Lattice Points in Polytopes

Richard P. Stanley

M.I.T.

Lattice Points in Polytopes – p. 1 A lattice polygon

Georg Alexander Pick (1859–1942) P : lattice polygon in R2 (vertices Z2, no self-intersections) ∈

Lattice Points in Polytopes – p. 2 Boundary and interior lattice points

Lattice Points in Polytopes – p. 3 Pick’s theorem

A = area of P I = # interior points of P (= 4) B = #boundary points of P (= 10) Then 2I + B 2 A = − . 2

Lattice Points in Polytopes – p. 4 Pick’s theorem

A = area of P I = # interior points of P (= 4) B = #boundary points of P (= 10) Then 2I + B 2 A = − . 2 Example on previous slide:

2 4 + 10 2 · − = 9. 2 Lattice Points in Polytopes – p. 4 Two tetrahedra

Pick’s theorem (seemingly) fails in higher dimensions. For example, let T1 and T2 be the tetrahedra with vertices v(T ) = (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1) 1 { } v(T ) = (0, 0, 0), (1, 1, 0), (1, 0, 1), (0, 1, 1) . 2 { }

Lattice Points in Polytopes – p. 5 Failure of Pick’s theorem in dim 3

Then I(T1)= I(T2) = 0

B(T1)= B(T2) = 4

A(T1) = 1/6, A(T2) = 1/3.

Lattice Points in Polytopes – p. 6 Polytope dilation

Let P be a convex polytope (convex hull of a finite set of points) in Rd. For n 1, let ≥ nP = nα : α . { ∈ P}

Lattice Points in Polytopes – p. 7 Polytope dilation

Let P be a convex polytope (convex hull of a finite set of points) in Rd. For n 1, let ≥ nP = nα : α . { ∈ P}

P 3P Lattice Points in Polytopes – p. 7 i(P,n)

Let i(P,n) = #(n Zd) P ∩ = # α : nα Zd , { ∈ P ∈ } the number of lattice points in n . P

Lattice Points in Polytopes – p. 8 ¯i(P,n)

Similarly let P◦ = interior of = ∂ P P− P

d ¯i(P,n) = #(n ◦ Z ) P ∩ d = # α ◦ : nα Z , { ∈ P ∈ } the number of lattice points in the interior of n . P

Lattice Points in Polytopes – p. 9 ¯i(P,n)

Similarly let P◦ = interior of = ∂ P P− P

d ¯i(P,n) = #(n ◦ Z ) P ∩ d = # α ◦ : nα Z , { ∈ P ∈ } the number of lattice points in the interior of n . P Note. Could use any lattice L instead of Zd.

Lattice Points in Polytopes – p. 9 An example

P 3P

i( ,n) = (n + 1)2 P ¯i( ,n) = (n 1)2 = i( , n). P − P − Lattice Points in Polytopes – p. 10 Reeve’s theorem lattice polytope: polytope with integer vertices Theorem (Reeve, 1957). Let be a three-dimensional lattice polytope.P Then the volume V ( ) is a certain (explicit) function of P i( , 1), ¯i( , 1), and i( , 2). P P P

Lattice Points in Polytopes – p. 11 The main result

Theorem (Ehrhart 1962, Macdonald 1963). Let

P = lattice polytope in RN , dim = d. P Then i( ,n) is a polynomial (the Ehrhart polynomialP of ) in n of degree d. P

Lattice Points in Polytopes – p. 12 Reciprocity and volume

Moreover, i( , 0) = 1 P ¯i( ,n) = ( 1)di( , n), n> 0 P − P − (reciprocity).

Lattice Points in Polytopes – p. 13 Reciprocity and volume

Moreover, i( , 0) = 1 P ¯i( ,n) = ( 1)di( , n), n> 0 P − P − (reciprocity). If d = N then i( ,n)= V ( )nd + lower order terms, P P where V (P) is the volume of . P

Lattice Points in Polytopes – p. 13 Eugène Ehrhart

April 29, 1906: born in Guebwiller, France 1932: begins teaching career in lycées 1959: Prize of French Sciences Academy 1963: begins work on Ph.D. thesis 1966: obtains Ph.D. thesis from Univ. of Strasbourg 1971: retires from teaching career January 17, 2000: dies

