Lattice Points in Polytopes Richard P. Stanley U. Miami & - PowerPoint PPT Presentation

i ( Π d , n ) Theorem. i ( Π d , n ) = ∑ d − 1 k = 0 f k ( d ) n k , where f k ( d ) = # { forests with k edges on vertices 1 ,... , d } 1 2 3 i ( Π 3 , n ) = 3 n 2 + 3 n + 1 Can be greatly generalized ( Postnikov , et al.).

Application to graph theory Let G be a graph (with no loops or multiple edges) on the vertex set V ( G ) = { 1 , 2 ,... , n } . Let d i = degree (# incident edges) of vertex i . Define the ordered degree sequence d ( G ) of G by d ( G ) = ( d 1 ,... , d n ) .

Example of d ( G ) Example. d ( G ) = ( 2 , 4 , 0 , 3 , 2 , 1 ) 1 2 3 4 5 6

# ordered degree sequences Let f ( n ) be the number of distinct d ( G ) , where V ( G ) = { 1 , 2 ,... , n } .

f ( n ) for n ≤ 4 Example. If n ≤ 3, all d ( G ) are distinct, so f ( 1 ) = 1, f ( 2 ) = 2 1 = 2, f ( 3 ) = 2 3 = 8. For n ≥ 4 we can have G ≠ H but d ( G ) = d ( H ) , e.g., 1 2 1 2 1 2 3 4 3 4 3 4 In fact, f ( 4 ) = 54 < 2 6 = 64.

The polytope of degree sequences Let conv denote convex hull, and D n = conv { d ( G ) ∶ V ( G ) = { 1 ,... , n }} ⊂ R n , the polytope of degree sequences ( Perles, Koren ).

The polytope of degree sequences Let conv denote convex hull, and D n = conv { d ( G ) ∶ V ( G ) = { 1 ,... , n }} ⊂ R n , the polytope of degree sequences ( Perles, Koren ). Easy fact. Let e i be the i th unit coordinate vector in R n . E.g., if n = 5 then e 2 = ( 0 , 1 , 0 , 0 , 0 ) . Then D n = Z ( e i + e j ∶ 1 ≤ i < j ≤ n ) .

The Erd˝ os-Gallai theorem Theorem. Let α = ( a 1 ,... , a n ) ∈ Z n . Then α = d ( G ) for some G if and only if α ∈ D n a 1 + a 2 + ⋯ + a n is even.

A generating function Enumerative techniques leads to: Theorem. Let F ( x ) f ( n ) x n = ∑ n ! n ≥ 0 1 + x + 2 x 2 2! + 8 x 3 3! + 54 x 4 4! + ⋯ . = Then:

A formula for F ( x ) ⎡ ⎢ ⎢ 1 / 2 F ( x ) = 1 ( 1 + 2 ∑ n n x n n ! ) ⎢ ⎢ ⎣ 2 n ≥ 1 × ( 1 − ∑ ( n − 1 ) n − 1 x n n ! ) + 1 ] n ≥ 1 ( 0 0 = 1 ) n n − 2 x n × exp ∑ n ! n ≥ 1

Coefficients of i (P , n ) Let P denote the tetrahedron with vertices ( 0 , 0 , 0 ) , ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 1 , 1 , 13 ) . Then 6 n 3 + n 2 − 1 i (P , n ) = 13 6 n + 1 .

The “bad” tetrahedron y x z

The “bad” tetrahedron y x z Thus in general the coefficients of Ehrhart polynomials are not “nice.” Is there a “better” basis?

The h ∗ -vector of i (P , n ) Let P be a lattice polytope of dimension d . Since i (P , n ) is a polynomial of degree d , ∃ h i ∈ Z such that i (P , n ) x n = h 0 + h 1 x + ⋯ + h d x d ∑ ( 1 − x ) d + 1 . n ≥ 0

The h ∗ -vector of i (P , n ) Let P be a lattice polytope of dimension d . Since i (P , n ) is a polynomial of degree d , ∃ h i ∈ Z such that i (P , n ) x n = h 0 + h 1 x + ⋯ + h d x d ∑ ( 1 − x ) d + 1 . n ≥ 0 Definition. Define h ∗ (P) = ( h 0 , h 1 ,... , h d ) , the h ∗ -vector of P .

