# Lattice Points in Polytopes - MIT Mathematics rstan/transparencies/lp-  · PDF fileBoundary and interior lattice points Lattice Points in Polytopes – p. 3. ... Birkhoff polytope

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• Lattice Points in Polytopes

Richard P. Stanley

M.I.T.

Lattice Points in Polytopes p. 1

• A lattice polygon

Georg Alexander Pick (18591942)

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

• Picks theorem

A = area of PI = # interior points of P (= 4)B = #boundary points of P (= 10)

Then

A =2I + B 2

2.

Example on previous slide:

2 4 + 10 22

= 9.

Lattice Points in Polytopes p. 4

• Picks theorem

A = area of PI = # interior points of P (= 4)B = #boundary points of P (= 10)

Then

A =2I + B 2

2.

Example on previous slide:

2 4 + 10 22

= 9.

Lattice Points in Polytopes p. 4

• Two tetrahedra

Picks theorem (seemingly) fails in higherdimensions. For example, let T1 and T2 be thetetrahedra with vertices

v(T1) = {(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)}

v(T2) = {(0, 0, 0), (1, 1, 0), (1, 0, 1), (0, 1, 1)}.

Lattice Points in Polytopes p. 5

• Failure of Picks theorem in dim 3

ThenI(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 afinite set of points) in Rd. For n 1, let

nP = {n : P}.

3PP

Lattice Points in Polytopes p. 7

• Polytope dilation

Let P be a convex polytope (convex hull of afinite set of points) in Rd. For n 1, let

nP = {n : P}.

3PPLattice Points in Polytopes p. 7

• i(P, n)

Let

i(P, n) = #(nP Zd)

= #{ P : n Zd},

the number of lattice points in nP.

Lattice Points in Polytopes p. 8

• i(P, n)

Similarly let

P = interior of P = P P

i(P, n) = #(nP Zd)

= #{ P : n Zd},

the number of lattice points in the interior of nP.

Note. Could use any lattice L instead of Zd.

Lattice Points in Polytopes p. 9

• i(P, n)

Similarly let

P = interior of P = P P

i(P, n) = #(nP Zd)

= #{ P : n Zd},

the number of lattice points in the interior of nP.

Note. Could use any lattice L instead of Zd.

Lattice Points in Polytopes p. 9

• An example

P 3P

i(P, n) = (n + 1)2

i(P, n) = (n 1)2 = i(P,n).

Lattice Points in Polytopes p. 10

• Reeves theorem

lattice polytope: polytope with integer vertices

Theorem (Reeve, 1957). Let P be athree-dimensional lattice polytope. Then thevolume V (P) is a certain (explicit) function ofi(P, 1), i(P, 1), and i(P, 2).

Lattice Points in Polytopes p. 11

• The main result

Theorem (Ehrhart 1962, Macdonald 1963) Let

P = lattice polytope in RN , dimP = d.

Then i(P, n) is a polynomial (the Ehrhartpolynomial of P) in n of degree d.

Lattice Points in Polytopes p. 12

• Reciprocity and volume

Moreover,

i(P, 0) = 1

i(P, n) = (1)di(P,n), n > 0

(reciprocity).

If d = N then

i(P, n) = V (P)nd + lower order terms,

where V (P) is the volume of P.

Lattice Points in Polytopes p. 13

• Reciprocity and volume

Moreover,

i(P, 0) = 1

i(P, n) = (1)di(P,n), n > 0

(reciprocity).

If d = N then

i(P, n) = V (P)nd + lower order terms,

where V (P) is the volume of P.

Lattice Points in Polytopes p. 13

• Generalized Picks theorem

Corollary. Let P Rd and dimP = d. Knowingany d of i(P, n) or i(P, n) for n > 0 determinesV (P).

Proof. Together with i(P, 0) = 1, this datadetermines d + 1 values of the polynomial i(P, n)of degree d. This uniquely determines i(P, n)and hence its leading coefficient V (P).

Lattice Points in Polytopes p. 14

• Generalized Picks theorem

Corollary. Let P Rd and dimP = d. Knowingany d of i(P, n) or i(P, n) for n > 0 determinesV (P).

Proof. Together with i(P, 0) = 1, this datadetermines d + 1 values of the polynomial i(P, n)of degree d. This uniquely determines i(P, n)and hence its leading coefficient V (P).

Lattice Points in Polytopes p. 14

• An example: Reeves theorem

Example. When d = 3, V (P) is determined by

i(P, 1) = #(P Z3)

i(P, 2) = #(2P Z3)

i(P, 1) = #(P Z3),

which gives Reeves theorem.

Lattice Points in Polytopes p. 15

• Birkhoff polytope

Example. Let BM RMM be the Birkhoffpolytope of all M M doubly-stochasticmatrices A = (aij), i.e.,

aij 0

i

aij = 1 (column sums 1)

j

aij = 1 (row sums 1).