Lattice Points in Polytopes – p. 14 Photo of Ehrhart

Lattice Points in Polytopes – p. 15 Self-portrait

Lattice Points in Polytopes – p. 16 Generalized Pick’s theorem

Corollary. Let Rd and dim = d. Knowing any d of i( ,n)Por ⊂¯i( ,n) for n>P0 determines V ( ). P P P

Lattice Points in Polytopes – p. 17 Generalized Pick’s theorem

Corollary. Let Rd and dim = d. Knowing any d of i( ,n)Por ⊂¯i( ,n) for n>P0 determines V ( ). P P P Proof. Together with i( , 0) = 1, this data determines d + 1 valuesP of the polynomial i( ,n) of degree d. This uniquely determines i( ,nP) and hence its leading coefficient V ( ). P P

Lattice Points in Polytopes – p. 17 An example: Reeve’s theorem

Example. When d = 3, V ( ) is determined by P i( , 1) = #( Z3) P P ∩ i( , 2) = #(2 Z3) P P ∩ 3 ¯i( , 1) = #( ◦ Z ), P P ∩ which gives Reeve’s theorem.

Lattice Points in Polytopes – p. 18 Birkhoff polytope

M M Example. Let BM R × be the Birkhoff polytope of all M ⊂M doubly-stochastic × matrices A = (aij), i.e., a 0 ij ≥

aij = 1 (column sums 1) i X

aij = 1 (row sums 1). j X

Lattice Points in Polytopes – p. 19 (Weak) magic squares

M M Note. B = (b ) n Z × if and only if ij ∈ BM ∩ b N = 0, 1, 2,... ij ∈ { }

bij = n i X

bij = n. j X

Lattice Points in Polytopes – p. 20 Example of a magic square

2104 3112   (M = 4, n = 7) 1321    1240     

Lattice Points in Polytopes – p. 21 Example of a magic square

2104 3112   (M = 4, n = 7) 1321    1240      7 ∈ B4

Lattice Points in Polytopes – p. 21 HM (n)

HM (n) := # M M N-matrices, line sums n { × } = i( ,n) BM

Lattice Points in Polytopes – p. 22 HM (n)

HM (n) := # M M N-matrices, line sums n { × } = i( ,n) BM

H1(n) = 1

H2(n) = ??

Lattice Points in Polytopes – p. 22 HM (n)

HM (n) := # M M N-matrices, line sums n { × } = i( ,n) BM

H1(n) = 1

H2(n) = n + 1

a n a − , 0 a n. n a a ≤ ≤ " − #

Lattice Points in Polytopes – p. 22 The case M = 3

n + 2 n + 3 n + 4 H (n) = + + 3 4 4 4       (MacMahon)

Lattice Points in Polytopes – p. 23 Values for small n

HM (0) = ??

Lattice Points in Polytopes – p. 24 Values for small n

HM (0) = 1

Lattice Points in Polytopes – p. 24 Values for small n

HM (0) = 1

HM (1) = ??

Lattice Points in Polytopes – p. 24 Values for small n

HM (0) = 1

HM (1) = M! (permutation matrices)

Lattice Points in Polytopes – p. 24 Values for small n

HM (0) = 1

HM (1) = M! (permutation matrices)

Anand-Dumir-Gupta, 1966:

xM H (2) =?? M M!2 M 0 X≥

Lattice Points in Polytopes – p. 24 Values for small n

HM (0) = 1

HM (1) = M! (permutation matrices)

Anand-Dumir-Gupta, 1966:

xM ex/2 HM (2) = M!2 √1 x M 0 X≥ −

Lattice Points in Polytopes – p. 24 Anand-Dumir-Gupta conjecture

Theorem (Birkhoff-von Neumann). The vertices of M consist of the M! M M permutation matrices.B Hence is a lattice× polytope. BM

Lattice Points in Polytopes – p. 25 Anand-Dumir-Gupta conjecture

Theorem (Birkhoff-von Neumann). The vertices of M consist of the M! M M permutation matrices.B Hence is a lattice× polytope. BM Corollary (Anand-Dumir-Gupta conjecture). H (n) is a polynomial in n (of degree (M 1)2). M −

Lattice Points in Polytopes – p. 25 H4(n)

1 Example. H (n)= 11n9 + 198n8 + 1596n7 4 11340 +7560n6 + 23289n5 + 48762n5 + 70234n4 + 68220n2 +40950n + 11340) .

Lattice Points in Polytopes – p. 26 Reciprocity for magic squares

Reciprocity H ( n)= ⇒± M − # M M matrices B of positive integers, line sum n { × } But every such B can be obtained from an M M matrix A of nonnegative integers by adding× 1 to each entry.