Example of an h ∗ -vector Example. Recall i (B 4 , n ) = 11340 ( 11 n 9 1 + 198 n 8 + 1596 n 7 + 7560 n 6 + 23289 n 5 + 48762 n 5 + 70234 n 4 + 68220 n 2 + 40950 n + 11340 ) .

Example of an h ∗ -vector Example. Recall i (B 4 , n ) = 11340 ( 11 n 9 1 + 198 n 8 + 1596 n 7 + 7560 n 6 + 23289 n 5 + 48762 n 5 + 70234 n 4 + 68220 n 2 + 40950 n + 11340 ) . Then h ∗ (B 4 ) = ( 1 , 14 , 87 , 148 , 87 , 14 , 1 , 0 , 0 , 0 ) .

Two terms of h ∗ (P) h 0 = 1 h d = (− 1 ) dim P i (P , − 1 ) = I (P)

Main properties of h ∗ (P) Theorem A (nonnegativity) . ( McMullen, RS ) h i ≥ 0.

Main properties of h ∗ (P) Theorem A (nonnegativity) . ( McMullen, RS ) h i ≥ 0. Theorem B (monotonicity) . ( RS ) If P and Q are lattice polytopes and Q ⊆ P , then h i ( Q ) ≤ h i ( P ) ∀ i .

Main properties of h ∗ (P) Theorem A (nonnegativity) . ( McMullen, RS ) h i ≥ 0. Theorem B (monotonicity) . ( RS ) If P and Q are lattice polytopes and Q ⊆ P , then h i ( Q ) ≤ h i ( P ) ∀ i . B ⇒ A: take Q = ∅ .

Proofs: the Ehrhart ring P : (convex) lattice polytope in R d with vertex set V x β = x β 1 ⋯ x β d , β ∈ Z d Ehrhart ring (over Q ): R P = Q [ x β y n ∶ β ∈ Z d , n ∈ P , β n ∈ P ] deg x β y n = n

Proofs: the Ehrhart ring P : (convex) lattice polytope in R d with vertex set V x β = x β 1 ⋯ x β d , β ∈ Z d Ehrhart ring (over Q ): R P = Q [ x β y n ∶ β ∈ Z d , n ∈ P , β n ∈ P ] deg x β y n = n R P = ( R P ) 0 ⊕ ( R P ) 1 ⊕ ⋯

Simple properties of R P Hilbert function of R P : H ( R P , n ) = dim Q ( R P ) n .

Simple properties of R P Hilbert function of R P : H ( R P , n ) = dim Q ( R P ) n . Theorem (easy). H ( R P , n ) = i ( P , n )

Simple properties of R P Hilbert function of R P : H ( R P , n ) = dim Q ( R P ) n . Theorem (easy). H ( R P , n ) = i ( P , n ) Q [ V ] : subalgebra of R P generated by x α y , α ∈ V .

Simple properties of R P Hilbert function of R P : H ( R P , n ) = dim Q ( R P ) n . Theorem (easy). H ( R P , n ) = i ( P , n ) Q [ V ] : subalgebra of R P generated by x α y , α ∈ V . Theorem (easy). R P is a finitely-generated Q [ V ] -module.

The Cohen-Macaulay property Theorem ( Hochster , 1972). R P is a Cohen-Macaulay ring.

The Cohen-Macaulay property Theorem ( Hochster , 1972). R P is a Cohen-Macaulay ring. This means (using finiteness of R P over Q [ V ] ): if dim P = m then there exist algebraically independent θ 1 ,... ,θ m ∈ ( R P ) 1 such that R P is a finitely-generated free Q [ θ 1 ,... ,θ m ] -module. θ 1 ,... ,θ m is a homogeneous system of parameters ( h.s.o.p. ).

The Cohen-Macaulay property Theorem ( Hochster , 1972). R P is a Cohen-Macaulay ring. This means (using finiteness of R P over Q [ V ] ): if dim P = m then there exist algebraically independent θ 1 ,... ,θ m ∈ ( R P ) 1 such that R P is a finitely-generated free Q [ θ 1 ,... ,θ m ] -module. θ 1 ,... ,θ m is a homogeneous system of parameters ( h.s.o.p. ). Thus R P = ⊕ r j = 1 η j Q [ θ 1 ,... ,θ m ] , where η j ∈ ( R P ) e j .

