Weighted lattice point sums in lattice polytopes Paul Gunnels - PowerPoint PPT Presentation

Weighted lattice point sums in lattice polytopes Paul Gunnels Matthias Beck University of Massachusetts San Francisco State University Freie Universit¨ at Berlin Evgeny Materov math.sfsu.edu/beck Siberian Fire and Rescue Academy of EMERCOM of Russia

The Bott–Brion–Dehn–Ehrhart–Euler– Khovanskii–Maclaurin–Pukhlikov– Sommerville–Vergne formula for simple lattice polytopes Paul Gunnels Matthias Beck University of Massachusetts San Francisco State University Freie Universit¨ at Berlin Evgeny Materov math.sfsu.edu/beck Siberian Fire and Rescue Academy of EMERCOM of Russia

The Menu Lattice-point counting in lattice polytopes: (weighted) Ehrhart ◮ polynomials and their reciprocity Face-counting for simple polytopes: (generalized) Dehn–Sommerville ◮ relations Our goal Give a unifying reciprocity theorem Secondary goal Entice (some of) you to study weighted Ehrhart polynomials Weighted lattice point sums in lattice polytopes Matthias Beck 2

Ehrhart–Macdonald Reciprocity V — real vector space of dimension n equipped with a lattice M ⊂ V P ⊂ V — ( n -dimensional) lattice polytope (i.e., vertices in M ) For t ∈ Z > 0 let E P ( t ) := | M ∩ tP | Ehrhart–Macdonald (1960s) E P ( t ) is a polynomial in t (of degree dim( P ) and with constant term 1 ) that satisfies E P ( − t ) = ( − 1) dim( P ) E P ◦ ( t ) . Example P = conv { ( ± 1 , ± 1 , 1) , (0 , 0 , 1) } 3 t 3 + 4 t 2 + 11 4 E P ( t ) = 3 t + 1 Weighted lattice point sums in lattice polytopes Matthias Beck 3

Ehrhart–Macdonald Reciprocity V — real vector space of dimension n equipped with a lattice M ⊂ V P ⊂ V — ( n -dimensional) lattice polytope (i.e., vertices in M ) For t ∈ Z > 0 let E P ( t ) := | M ∩ tP | Ehrhart–Macdonald (1960s) E P ( t ) is a polynomial in t (of degree dim( P ) and with constant term 1 ) that satisfies E P ( − t ) = ( − 1) dim( P ) E P ◦ ( t ) . In the dictionary P ← → toric variety (if P is a very ample), E P ( t ) ◮ equals the Hilbert polynomial of this toric variety under the projective embedding given by the very ample divisor associated with P . Ehrhart–Macdonald is part of an illustrious series of combinatorial ◮ reciprocity theorems Weighted lattice point sums in lattice polytopes Matthias Beck 3

Ehrhart Polynomials V — real vector space of dimension n equipped with a lattice M ⊂ V P ⊂ V — ( n -dimensional) lattice polytope (i.e., vertices in M ) For t ∈ Z > 0 let E P ( t ) := | M ∩ tP | Ehrhart–Macdonald (1960s) E P ( t ) is a polynomial in t (of degree dim( P ) and with constant term 1 ). Natural, currently en vogue questions: (Sub-)Classification of Ehrhart polynomials ◮ Families of polytopes with positive/unimodal/... Ehrhart coefficients ◮ Weighted lattice point sums in lattice polytopes Matthias Beck 4

Weighted Ehrhart–Macdonald Reciprocity V — real vector space of dimension n equipped with a lattice M ⊂ V P ⊂ V — ( n -dimensional) lattice polytope (i.e., vertices in M ) For t ∈ Z > 0 let E P ( t ) := | M ∩ tP | Ehrhart–Macdonald (1960s) E P ( t ) is a polynomial in t (of degree dim( P ) and with constant term 1 ) that satisfies E P ( − t ) = ( − 1) dim( P ) E P ◦ ( t ) . � For a homogeneous polynomial ϕ let E ϕ,P ( t ) := ϕ ( m ) m ∈ M ∩ tP Brion–Vergne (1997) E ϕ,P ( t ) is a polynomial in t (of degree dim( P ) + deg( ϕ ) and with constant term ϕ (0) ) that satisfies E ϕ,P ( − t ) = ( − 1) dim( P )+deg( ϕ ) E ϕ,P ◦ ( t ) . Weighted lattice point sums in lattice polytopes Matthias Beck 5