Lattice Points in Polytopes – p. 27 Reciprocity for magic squares

Reciprocity H ( n)= ⇒± M − # M M matrices B of positive integers, line sum n { × } But every such B can be obtained from an M M matrix A of nonnegative integers by adding× 1 to each entry. Corollary. H ( 1) = H ( 2) = = H ( M + 1) = 0 M − M − ··· M − M 1 H ( M n) = ( 1) − H (n) M − − − M

Lattice Points in Polytopes – p. 27 Two remarks

Reciprocity greatly reduces computation. Applications of magic squares, e.g., to statistics (contingency tables).

Lattice Points in Polytopes – p. 28 Zeros of H9(n) in complex plane

Zeros of H_9(n) 3

2

1

–8 –6 –4 –2 0

–1

–2

–3

Lattice Points in Polytopes – p. 29 Zeros of H9(n) in complex plane

Zeros of H_9(n) 3

2

1

–8 –6 –4 –2 0

–1

–2

–3

No explanation known.

Lattice Points in Polytopes – p. 29 Zonotopes

d Let v1,...,vk R . The zonotope Z(v ,...,v ) ∈ 1 k generated by v1,...,vk:

Z(v1,...,vk) = λ v + + λ v : 0 λ 1 { 1 1 ··· k k ≤ i ≤ }

Lattice Points in Polytopes – p. 30 Zonotopes

d Let v1,...,vk R . The zonotope Z(v ,...,v ) ∈ 1 k generated by v1,...,vk:

Z(v1,...,vk) = λ v + + λ v : 0 λ 1 { 1 1 ··· k k ≤ i ≤ } Example. v1 = (4, 0), v2 = (3, 1), v3 = (1, 2)

(1,2) (3,1)

(0,0) (4,0)

Lattice Points in Polytopes – p. 30 Lattice points in a zonotope

Theorem. Let Z = Z(v ,...,v ) Rd, 1 k ⊂ where v Zd. Then i ∈ i(Z, 1) = h(X), XX where X ranges over all linearly independent subsets of v1,...,vk , and h(X) is the gcd of all j j minors{ (j = #X}) of the matrix whose rows are× the elements of X.

Lattice Points in Polytopes – p. 31 An example

Example. v1 = (4, 0), v2 = (3, 1), v3 = (1, 2)

(1,2) (3,1)

(0,0) (4,0)

Lattice Points in Polytopes – p. 32 Computation of i(Z, 1)

4 0 4 0 3 1 i(Z, 1) = + + 3 1 1 2 1 2

+gcd(4 , 0) + gcd(3 , 1) +gcd(1 , 2) + det( ) ∅ = 4+8+5+4+1+1+1 = 24.

Lattice Points in Polytopes – p. 33 Computation of i(Z, 1)

4 0 4 0 3 1 i(Z, 1) = + + 3 1 1 2 1 2

+gcd(4 , 0) + gcd(3 , 1) +gcd(1 , 2) + det( ) ∅ = 4+8+5+4+1+1+1 = 24.

Lattice Points in Polytopes – p. 33 Corollaries

Corollary. If Z is an integer zonotope generated by integer vectors, then the coefficients of i(Z,n) are nonnegative integers.

Lattice Points in Polytopes – p. 34 Corollaries

Corollary. If Z is an integer zonotope generated by integer vectors, then the coefficients of i(Z,n) are nonnegative integers. Neither property is true for general integer polytopes. There are numerous conjectures concerning special cases.

Lattice Points in Polytopes – p. 34 The permutohedron

d Πd = conv (w(1),...,w(d)): w S R { ∈ d}⊂

Lattice Points in Polytopes – p. 35 The permutohedron

d Πd = conv (w(1),...,w(d)): w S R { ∈ d}⊂

d + 1 dim Π = d 1, since w(i)= d − 2 X  

Lattice Points in Polytopes – p. 35 The permutohedron

d Πd = conv (w(1),...,w(d)): w S R { ∈ d}⊂

d + 1 dim Π = d 1, since w(i)= d − 2 X   Π Z(e e : 1 i < j d) d ≈ i − j ≤ ≤

Lattice Points in Polytopes – p. 35 Π3

123

132 213

Π3 222 231 312

321

Lattice Points in Polytopes – p. 36 Π3

123

132 213

Π3 222 231 312

321

2 i(Π3,n) = 3n + 3n + 1

Lattice Points in Polytopes – p. 36 Π4

(truncated octahedron)

