Euler-Maclaurin formula for polytopes.

Euler-Maclaurin formula for polytopes.

- Common work with Nicole Berline.

Preprint in arkiv; CO/0507256.

- Thanks to Charles Cochet for drawings (the good ones).

. – p.1

Euler-Maclaurin formula in dimension 1.

Consider the interval [0, k ] and the function x 7→ x

on R . Compute the sum of x

over the first k + 1 integers, and compare it to the integral R

0

x

dx =

.

0

+ 1

+ 2

+ · · · + k

= k + 1.

0 + 1 + 2 + 3 + · · · + k = k (k + 1)

2 .

0

+ 1

+ 2

+ 3

+ · · · + k

= k

3 + k

2 + k 6 .

· · ·

Bernoulli polynomials

Theorem: (Jakob BERNOULLI -(1654-1705)) For any non negative integer m , the sum

S

(k) =

X

a

of the m

-powers of the k + 1 first integers is given by a polynomial in k .

. – p.3

IMPORTANT for computational purposes: Fix m . Then the polynomial behaviour in k of S

(k) =

+ · · · implies

that to compute the value of S

(k) (a sum of k numbers),

we need only a number of operations only less or equal to

C ∗ Log(k )

. In other words, we can compute this sum

in "polynomial time" with respect to k .

Bernoulli numbers

Define the analytic function Ber (u)

= 1

(1 − exp(u)) + 1

u = 1

2 − u

12 + · · · = −

X

B

u

(j + 1)!

B

Bernoulli numbers. B

= 0 if j is odd, except B

= 1/2 .

. – p.5

Euler-Maclaurin formula for the interval

Let φ be a polynomial function over R , A and B two integers.

S (φ, A, B ) =

X

φ(a), a runs over integers between A , B . Then

S (φ, A, B ) =

Z

φ(x)dx + R

R sum of derivatives of φ at boundary of interval R := 1

2 (φ(A) + φ(B )) +

X

B

2j ! (φ

(A) − φ

(B ))

Euler-Maclaurin formula in dimension 1 More condensed formula:

S (φ, A, B ) =

Z

φ(x)dx+(Ber(−∂

)φ)(B )+(Ber (∂

)φ)(A).

. – p.7

We want to generalize this formula

Let P be a polytope in R

, with vertices in Z

, φ a polynomial function: compute

X

a∈P∩L

φ(a) = X

F f aces de P

Z

F

(D

φ(x))dx

with D

différential operators with constant coefficients.

We will impose conditions on D

that I will describe later.

Idea of the proof of Euler-Maclaurin formula

Do it for cones and decompose a polytope in cones.

. – p.9

B B

B B

A A

A

A

Important formulae

(1 − e

)

X

e

= 1

(1 − e

)

X

e

= 0 Continuous version

u

Z

x=0

e

dx = 1 u

Z

x=−∞

e

dx = 0

. – p.11

Thus explicit formula for summing exponentials

u négatif.

S (u) :=

X

e

= 1 1 − e

I (u) :=

Z

0

e

dx = 1

−u Thus

X

e

= e

1 − e

+ e

1 − e

AS

X

e

= 0.

Z

A

e

dx = e

−u + e

u AS

Z

−∞

e

dx = 0.

Comparison between sums and integrals for cones We want to write

X

e

in function of the integral and of a sum of derivatives of e

at the boundary.

Compare S (u) =

et I (u) =

. ??:

S (u) = I (u) + Ber (u) with Ber (u) =

+

ANALYTIQUE.

S (u) = I (u) + (Ber (∂

)e

)

.

. – p.13

Any dimension.

Polytope P ⊂ R

with vertices in Z

. φ polynomial function on R

:

Sum(φ, P ) = X

a∈Z^d

φ(a).

Theorem: Ehrhart (generalizes d Bernoulli) E (k ) = X

a∈kP∩Z^d

φ(a) is a polynomial in k

= Z

kP

φ(x)dx + · · ·

Ehrhart

Example: φ = 1 .

The number card(kP ∩ Z

) of integral points in kP is a polynômial in k : Ehrhart polynomial.

