Structure and Classifications of Lattice Polytopes Andreas Paffenholz Technische Universität Darmstadt (Classifications of) Lattice Polytopes d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi 2 Z August 2, 2016 | TUD | Andreas Paffenholz | 1 (Classifications of) Lattice Polytopes d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi 2 Z I Classifications of subfamilies . by dimension . number of (interior) lattice points . structural properties: . many vertices, h∗-vector, . August 2, 2016 | TUD | Andreas Paffenholz | 1 (Classifications of) Lattice Polytopes d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi 2 Z I Classifications of subfamilies . by dimension . number of (interior) lattice points . structural properties: . many vertices, h∗-vector, . I Why? . experimentation . obtain/test conjectures August 2, 2016 | TUD | Andreas Paffenholz | 1 (Classifications of) Lattice Polytopes d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi 2 Z I Classifications of subfamilies . by dimension . number of (interior) lattice points . structural properties: . many vertices, h∗-vector, . I Why? . experimentation . obtain/test conjectures I How? . computational . enumerate complete subfamilies . PALP, polymake . Graded Rings Database, polymake database polyDB August 2, 2016 | TUD | Andreas Paffenholz | 1 (Classifications of) Lattice Polytopes d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi 2 Z I Classifications of subfamilies . by dimension . number of (interior) lattice points . structural properties: . many vertices, h∗-vector, . I Why? . experimentation . obtain/test conjectures I How? . computational . structural . enumerate complete subfamilies . standard polytope constructions . PALP, polymake . projections, liftings . Graded Rings Database, . common properties polymake database polyDB . ... August 2, 2016 | TUD | Andreas Paffenholz | 1 Lattice Polytopes d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi 2 Z August 2, 2016 | TUD | Andreas Paffenholz | 2 Lattice Polytopes d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi 2 Z I hyperplane description: d Q = f x j hai , xi ≤ bi g ai 2 Z primitive, bi 2 Z August 2, 2016 | TUD | Andreas Paffenholz | 2 Lattice Polytopes d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi 2 Z I hyperplane description: d Q = f x j hai , xi ≤ bi g ai 2 Z primitive, bi 2 Z . assume irredundant: Each ai defines a facet Fi := fx 2 Q j hai , xi = bi g of Q August 2, 2016 | TUD | Andreas Paffenholz | 2 Lattice Polytopes d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi 2 Z I hyperplane description: d Q = f x j hai , xi ≤ bi g ai 2 Z primitive, bi 2 Z . assume irredundant: Each ai defines a facet Fi := fx 2 Q j hai , xi = bi g of Q I normal fan of Q: d ∗ . fan Σ ⊆ (R ) with rays a1, ... , am . rays form cone σ if corresponding facets define face of Q August 2, 2016 | TUD | Andreas Paffenholz | 2 Lattice Polytopes d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi 2 Z I hyperplane description: d Q = f x j hai , xi ≤ bi g ai 2 Z primitive, bi 2 Z . assume irredundant: Each ai defines a facet Fi := fx 2 Q j hai , xi = bi g of Q I normal fan of Q: d ∗ . fan Σ ⊆ (R ) with rays a1, ... , am . rays form cone σ if corresponding facets define face of Q I polar (dual) polytope: Q_ = f x j hx, vi ≤ 1 8v 2 Q g August 2, 2016 | TUD | Andreas Paffenholz | 2 Lattice Polytopes d . Ehrhart polynomial ehrP (k) := j k · P \ Z j polynomial of degree d 1 3 ehr (k) = k2 + k + 1 P 2 2 August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes d . Ehrhart polynomial ehrP (k) := j k · P \ Z j polynomial of degree d 1 3 ehr (k) = k2 + k + 1 P 2 2 August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes d . Ehrhart polynomial ehrP (k) := j k · P \ Z j polynomial of degree d 1 3 ehr (k) = k2 + k + 1 P 2 2 August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes d . Ehrhart polynomial ehrP (k) := j k · P \ Z j polynomial of degree d ∗ ∗ X k h (t) h -polynomial of P: ehrP (k)t = I (1 − t)d+1 k≥0 1 3 ehr (k) = k2 + k + 1 P 2 2 August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes d . Ehrhart polynomial ehrP (k) := j k · P \ Z j polynomial of degree d ∗ ∗ X k h (t) h -polynomial of P: ehrP (k)t = I (1 − t)d+1 k≥0 . h∗ has integral nonnegative coefficients ∗ ∗ ∗ ∗ . h -vector (h0, h1, ... , hd ) 1 3 ehr (k) = k2 + k + 1 P 2 2 August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes d . Ehrhart polynomial ehrP (k) := j k · P \ Z j polynomial of degree d ∗ ∗ X k h (t) h -polynomial of P: ehrP (k)t = I (1 − t)d+1 k≥0 . h∗ has integral nonnegative coefficients ∗ ∗ ∗ ∗ . h -vector (h0, h1, ... , hd ) ∗ . h0 = 1 ∗ d . h1 = jP \ Z j − d − 1 ∗ d . hd = j int P \ Z j 1 3 ehr (k) = k2 + k + 1 P 2 2 August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes d . Ehrhart polynomial ehrP (k) := j k · P \ Z j polynomial of degree d ∗ ∗ X k h (t) h -polynomial of P: ehrP (k)t = I (1 − t)d+1 k≥0 . h∗ has integral nonnegative coefficients ∗ ∗ ∗ ∗ . h -vector (h0, h1, ... , hd ) ∗ . h0 = 1 ∗ d . h1 = jP \ Z j − d − 1 ∗ d . hd = j int P \ Z j I lattice polytope Q with normal fan Σ ! 1 2 3 ehrP (k) = k + k + 1 projective toric variety XΣ 2 2 with ample divisior LP August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes d . Ehrhart polynomial ehrP (k) := j k · P \ Z j polynomial of degree d ∗ ∗ X k h (t) h -polynomial of P: ehrP (k)t = I (1 − t)d+1 k≥0 . h∗ has integral nonnegative coefficients ∗ ∗ ∗ ∗ . h -vector (h0, h1, ... , hd ) ∗ . h0 = 1 ∗ d . h1 = jP \ Z j − d − 1 ∗ d . hd = j int P \ Z j I lattice polytope Q with normal fan Σ ! 1 2 3 ehrP (k) = k + k + 1 projective toric variety XΣ 2 2 with ample divisior LP . toric dictionary: properties of variety correpond to properties of polytope August 2, 2016 | TUD | Andreas Paffenholz | 3 Classification of Lattice Polytopes Classify or enumerate complete subfamilies of lattice polytopes August 2, 2016 | TUD | Andreas Paffenholz | 4 Classification of Lattice Polytopes Classify or enumerate complete subfamilies of lattice polytopes . by dimension . my number of (interior) lattice points . by properties of the polytopes . ... August 2, 2016 | TUD | Andreas Paffenholz | 4 Classification of Lattice Polytopes Classify or enumerate complete subfamilies of lattice polytopes . by dimension . my number of (interior) lattice points . by properties of the polytopes . ... Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points. lattice equivalence: d transformations with affine maps x 7−! Mx + t, M unimodular, t 2 Z . August 2, 2016 | TUD | Andreas Paffenholz | 4 Classification of Lattice Polytopes Classify or enumerate complete subfamilies of lattice polytopes . by dimension . my number of (interior) lattice points . by properties of the polytopes . ... Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points. lattice equivalence: d transformations with affine maps x 7−! Mx + t, M unimodular, t 2 Z . d ∗ I P empty polytope: int P \ Z = ? () hd = 0. August 2, 2016 | TUD | Andreas Paffenholz | 4 Classification of Lattice Polytopes Classify or enumerate complete subfamilies of lattice polytopes . by dimension . my number of (interior) lattice points . by properties of the polytopes . ... Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points. lattice equivalence: d transformations with affine maps x 7−! Mx + t, M unimodular, t 2 Z . d ∗ I P empty polytope: int P \ Z = ? () hd = 0. −! different approaches for classifications of empty polytopes and those with interior lattice points. August 2, 2016 | TUD | Andreas Paffenholz | 4 Empty Lattice Polytopes ∗ d ∗ I hd = j int P \ Z j, so hd = 0 iff P is empty −! empty polytopes are filtered by the degree of h∗ (number of trailing zeros in h∗) August 2, 2016 | TUD | Andreas Paffenholz | 5 Empty Lattice Polytopes ∗ d ∗ I hd = j int P \ Z j, so hd = 0 iff P is empty −! empty polytopes are filtered by the degree of h∗ (number of trailing zeros in h∗) −! known classifications for . polytopes for deg h∗ 2 f0, 1g . h∗-vectors for deg h∗ = 2 . h∗-vectors with few nonzero entries August 2, 2016 | TUD | Andreas Paffenholz | 5 Empty Lattice Polytopes ∗ d ∗ I hd = j int P \ Z j, so hd = 0 iff P is empty −! empty polytopes are filtered by the degree of h∗ (number of trailing zeros in h∗) −! known classifications for . polytopes for deg h∗ 2 f0, 1g . h∗-vectors for deg h∗ = 2 . h∗-vectors with few nonzero entries ∗ d . d + 1 − max(k j hk 6= 0) = min(k j int(kP) \ Z 6= ?) August 2, 2016 | TUD | Andreas Paffenholz | 5 Empty Lattice Polytopes ∗ d ∗ I hd = j int P \ Z j, so hd = 0 iff P is empty −! empty polytopes are filtered by the degree of h∗ (number of trailing zeros in h∗) −! known classifications for .