Weighted Ehrhart–Macdonald Reciprocity V — real vector space of dimension n equipped with a lattice M ⊂ V P ⊂ V — ( n -dimensional) lattice polytope (i.e., vertices in M ) � For a homogeneous polynomial ϕ let E ϕ,P ( t ) := ϕ ( m ) m ∈ M ∩ tP Brion–Vergne (1997) E ϕ,P ( t ) is a polynomial in t (of degree dim( P ) + deg( ϕ ) and with constant term ϕ (0) ). Possible (and possibly en vogue ) questions: a la h ∗ Structural theorems ( ` P ≥ 0 ) under certain conditions ◮ Families of polytopes with positive/unimodal/... Ehrhart coefficients ◮ Special cases, e.g., ϕ ( m ) = m 1 or ϕ ( m ) = m 1 + m 2 + · · · + m n ◮ Weighted lattice point sums in lattice polytopes Matthias Beck 5

Dehn–Sommerville Relations P is simple if each vertex meets n edges F — set of faces of P � ( y − 1) dim( F ) The h -polynomial of P is h P ( y ) := F ∈F Dehn–Sommerville (early 1900s) If P is simple then y n h P ( 1 y ) = h P ( y ) . Example If P is a simplex then h P ( y ) = y n + y n − 1 + · · · + 1 Weighted lattice point sums in lattice polytopes Matthias Beck 6

Dehn–Sommerville Relations P is simple if each vertex meets n edges F — set of faces of P � ( y − 1) dim( F ) The h -polynomial of P is h P ( y ) := F ∈F Dehn–Sommerville (early 1900s) If P is simple then y n h P ( 1 y ) = h P ( y ) . Example If P is a simplex then h P ( y ) = y n + y n − 1 + · · · + 1 In the dictionary P ← → toric variety, Dehn–Sommerville corresponds ◮ to Poincar´ e duality for the rational cohomology of the toric variety attached to P . Combinatorially, Dehn–Sommerville follows from the fact that the face ◮ lattice of a polytope is Eulerian and thus its zeta polynomial is even/odd. Weighted lattice point sums in lattice polytopes Matthias Beck 6

Dehn–Sommerville Relations P is simple if each vertex meets n edges F — set of faces of P � ( y − 1) dim( F ) The h -polynomial of P is h P ( y ) := F ∈F Dehn–Sommerville (early 1900s) If P is simple then y n h P ( 1 y ) = h P ( y ) . Example If P is a simplex then h P ( y ) = y n + y n − 1 + · · · + 1 Natural, equally en vogue questions: (Sub-)Classification of h -polynomials ◮ Extensions to simplicial/polyhedral/... complexes ◮ Weighted lattice point sums in lattice polytopes Matthias Beck 6

Main Theorem (1st Version) � E ϕ,P ( − t ) = ( − 1) dim( P )+deg( ϕ ) E ϕ,P ◦ ( t ) E ϕ,P ( t ) := ϕ ( m ) m ∈ M ∩ tP � ( y − 1) dim( F ) y n h P ( 1 h P ( y ) := y ) = h P ( y ) F ∈F Let G ϕ,P ( t, y ) := ( y + 1) deg( ϕ ) � ( y + 1) dim( F ) ( − y ) codim( F ) E ϕ,F ( t ) F ∈F Theorem (M B–Gunnels–Materov) If P is a simple lattice polytope then G ϕ,P ( t, y ) = ( − y ) dim( P )+deg( ϕ ) G ϕ,P ( − t, 1 y ) . Weighted lattice point sums in lattice polytopes Matthias Beck 7