Lattice Points in Polytopes – p. 37 i(Πd,n)

d 1 k Theorem. i(Πd,n)= k−=0 fk(d)x , where fk(d) = # forests withPk edges on vertices 1,...,d { }

Lattice Points in Polytopes – p. 38 i(Πd,n)

d 1 k Theorem. i(Πd,n)= k−=0 fk(d)x , where fk(d) = # forests withPk edges on vertices 1,...,d { } 1 2

3

2 i(Π3,n) = 3n + 3n + 1

Lattice Points in Polytopes – p. 38 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 { }

di = degree (# incident edges) of vertex i. Define the ordered degree sequence d(G) of G by d(G) = (d1,...,dn).

Lattice Points in Polytopes – p. 39 Example of d(G)

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

4 5 6

Lattice Points in Polytopes – p. 40 # ordered degree sequences

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

Lattice Points in Polytopes – p. 41 f(n) for n ≤ 4

Example. If n 3, all d(G) are distinct, so ≤ f(1) = 1, f(2) = 21 = 2, f(3) = 23 = 8. For n 4 we can have G = H but d(G)= d(H), e.g., ≥ 6 12 1 2 1 2

3 4 3 4 3 4

In fact, f(4) = 54 < 26 = 64.

Lattice Points in Polytopes – p. 42 The polytope of degree sequences

Let conv denote convex hull, and

n Dn = conv d(G) : V (G)= 1,...,n R , { { }} ⊂ the polytope of degree sequences (Perles, Koren).

Lattice Points in Polytopes – p. 43 The polytope of degree sequences

Let conv denote convex hull, and

n Dn = conv d(G) : V (G)= 1,...,n R , { { }} ⊂ the polytope of degree sequences (Perles, Koren).

Easy fact. Let ei be the ith unit coordinate vector n in R . E.g., if n = 5 then e2 = (0, 1, 0, 0, 0). Then = Z(e + e : 1 i < j n). Dn i j ≤ ≤

Lattice Points in Polytopes – p. 43 The Erdos-Gallai˝ theorem

Theorem. Let α = (a ,...,a ) Zn. 1 n ∈ Then α = d(G) for some G if and only if α ∈ Dn a + a + + a is even. 1 2 ··· n

Lattice Points in Polytopes – p. 44 A generating function

Enumerative techniques leads to: Theorem. Let xn F (x) = f(n) n! n 0 X≥ x2 x3 x4 = 1+ x + 2 + 8 + 54 + . 2! 3! 4! ··· Then:

Lattice Points in Polytopes – p. 45 A formula for F (x)

1/2 1 xn F (x)= 1 + 2 nn 2  n! n 1 ! X≥  n n 1 x 1 (n 1) − + 1 × − − n! n 1 ! # X≥ n n 2 x 0 exp n − (0 = 1) × n! n 1 X≥

Lattice Points in Polytopes – p. 46 Coefficients of i(P,n)

Let denote the tetrahedron with vertices (0, 0P, 0), (1, 0, 0), (0, 1, 0), (1, 1, 13). Then 13 1 i( ,n)= n3 + n2 n + 1. P 6 − 6

Lattice Points in Polytopes – p. 47 The “bad” tetrahedron

y

x

z

Lattice Points in Polytopes – p. 48 The “bad” tetrahedron

y

x

z

Thus in general the coefficients of Ehrhart polynomials are not “nice.” Is there a “better” basis?

Lattice Points in Polytopes – p. 48 The h∗-vector of i(P,n)

Let be a lattice polytope of dimension d. Since P i( ,n) is a polynomial of degree d, hi Z such thatP ∃ ∈ h + h x + + h xd i( ,n)xn = 0 1 ··· d . P (1 x)d+1 n 0 X≥ −

Lattice Points in Polytopes – p. 49 The h∗-vector of i(P,n)

Let be a lattice polytope of dimension d. Since P i( ,n) is a polynomial of degree d, hi Z such thatP ∃ ∈ h + h x + + h xd i( ,n)xn = 0 1 ··· d . P (1 x)d+1 n 0 X≥ − Definition. Define

h(P) = (h0,h1,...,hd), the h∗-vector of . P

Lattice Points in Polytopes – p. 49 Example of an h∗-vector

Example. Recall 1 i( ,n)= (11n9 B4 11340 +198n8 + 1596n7 + 7560n6 + 23289n5 +48762n5 + 70234n4 + 68220n2 +40950n + 11340).

