
متن کاملSpanning trees in complete uniform hypergraphs and a connection to extended Shi hyperplane arrangements
We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r ≥ 3. Similar to the bijection between spanning trees in complete graphs and Parking functions, we derive a bijection from spanning trees of the complete (r + 1)-uniform hypergraph which arise from a fixed r-perfect matching (see Section 2) and r-Parking functions. We observe a simp...
متن کاملCounting External Facets of Simple Hyperplane Arrangements
The number of external facets of a simple arrangement depends on its combinatorial type. A computation framework for counting the number of external facets is introduced and improved by exploiting the combinatorial structure of the set of sign vectors of the cells of the arrangement. 1 Background and introduction n hyperplanes in dimension d form a hyperplane arrangement. An hyperplane arrangem...
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ژورنال
عنوان ژورنال: Journal of Algebra
سال: 2015
ISSN: 0021-8693
DOI: 10.1016/j.jalgebra.2014.09.011