Main Theorem (1st Version) Let G ϕ,P ( t, y ) := ( y + 1) deg( ϕ ) � ( y + 1) dim( F ) ( − y ) codim( F ) E ϕ,F ( t ) F ∈F Theorem (M B–Gunnels–Materov) If P is a simple lattice polytope then G ϕ,P ( t, y ) = ( − y ) dim( P )+deg( ϕ ) G ϕ,P ( − t, 1 y ) . If ϕ = 1 then E ϕ,F ( t ) = E F ( t ) and the constant terms (in y ) of � ( − 1) dim( P ) G ϕ =1 ,P ( − t, y ) ( y + 1) dim( F ) ( − y ) codim( F ) E F ( − t ) = F ∈F � y dim( P ) G ϕ =1 ,P ( t, 1 ( y + 1) dim( F ) ( − 1) codim( F ) E F ( t ) y ) = F ∈F � � ( − 1) codim( F ) E F ( t ) = E P ◦ ( t ) are E F ◦ ( t ) = E P ( t ) and F ∈F F ∈F Weighted lattice point sums in lattice polytopes Matthias Beck 7

Main Theorem (1st Version) Let G ϕ,P ( t, y ) := ( y + 1) deg( ϕ ) � ( y + 1) dim( F ) ( − y ) codim( F ) E ϕ,F ( t ) F ∈F Theorem (M B–Gunnels–Materov) If P is a simple lattice polytope then G ϕ,P ( t, y ) = ( − y ) dim( P )+deg( ϕ ) G ϕ,P ( − t, 1 y ) . If ϕ = 1 and t = 0 then � ( y + 1) dim( F ) ( − y ) codim( F ) = ( − y ) dim( P ) h P ( − 1 G ϕ =1 ,P (0 , y ) = y ) F ∈F and � ( − 1 − y ) dim( F ) = h P ( − y ) ( − y ) dim( P ) G ϕ,P (0 , 1 y ) = F ∈F Weighted lattice point sums in lattice polytopes Matthias Beck 7

g -Polynomials We define polynomials f P ( x ) and g P ( x ) recursively by dimension: f ∅ ( x ) = g ∅ ( x ) = 1 ◮ dim( P ) � � g F ( x )( x − 1) n − dim( F ) − 1 = f j x j f P ( x ) = ◮ j =0 F ∈F\{ P } g P ( x ) = f 0 + ( f 1 − f 0 ) x + ( f 2 − f 1 ) x 2 + · · · + ( f m − f m − 1 ) x m where m = ⌊ dim( P ) ⌋ 2 Master Duality Theorem (Stanley 1974) f P ( x ) = x dim( P ) f P ( 1 x ) Weighted lattice point sums in lattice polytopes Matthias Beck 8

g -Polynomials We define polynomials f P ( x ) and g P ( x ) recursively by dimension: f ∅ ( x ) = g ∅ ( x ) = 1 ◮ dim( P ) � � g F ( x )( x − 1) n − dim( F ) − 1 = f j x j f P ( x ) = ◮ j =0 F ∈F\{ P } g P ( x ) = f 0 + ( f 1 − f 0 ) x + ( f 2 − f 1 ) x 2 + · · · + ( f m − f m − 1 ) x m where m = ⌊ dim( P ) ⌋ 2 Master Duality Theorem (Stanley 1974) f P ( x ) = x dim( P ) f P ( 1 x ) This definition of f P ( x ) is dual to that of the h -polynomial. It favors ◮ simplicial polytopes, in that Dehn–Sommerville holds with no g P ( x ) corrections. In the dictionary P ← → toric variety, g P ( x ) takes into account the ◮ intersection cohomology of the variety. Weighted lattice point sums in lattice polytopes Matthias Beck 8

Main Theorem (2nd Version) Let � F be the dual face of F in the polar polytope of P and � g F ( x ) := g � F ( x ) G ϕ,P ( t, y ) := ( y + 1) deg( ϕ ) � ( y + 1) dim( F ) ( − y ) codim( F ) E ϕ,F ( t ) � g F ( − 1 y ) F ∈F Remark If P is simple then � F is a simplex for every proper face F and thus g F ( x ) = 1 , recovering our earlier definition. � Theorem (M B–Gunnels–Materov) If P is a lattice polytope then G ϕ,P ( t, y ) = ( − y ) dim( P )+deg( ϕ ) G ϕ,P ( − t, 1 y ) . Weighted lattice point sums in lattice polytopes Matthias Beck 9