Lecture Notes on Lattice Polytopes (preliminary version of December 7, 2012) Winter 2012 Fall School on Polyhedral Combinatorics TU Darmstadt Christian Haase Benjamin Nill Andreas Paffenholz • • text on this page — prevents rotating Chapter Contents 1 Polytopes, Cones, and Lattices ..................................... 1 1.1 Cones........................................................2 1.2 Polytopes.....................................................6 1.3 Lattices....................................................... 10 Problems.......................................................... 18 2 An invitation to lattice polytopes .................................. 19 2.1 Lattice polytopes and unimodular equivalence................... 20 2.2 Lattice polygons............................................... 21 2.3 Volume of lattice polytopes..................................... 24 2.4 Problems..................................................... 26 3 Ehrhart Theory .................................................... 27 3.1 Motivation.................................................... 27 3.1.1 Why do we count lattice points?......................... 27 3.1.2 First Ehrhart polynomials............................... 29 3.2 Triangulations and Half-open Decompositions................... 30 3.3 EHRHART’s Theorem........................................... 33 3.3.1 Encoding Points in Cones: Generating Functions.......... 33 3.3.2 Counting Lattice Points in Polytopes..................... 37 3.3.3 Counting the Interior: Reciprocity....................... 41 3.3.4 Ehrhart polynomials of lattice polygons.................. 45 3.4 The Theorem of Brion......................................... 46 3.5 Computing the Ehrhart Polynomial: Barvinok’s Algorithm........ 48 3.5.1 Basic Version of the Algorithm........................... 49 3.5.2 A versatile tool: LLL.................................... 52 3.6 Problems..................................................... 57 4 Geometry of Numbers ............................................. 61 4.1 Minkowski’s Theorems......................................... 61 4.2 Lattice packing and covering................................... 64 4.3 The Flatness Theorem......................................... 67 4.4 Problems..................................................... 68 5 Reflexive and Gorenstein polytopes ................................ 69 5.1 Reflexive polytopes............................................ 69 5.1.1 Dimension 2 and the number 12......................... 71 5.1.2 Dimension 3 and the number 24......................... 72 5.2 Gorenstein polytopes.......................................... 74 5.3 The combinatorics of simplicial reflexive polytopes.............. 77 5.3.1 The maximal number of vertices......................... 77 — vii — viii 5.3.2 The free sum construction............................... 78 5.3.3 The addition property.................................. 78 5.3.4 Vertices between parallel facets.......................... 79 5.3.5 Special facets.......................................... 80 5.4 Problems..................................................... 81 6 Unimodular Triangulations ........................................ 83 6.1 Regular Triangulations......................................... 83 6.2 Pulling Triangulations......................................... 84 6.3 Compressed Polytopes......................................... 84 6.4 Special Simplices in Gorenstein Polytopes....................... 86 6.5 Dilations...................................................... 88 6.5.1 Composite Volume...................................... 88 6.5.2 Prime Volume.......................................... 89 6.6 Problems..................................................... 90 References ............................................................. 91 Index .................................................................. 93 Name Index ........................................................... 97 Polytopes, Cones, and Lattices 1 In this chapter we want to introduce the basic objects that we will look at for the rest of the semester. We will start with polyhedral cones, which are the intersection of a finite set of linear half spaces. Generalizing to intersections of affine half spaces leads to polyhedra. We are mainly interested in the subset of bounded polyhedra, the polytopes. Specializing further, we will deal with integral polytopes. We will not prove all theorems in this chapter. For more on polytopes you may consult the book of Ziegler [28]. In the second part of this chapter we link integral polytopes to lattices, discrete subgroups of the additive group Rd . This gives a connection to commutative al- gebra by interpreting a point v Zd as the exponent vector of a monomial in d variables. 