Lattice Points in Polytopes – p. 50 Example of an h∗-vector

Example. Recall 1 i( ,n)= (11n9 B4 11340 +198n8 + 1596n7 + 7560n6 + 23289n5 +48762n5 + 70234n4 + 68220n2 +40950n + 11340). Then

h∗( ) = (1, 14, 87, 148, 87, 14, 1, 0, 0, 0). B4

Lattice Points in Polytopes – p. 50 Elementary properties of h∗(P)

h0∗ = 1 dim h∗ = ( 1) Pi( , 1) = I( ) d − P − P max i : h∗ = 0 = min j 0 : { i 6 } { ≥ i( , 1) = i( , 2) = = i( , (d j)) = 0 P − P − ··· P − − }

E.g., h∗( ) = (h0∗,...,hd∗ 2, 0, 0) i( , 1) = P − ⇔ P − i( , 2) = 0. P −

Lattice Points in Polytopes – p. 51 Another property

i( , n k) = ( 1)d i( ,n) n P − − − P ∀ ⇔ hi∗ = hd∗+1 k i i, and − − ∀

hd∗+2 k i = hd∗+3 k i = = hd∗ = 0 − − − − ···

Lattice Points in Polytopes – p. 52 Back to B4

Recall:

h∗( ) = (1, 14, 87, 148, 87, 14, 1, 0, 0, 0). B4 Thus i( , 1) = i( , 2) = i( , 3) = 0 B4 − B4 − B4 − i( , n 4) = i( ,n). B4 − − − B4

Lattice Points in Polytopes – p. 53 Main properties of h∗(P)

Theorem A (nonnegativity). (McMullen, RS) h∗ 0. i ≥

Lattice Points in Polytopes – p. 54 Main properties of h∗(P)

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

Lattice Points in Polytopes – p. 54 Main properties of h∗(P)

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

B A: take = . ⇒ Q ∅

Lattice Points in Polytopes – p. 54 Proofs

Both theorems can be proved geometrically. There are also elegant algebraic proofs based on commutative algebra.

Lattice Points in Polytopes – p. 55 Further directions

I. Zeros of Ehrhart polynomials Sample theorem (de Loera, Develin, Pfeifle, RS). Let be a lattice d-polytope. Then P i( , α) = 0, α R d α d/2 . P ∈ ⇒− ≤ ≤ ⌊ ⌋

Lattice Points in Polytopes – p. 56 Further directions

I. Zeros of Ehrhart polynomials Sample theorem (de Loera, Develin, Pfeifle, RS). Let be a lattice d-polytope. Then P i( , α) = 0, α R d α d/2 . P ∈ ⇒− ≤ ≤ ⌊ ⌋ Theorem. Let d be odd. There exists a 0/1 d-polytope d and a real zero αd of i( d,n) such that P P α 1 lim d = = 0.0585 . d→∞ d odd d 2πe ···

Lattice Points in Polytopes – p. 56 An open problem

Open. Is the set of all complex zeros of all Ehrhart polynomials of lattice polytopes dense in C? (True for chromatic polynomials of graphs.)

Lattice Points in Polytopes – p. 57 II. Brion’s theorem

Example. Let P be the polytope [2, 5] in R, so is defined by P (1) x 2, (2) x 5. ≥ ≤

Lattice Points in Polytopes – p. 58 II. Brion’s theorem

Example. Let P be the polytope [2, 5] in R, so is defined by P (1) x 2, (2) x 5. ≥ ≤ Let t2 F (t) = tn = 1 1 t n 2 − Xn≥Z ∈ 5 n t F2(t) = t = . 1 1 n 5 t Xn≤Z − ∈ Lattice Points in Polytopes – p. 58 F1(t) + F2(t)

t2 t5 F1(t)+ F2(t) = + 1 t 1 1 − − t = t2 + t3 + t4 + t5 = tm. m Z X∈P∩

Lattice Points in Polytopes – p. 59 Cone at a vertex

N P: Z-polytope in R with vertices v1,..., vk

Ci: cone at vertex v supporting i P

Lattice Points in Polytopes – p. 60 Cone at a vertex

N P: Z-polytope in R with vertices v1,..., vk

Ci: cone at vertex v supporting i P v

C (v)