The Cohen-Macaulay property Theorem ( Hochster , 1972). R P is a Cohen-Macaulay ring. This means (using finiteness of R P over Q [ V ] ): if dim P = m then there exist algebraically independent θ 1 ,... ,θ m ∈ ( R P ) 1 such that R P is a finitely-generated free Q [ θ 1 ,... ,θ m ] -module. θ 1 ,... ,θ m is a homogeneous system of parameters ( h.s.o.p. ). Thus R P = ⊕ r j = 1 η j Q [ θ 1 ,... ,θ m ] , where η j ∈ ( R P ) e j . x n = x e 1 + ⋯ + x e r , so h ∗ ( P ) ≥ 0. Corollary. ∑ n ≥ 0 H ( R P , n ) ( 1 − x ) m �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� i (P , n )

Monotonicity The result Q ⊆ P ⇒ h ∗ ( Q ) ≤ h ∗ ( P ) is proved similarly. We have R Q ⊂ R P . The key fact is that we can find an h.s.o.p. θ 1 ,... ,θ k for R Q that extends to an h.s.o.p. for R P .

The canonical module Let R = R 0 ⊕ R 1 ⊕ ⋯ be a Cohen-Macaulay graded algebra over a field K = R 0 , with Krull dimension m and Hilbert series ( dim K R n ) x n = ∑ r j = 1 x e j ∑ ( 1 − x d 1 ) ⋯ ( 1 − x d m ) . n ≥ 0 Let R ≅ A / I , where A = K [ x 1 ,... , x t ] .

The canonical module Let R = R 0 ⊕ R 1 ⊕ ⋯ be a Cohen-Macaulay graded algebra over a field K = R 0 , with Krull dimension m and Hilbert series ( dim K R n ) x n = ∑ r j = 1 x e j ∑ ( 1 − x d 1 ) ⋯ ( 1 − x d m ) . n ≥ 0 Let R ≅ A / I , where A = K [ x 1 ,... , x t ] . canonical module : Ω ( R ) = Ext t − m ( R , A ) , a graded R -module. A

Reciprocity redux Basic result in commutative/homological algebra: x c ∑ r ( dim K Ω ( R ) n ) x n = j = 1 x − e j ∑ ( 1 − x d 1 ) ⋯ ( 1 − x d m ) . n ≥ 0

Reciprocity redux Basic result in commutative/homological algebra: x c ∑ r ( dim K Ω ( R ) n ) x n = j = 1 x − e j ∑ ( 1 − x d 1 ) ⋯ ( 1 − x d m ) . n ≥ 0 Theorem. Ω ( R P ) = span Q { x β y n ∶ β ∈ Z d , n ∈ P , β n ∈ interior ( P )}

Reciprocity redux Basic result in commutative/homological algebra: x c ∑ r ( dim K Ω ( R ) n ) x n = j = 1 x − e j ∑ ( 1 − x d 1 ) ⋯ ( 1 − x d m ) . n ≥ 0 Theorem. Ω ( R P ) = span Q { x β y n ∶ β ∈ Z d , n ∈ P , β n ∈ interior ( P )} i ( P , n ) = ( − 1 ) d i ( P , n ) . Corollary. ¯

Further properties: I. Brion’s theorem Example. Let P be the polytope [ 2 , 5 ] in R , so P is defined by ( 1 ) x ≥ 2 , ( 2 ) x ≤ 5 .

Further properties: I. Brion’s theorem Example. Let P be the polytope [ 2 , 5 ] in R , so P is defined by ( 1 ) x ≥ 2 , ( 2 ) x ≤ 5 . Let t n = t 2 = F 1 ( t ) ∑ 1 − t n ≥ 2 n ∈ Z t n = t 5 = F 2 ( t ) ∑ 1 − 1 . n ≤ 5 t n ∈ Z

F 1 ( t ) + F 2 ( t ) F 1 ( t ) + F 2 ( t ) 1 − t + t 2 t 5 = 1 − 1 t 2 + t 3 + t 4 + t 5 t = = t m . ∑ m ∈P∩ Z

Cone at a vertex P : Z -polytope in R N with vertices v 1 ,... , v k C i : cone at vertex v i supporting P