2 We use Z, Q, R and C to denote the integer, rational, real and complex num- bers. We also use Z>, Z , Z<, Z , R<, R , R>, R . ≥ ≤ ≤ ≥ k 1.0.1 Definition. Let x ,..., x n, and λ ,..., λ . Then P λ x is 1 k R 1 k R i=1 i i called a linear combination of the2 vectors x1,..., xk. It is2 further a (1) conic combination, if λi 0, k (2) affine combination, if P≥ λ 1, and a i=1 i = (3) convex combination, if it is conic and affine. The linear (conic, affine, convex) hull of a set X Rn is the set of all points that are a linear (conic, affine, convex) combination2 of some finite subset of X . It is denoted by lin(X ) (or, cone(X ), aff(X ), conv(X ), respectively). X is a linear space (cone, affine space, convex set if X equals its linear hull (or conic hull, affine hull, convex hull, respectively). d ? 1.0.2 Definition (hyperplanes and half-spaces). For any non-zero α (R ) and δ R the set 2 2 1 Lecture Notes Fall School “Polyhedral Combinatorics” — Darmstadt 2012 (preliminary version of December 7, 2012) Hα,δ := x α(x) 0 is an affine hyperplane, and f j ≤ g Hα := x α(x) δ is a linear hyperplane. f j ≤ g The corresponding positive and negative half-spaces are + Hα,δ := x α(x) δ Hα−,δ := x α(x) δ f j ≥ g f j ≤ g + Hα := x α(x) 0 Hα− := x α(x) 0 . f j ≥ g f j ≤ g + d Then Hα,δ Hα−,δ = Hα,δ. Let H := Hα,b R be a hyperplane. We say that a point d \ ⊆ y R is beneath H if α(y) < b and beyond H if α(y) > b. 2 1.1 Cones Cones are the basic objects for most of what we will study in these notes. In this section we will introduce two definitions of polyhedral cones. The WEYL- MINKOWSKI Theorem will tell us that these two definitions coincide. In the next section we will use this to study polytopes. Cones will reappear prominently when we start counting lattice points in polytopes. In the next chapter we will learn that counting in polytopes is best be done by studying either the cone over the polytope, or the vertex cones of the polytope. 1.1.1 Definition. A subset C Rd is a cone if for all x, y C and λ, µ R also ⊆ 2 2 ≥ λx +µy C. A cone C is polyhedral (finitely constrained) if there are α1,..., αm d ? (R ) such2 that 2 m \ C H x n α x 0 for 1 i m . (1.1.1) = α−i = R i( ) i=1 f 2 j ≤ ≤ ≤ g n A cone C is called finitely generated by vectors v1,..., vr R if 2 ( n ) X C = cone(v1,..., vn) := λi vi λi 0 for 1 i n . (1.1.2) i 1 j ≥ ≤ ≤ Figure missing = It is easy to check that any set of the form (1.1.1) or (1.1.2) indeed defines a Fig. 1.1 cone. 1.1.2 Example. See Figure 1.1. The two notions of a finitely generated and finitely constrained cone are in fact equivalent. This is the result of the WEYL-MINKOWSKI Duality for cones. 1.1.3 Theorem(W EYL-MINKOWSKI Duality for Cones). A cone is polyhedral if and only if it is finitely generated. We have to defer the proof a little bit until we know more about cones. 1.1.4 Lemma. Let C Rd+1 be a polyhedral cone and π : Rd+1 Rd the projec- tion onto the last d coordinates.⊆ Then also π(C) is a polyhedral cone.! Proof. We use a technique called FOURIER-MOTZKIN Elimination for this. Let C be defined by C = (x0, x) λi x0 + αi(x) 0 for 1 i n f j ≤ ≤ ≤ g d ? for some linear functionals αi (R ) and λi R, 1 i m. Then 2 2 ≤ ≤ — 2 — Haase, Nill, Paffenholz: Lattice Polytopes Polytopes, Cones, and Lattices (preliminary version of December 7, 2012) C 0 := π(C) = x x0 R : (x0, x) C . f j 9 2 2 g We can assume that there are a, b Z such that 2 ≥ 8 = 0 for 1 i a < ≤ ≤ λi = > 0 for a + 1 i b : ≤ ≤ < 0 for b + 1 i m . ≤ ≤ Define functionals βi j := λiαj λjαi for a < i b < j m. Then − ≤ ≤ C 0 D := x αi(x) 0, 1 i a, βi j(x) 0, a < i b < j m . ⊆ f j ≤ ≤ ≤ ≤ ≤ ≤ g We want to show D C 0. Let x D. Then for any x0 R and 1 i a ⊆ 2 2 ≤ ≤ λi x0 + αi(x) 0 , ≤ as λi = 0. Further, βi j(x) 0 implies ≤ 1 1 αj(x) αi(x) λj ≥ λi for all a < i b < j m. Hence, there is x0 such that ≤ ≤ 1 1 min αj(x) x0 max αi(x) . a+1 i b λj b+1 j m λi ≤ ≤ ≤ − ≤ ≥ ≤ This means λi x0 + α(x) 0 for a + 1 i b ≤ ≤ ≤ λj x0 + α(x) 0 for b + 1 j m . ≤ ≤ ≤ Hence, (x0, x) in C, so x C 0. 2 ut This suffices to prove one direction of the WEYL-MINKOWSKI Theorem. 1.1.5 Theorem(W EYL’s Theorem). Let C be a finitely generated cone. Then C is polyhedral. d Proof. Let v1,..., vn R be generators of C, i.e. 2 ( n ) X C := λi vi λi 0 for 1 i n . i=1 j ≥ ≤ ≤ Then ( n ) d X C = x R λ1,..., λn R : x λi vi = 0, λ1,..., λn 0 . 2 j 9 2 − i=1 ≥ The cone C is the projection onto the last d coordinates of the set ( n ) X C : λ, x x λ v 0, λ ,..., λ 0 . 0 = ( ) i i = 1 n Figure missing j − i=1 ≥ This is clearly a polyhedral cone. By Lemma 1.1.4 C is polyhedral. Fig. 1.2 ut Haase, Nill, Paffenholz: Lattice Polytopes — 3 — Lecture Notes Fall School “Polyhedral Combinatorics” — Darmstadt 2012 (preliminary version of December 7, 2012) d 1.1.6 Theorem(F ARKAS Lemma). Let a cone C be generated by v1,..., vn R . Then for x Rd exactly one of the following holds. 2 2 (1)x C, or d ? (2) there2 is α (R ) such that α(y) 0 for all y C and α(x) > 0.