Talk:Ehrhart polynomial

The polytope being "integral" means that all its corners are lattice points? AxelBoldt 20:03, 16 Feb 2004 (UTC)

Anyway, I blanked the page since it was a copyright violation; the text was a verbatim copy of http://mathworld.wolfram.com/EhrhartPolynomial.html . Somebody should rephrase that material and write it up again. AxelBoldt 02:25, 18 Feb 2004 (UTC)

Ehrhart's theorem for non-convex polytopes[edit]

Usually Ehrhart's theorem is stated for convex polytopes, but here convexity is not stated. Is that correct? McKay 16:17, 4 August 2006 (UTC)[reply]

Good point--the polynomiality theorem still holds, but Ehrhart reciprocity needs a stronger condition, e.g., the polytope being homeomorphic to a manifold is good enough. For simplicity, I inserted the condition that the polytope is convex. Mattbeck 04:18, 14 August 2006 (UTC)[reply]

Thanks. Do you happen to know where more general conditions are investigated? McKay 08:01, 15 August 2006 (UTC)[reply]

Try Peter McMullen's papers, e.g., Valuations and Euler-type relations on certain classes of convex polytopes, Proc. London Math. Soc. (3) 35, no. 1 (1977), 113-135, or Lattice invariant valuations on rational polytopes, Arch. Math. 31, no. 5 (1978), 509-516. Another good starting point might be Richard Stanley, Combinatorial reciprocity theorems, Advances in Math. 14 (1974), 194-253. Mattbeck 23:43, 15 August 2006 (UTC)[reply]

Thanks, I'll check those out. --McKay 04:18, 16 August 2006 (UTC)[reply]

please can you define L(P,-t)? —Preceding unsigned comment added by Flandre (talk • contribs) 15:24, 7 May 2009 (UTC)[reply]

L(P,t) is a priori only defined for positive integers t. Ehrhart proved that L(P,t) is a quasipolynomial (if P is a rational polytope), and this quasipolynomial can be evaluated at other inputs, e.g., negative integers. Thus Ehrhart's reciprocity theorem gives an algebraic relation between two quasipolynomials, i.e., L(P,-t) is just an algebraic expression (it's the quasipolynomial L(P,t) where we plug in -t for t). Mattbeck (talk) 06:59, 14 September 2009 (UTC)[reply]

is the volume of P > the number of integer interior points? (if P has 01 vertices) —Preceding unsigned comment added by Flandre (talk • contribs) 15:12, 4 May 2009 (UTC)[reply]

Example image[edit]

Partially truncated cube (see here for the part cut away)

This polyhedron would probably make sense as an example.
The volume is 4³ - 18 = 46. The number of contained points is 5³ - 24 = 101. --Watchduck (quack) 19:41, 8 October 2021 (UTC)[reply]