Lattice invariant valuations on rational polytopes
Tóm tắt
LetΛ be a lattice ind-dimensional euclidean space
$$\mathbb{E}^d $$
, and
$$\bar \Lambda $$
the rational vector space it generates. Ifϕ is a valuation invariant underΛ, andP is a polytope with vertices in
$$\bar \Lambda $$
, then for non-negative integersn there is an expression
$$\varphi (n P) = \sum\limits_{r = 0}^d {n^r \varphi _r } (P, n)$$
, where the coefficientϕ(P, n) depends only on the congruence class ofn modulo the smallest positive integerk such that the affine hull of eachr-face ofk P is spanned by points ofΛ. Moreover,ϕ
r satisfies the Euler-type relation
$$\sum\limits_F {( - 1)^{\dim F} } \varphi _r (F, n) = ( - 1)^r \varphi _r ( - P, - n)$$
where the sum extends over all non-empty facesF ofP. The proof involves a specific representation of simple such valuations, analogous to Hadwiger's representation of weakly continuous valuations on alld-polytopes. An example of particular interest is the lattice-point enumeratorG, whereG(P) = card(P∩λ); the results of this paper confirm conjectures of Ehrhart concerningG.
Tài liệu tham khảo
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