Lattice Points in Polytopes – p. 60 The general result

m1 mN Let Fi(t1,...,tN )= t1 tN . N ··· (m1,...,mN ) i Z X∈C ∩

Lattice Points in Polytopes – p. 61 The general result

m1 mN Let Fi(t1,...,tN )= t1 tN . N ··· (m1,...,mN ) i Z X∈C ∩

Theorem (Brion). Each Fi is a rational function of t1,...,tN , and

k m1 mN Fi(t1,...,tN )= t1 tN N ··· i=1 (m1,...,mN ) Z X X∈P∩ (as rational functions).

Lattice Points in Polytopes – p. 61 III. Toric varieties

Given an integer polytope , can define a P projective algebraic variety XP ,a toric variety. Leads to deep connections with toric geometry, including new formulas for i( ,n). P

Lattice Points in Polytopes – p. 62 IV. Complexity

Computing i( ,n), or even i( , 1) is #P -completeP. Thus an “efficient”P (polynomial time) algorithm is extremely unlikely. However:

Lattice Points in Polytopes – p. 63 IV. Complexity

Computing i( ,n), or even i( , 1) is #P -completeP. Thus an “efficient”P (polynomial time) algorithm is extremely unlikely. However:

Theorem (A. Barvinok, 1994). For fixed dim , polynomial-time algorithm for computing i( ,nP).∃ P

Lattice Points in Polytopes – p. 63 V. Fractional lattice polytopes

Example. Let SM (n) denote the number of symmetric M M matrices of nonnegative integers, every× row and column sum n. Then

1 3 2 8(2n + 9n + 14n + 8), n even S3(n) = 1 3 2 ( 8(2n + 9n + 14n + 7), n odd 1 = (4n3 + 18n2 + 28n + 15 + ( 1)n). 16 −

Lattice Points in Polytopes – p. 64 V. Fractional lattice polytopes

Example. Let SM (n) denote the number of symmetric M M matrices of nonnegative integers, every× row and column sum n. Then

1 3 2 8(2n + 9n + 14n + 8), n even S3(n) = 1 3 2 ( 8(2n + 9n + 14n + 7), n odd 1 = (4n3 + 18n2 + 28n + 15 + ( 1)n). 16 −

Why a different polynomial depending on n modulo 2?

Lattice Points in Polytopes – p. 64 The symmetric Birkhoff polytope

TM : the polytope of all M M symmetric doubly-stochastic matrices.×

Lattice Points in Polytopes – p. 65 The symmetric Birkhoff polytope

TM : the polytope of all M M symmetric doubly-stochastic matrices.× M M Easy fact: S (n) = # n Z × M TM ∩

Lattice Points in Polytopes – p. 65 The symmetric Birkhoff polytope

TM : the polytope of all M M symmetric doubly-stochastic matrices.× M M Easy fact: S (n) = # n Z × M TM ∩ Fact: vertices of have the form 1(
P + P t), TM 2 where P is a permutation matrix.

Lattice Points in Polytopes – p. 65 The symmetric Birkhoff polytope

TM : the polytope of all M M symmetric doubly-stochastic matrices.× M M Easy fact: S (n) = # n Z × M TM ∩ Fact: vertices of have the form 1(
P + P t), TM 2 where P is a permutation matrix.

M M Thus if v is a vertex of then 2v Z × . TM ∈

Lattice Points in Polytopes – p. 65 SM (n) in general

Theorem. There exist polynomials PM (n) and QM (n) for which S (n)= P (n) + ( 1)nQ (n), n 0. M M − M ≥ M Moreover, deg PM (n)= 2 .

Lattice Points in Polytopes – p. 66 SM (n) in general

Theorem. There exist polynomials PM (n) and QM (n) for which S (n)= P (n) + ( 1)nQ (n), n 0. M M − M ≥ M Moreover, deg PM (n)= 2 . Difficult result (W. Dahmen
and C. A. Micchelli, 1988):

n 1 −2 1, n odd deg QM (n)= n 2 − ( − 1, n even. 2
−

Lattice Points in Polytopes – p. 66 The last slide

Lattice Points in Polytopes – p. 67 The last slide

Lattice Points in Polytopes – p. 68 The last slide

Lattice Points in Polytopes – p. 68