Lattice Points in Polytopes p. 16

• (Weak) magic squares

Note. B = (bij) nBM ZMM if and only if

bij N = {0, 1, 2, . . . }

i

bij = n

j

bij = n.

Lattice Points in Polytopes p. 17

• Example of a magic square

2 1 0 4

3 1 1 2

1 3 2 1

1 2 4 0

(M = 4, n = 7)

7B4

Lattice Points in Polytopes p. 18

• Example of a magic square

2 1 0 4

3 1 1 2

1 3 2 1

1 2 4 0

(M = 4, n = 7)

7B4

Lattice Points in Polytopes p. 18

• HM(n)

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

H1(n) = 1

H2(n) = n + 1

[

a n a

n a a

]

, 0 a n.

Lattice Points in Polytopes p. 19

• HM(n)

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

H1(n) = 1

H2(n) = n + 1

[

a n a

n a a

]

, 0 a n.

Lattice Points in Polytopes p. 19

• The case M = 3

H3(n) =

(

n + 2

4

)

+

(

n + 3

4

)

+

(

n + 4

4

)

(MacMahon)

Lattice Points in Polytopes p. 20

• The Anand-Dumir-Gupta conjecture

HM(0) = 1

HM(1) = M ! (permutation matrices)

Theorem (Birkhoff-von Neumann). The verticesof BM consist of the M ! M M permutationmatrices. Hence BM is a lattice polytope.

Corollary (Anand-Dumir-Gupta conjecture).HM(n) is a polynomial in n (of degree (M 1)2).

Lattice Points in Polytopes p. 21

• The Anand-Dumir-Gupta conjecture

HM(0) = 1

HM(1) = M ! (permutation matrices)

Theorem (Birkhoff-von Neumann). The verticesof BM consist of the M ! M M permutationmatrices. Hence BM is a lattice polytope.

Corollary (Anand-Dumir-Gupta conjecture).HM(n) is a polynomial in n (of degree (M 1)2).

Lattice Points in Polytopes p. 21

• H4(n)

Example. H4(n) =1

11340

(

11n9 + 198n8 + 1596n7

+7560n6 + 23289n5 + 48762n5 + 70234n4 + 68220n2

+40950n + 11340) .

Lattice Points in Polytopes p. 22

• Reciprocity for magic squares

Reciprocity HM(n) =

#{MM matrices B of positive integers, line sum n}.

But every such B can be obtained from anM M matrix A of nonnegative integers byadding 1 to each entry.

Corollary.HM(1) = HM(2) = = HM(M + 1) = 0

HM(M n) = (1)M1HM(n)

Lattice Points in Polytopes p. 23

• Reciprocity for magic squares

Reciprocity HM(n) =

#{MM matrices B of positive integers, line sum n}.

But every such B can be obtained from anM M matrix A of nonnegative integers byadding 1 to each entry.

Corollary.HM(1) = HM(2) = = HM(M + 1) = 0

HM(M n) = (1)M1HM(n)

Lattice Points in Polytopes p. 23

• Two remarks

Reciprocity greatly reduces computation.

Applications of magic squares, e.g., tostatistics (contingency tables).

Lattice Points in Polytopes p. 24

• Zeros of H9(n) in complex plane

Zeros of H_9(n)

3

2

1

0

1

2

3

8 6 4 2

No explanation known.

Lattice Points in Polytopes p. 25

• Zeros of H9(n) in complex plane

Zeros of H_9(n)

3

2

1

0

1

2

3

8 6 4 2

No explanation known.

Lattice Points in Polytopes p. 25

• Zonotopes

Let v1, . . . , vk Rd. The zonotope Z(v1, . . . , vk)generated by v1, . . . , vk:

Z(v1, . . . , vk) = {1v1 + + kvk : 0 i 1}

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

(4,0)

(3,1)(1,2)

(0,0)

Lattice Points in Polytopes p. 26

• Zonotopes

Let v1, . . . , vk Rd. The zonotope Z(v1, . . . , vk)generated by v1, . . . , vk:

Z(v1, . . . , vk) = {1v1 + + kvk : 0 i 1}

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

(4,0)

(3,1)(1,2)

(0,0)

Lattice Points in Polytopes p. 26

• Lattice points in a zonotope

Theorem. Let

Z = Z(v1, . . . , vk) Rd,

where vi Zd. Then

i(Z, 1) =

X

h(X),

where X ranges over all linearly independentsubsets of {v1, . . . , vk}, and h(X) is the gcd of allj j minors (j = #X) of the matrix whose rowsare the elements of X.

Lattice Points in Polytopes p. 27

• An example

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

(4,0)

(3,1)(1,2)

(0,0)

Lattice Points in Polytopes p. 28

• Computation of i(Z, 1)

i(Z, 1) =

4 0

3 1

+

4 0

1 2

+

3 1

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. 29

• Computation of i(Z, 1)

i(Z, 1) =

4 0

3 1

+

4 0

1 2

+

3 1

1 2

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