Lattice-point generating functions for free sums of polytopes

Transcription

1 Lattice-point generating functions for free sums of polytopes Tyrrell B. McAllister 1 Joint work with Matthias Beck 2 and Pallavi Jayawant 3 1 University of Wyoming 2 San Francisco State University 3 Bates College The University of Akron 20 October 2012 T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

2 Introduction Free Sums Notation: For S R n, let S Z = S Z n. Let P, Q R n be polytopes. P Q := conv(p Q) is a free sum if Example 0 P Q, span P span Q = R n as a direct sum of vector spaces, and (span P) Z (span Q) Z = Z n as a direct sum of lattices. 0 0 = 0 T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

3 Introduction Free Sums P Q is related to familiar operations: Dual to cartesian product: Let P be the polar dual of P (with respect to span(p)). Then (P Q) = P Q. Homogenizes as Minkowski sum: Let cone(p) be the positive span of P {1} R n+1. Then cone(p Q) = cone(p) + cone(q). T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

4 Introduction Ehrhart Theory P R n is rational (resp., lattice) if all vertices are in Q n (resp., Z n ). The denominator of P is den(p) := min {k Z 1 : kp is lattice}. The Ehrhart series of P is the formal power series Theorem (E. Ehrhart 1962) Ehr P (t) := (kp) Z t k. k=0 Ehr P (t) is a rational function of the form Ehr P (t) = where δ P (t) is the δ-polynomial of P. δ P (t) (1 t den(p) ) dim(p)+1, T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

5 Introduction Braun s Theorem Let V be the dual space of V := span(p). The polar dual of P is P := {ϕ V : ϕ(p) R 1 }. Let V Z := {ϕ V : ϕ(v Z ) Z} be the dual lattice of V Z. P is a lattice polyhedron in V if the vertices of P are in V Z. A polytope P is reflexive if P and P are both lattice polytopes. T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

6 Braun s Theorem Recall the δ-polynomial: Ehr P (t) = (kp) Z t k = k=0 δ P (t) (1 t den(p) ) dim(p)+1. Theorem (B. Braun 2006) Let P Q be a free sum. If P is a reflexive polytope and Q is a lattice polytope with 0 Q, then δ P Q (t) = δ P (t) δ Q (t). In terms of Ehrhart series, Braun s formula says Ehr P Q (t) = (1 t) Ehr P (t) Ehr Q (t). T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

7 Introduction cone(p) Given polytope P R n, let cone(p) R n+1 be the union of rays through P {1}. P {1} P T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

8 Introduction Lattice-point Generating Functions Let σ cone(p) (x) be the lattice-point generating function of cone(p): σ cone(p) (x) = σ cone(p) (x 1,..., x n+1 ) = where x m := x m 1 1 x m n+1 n+1. m cone(p) Z x m, The Ehrhart series is a specialization of the lattice-point generating function: Ehr P (t) = σ cone P (1,..., 1, t). T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

9 First Main Result Recall Braun s formula: If P Q is a free sum with P reflexive and Q a lattice polytope such that 0 Q, then Ehr P Q (t) = (1 t) Ehr P (t) Ehr Q (t). Theorem (Beck, M, Pallavi) Let P Q R n be a free sum of rational polytopes. Then σ cone(p Q) (x) = (1 x n+1 ) σ cone P (x) σ cone Q (x) if and only if either P or Q is a lattice polyhedron. Implies Braun s theorem. Condition is necessary and sufficient. Doesn t require P or Q to be lattice. 0 can be on boundary of P or Q. T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

10 Second Main Result What if P and Q are rational, but neither P nor Q is a lattice polyhedron? For integers i 0, define the shifted cones Theorem (Beck, M, Pallavi) cone i i P := cone P + den(p ) e n+1, i cone i Q := cone Q den(p ) e n+1. Let P Q be a free sum of rational polytopes. Then σ cone(p Q) = den(p ) 1 i=0 (σ cone i P σ cone i+1 P) σ conei Q. T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

11 Affine Free Sums What if P Q isn t the origin? Suppose now that P, Q R n are polytopes such that P Q = aff(p) aff(q) = {p}, p Q n. Previously, P Q was a free sum if Z n = (span P) Z (span Q) Z (lattice direct sum). Let Λ p R n be the lattice generated by Z n {p}. For S R n, let S Λ p := S Λ p. P Q := conv(p Q) is an affine free sum of P and Q if Λ p = span(p p) Λ p span(q p) Λ p. T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

12 Third Main Result Gorenstein polytopes A lattice polytope P is Gorenstein of index k if there exists a (unique) point m (kp) Z such that kp m is reflexive. Theorem (Beck, M., Pallavi) Let polytope P be Gorenstein of index k with kp m reflexive. Let Q R n be a polytope containing 1 k m P such that P Q is an affine free sum. Then ( σ cone(p Q) (x) = 1 x (m,k)) σ cone P (x) σ cone Q (x), where x (m,k) := x m 1 1 x mn n x k n+1. T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

13 Methods Lattice-theory Lemmas Recall: cone(p Q) = cone(p) + cone(q) Equivalently, cone(p Q) = (x + cone Q). x cone P The lower envelope of cone(p) is the subset of cone(p) that is vertically minimal. The lower lattice envelope Z cone(p) is the subset of points in cone(p) that are directly beneath lattice points. P {1} P T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

14 Methods Lattice-theory Lemmas Proposition Let P Q be a free sum of polytopes. Then cone(p Q) Z = (m + cone Q) Z, where denotes disjoint union. m Z cone P Q P T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

15 Methods Lattice-theory Lemmas Proposition Let P R n be a rational polytope containing the origin. Then the following are equivalent: P is a lattice polyhedron, Z cone P = ( cone P) Z, ( cone P) Z = (cone P) Z \ (cone P + e n+1 ) Z, σ cone P (x) = (1 x n+1 ) σ cone P (x). T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15

16 Open Questions 1 Is it possible to have Z cone P = ( cone P) Z when P is a convex set other than a rational polytope? 2 Dual to lattice was the right generalization of reflexive. What is the right generalization of Gorenstein? (Natural guess doesn t work.) T. B. McAllister (U. of Wyoming) Free sums of convex sets U. of Akron, 20 Oct / 15