Example: standard simplex in R

. Polynôme d’Ehrhart

d!

.

No formula in general, even for simplices.

Mordell example: simplex in R

with vertices

[0, 0, 0] [a, 0, 0], [0, b, 0], [0, 0, c].

. – p.15

o 7 8

1 2 3 4 5 6

1 2 3 4 5 6 7 8

k

o k

a b

volume(k∆) =

, |(k ∆ ∩ Z

)| =

Euler-MacLaurin formula in any dimension:

Let V be a real vector space with a lattice L , P a rational polytope in V , φ a polynomial function on V :

we want to write X

a∈P∩L

φ(a) = X

F f aces de P

Z

F

(D

φ(x))dx

with D

differential operators with constant coefficients.

Here dx is the canonical measure on faces determined by the lattice. WITH Conditions satisfied by D

:

. – p.17

Normal cone

Normal cone to a face of a polytope P : projection of the tangent cone in the vector space V /lin(F ) with lattice.

To see it: intersect the polytope by a the orthogonal space to F passing through a generic point of F .

Cone normal à l’arête.

normalcone

Conditions satisfied by D

Locality: Operators D

depends only of the normal cone.

Invariants: Invariants modulo translation by an element of the lattice.

: Explicit formula for D

, computable in polynomial time up to some order when the dimension and the order of D

are fixed.

Calculable: Function P

a∈P∩Z^d

φ(a) is computable in polynomial time for dimV fixed and degree of φ fixed

(BARVINOK 1994), so we want same computability for the D

.

. – p.19

What was known before ??.

Mac-Mullen: proof of the existence of the operators D

satisfying locality and invariance. But no construction.

Cappell-Shaneson, Brion-Vergne: Existence and

explicit construction of operators D

, but the operators where depending of the fan and are not local.

Pommersheim ( Danilov conjecture): For P integral, construction (via desingularisation of toric varieties) of rational and local numbers µ(F ) such that

card(P ∩ L) = X

F

µ(F )vol(F )

Theorem: Berline-Vergne

We give ourselves a rational scalar product on V . then X

a∈P∩L

φ(a) = X

F f aces de P

Z

F

(D

φ(x))dx

with D

differential operators with constant coefficients : -Local,invariant, rational, derivation with respect to normal directions to the face.

-explicit construction, computable in polynomial time .

. – p.21

Example for polygones in R

. Pick theorem:

Let P be a polygone in R

, with vertices with integral coordinates. then

Number of points in Z

∩ P

= Area P + 1 2

X

a=edges

lenght

(a) + 1 Pommersheim-Berline-Vergne:

= Area P + 1 2

X

a=edges

lenght

(a) + X

s=vertices

µ(s).

Example of constants µ(s) adding to 1 .

1 4 1

4

1 4

1 6 1 4

1 4

CS BV

1 4

1 4

1 3

3 8

1 4

1 8

The middle drawing is the constants found by

Cappell-Shaneson (CRAS). Clearly not local constants.

Berline-Vergne (If P is good):

µ(v ) = 1

4 + hα, β i

12 ( 1

hα, αi + 1

hβ, β i ).

Otherwise, formulae with Dedekind sums (computable by reciprocity).

. – p.23

Another example

µ([0,0]) = ₂₀₀⁸³

µ([3,1]) = ₁₀³

µ([4,4]) = ₄₀⁹ µ([3,4]) = ₅₀³

0 1 2 3 4

1 2 3 4

µ([0, 0]) = 83/200 , µ([3, 1]) = 3/10 , µ([4, 4]) = 9/40 , µ([3, 4]) = 3/50 ;

Number of points in Z

∩ P = 9 ;

Area of P = 6 , 4 edges of lenght 1 (with respect to the lattice):

. – p.24

D C

B

A

P

a∈(P∩Z²)

φ(a) = R

P

φ(x)dx + R

A

(D

φ)(x)dx + R

B

(D

φ)(x)dx + R

C

(D

φ)(x)dx + R

D

(D

φ)(x)dx

+(D

φ)(A) + (D

φ)(B ) + (D

φ)(C ) + (D

φ)(D ).

Remarque: opérateurs correspondant aux arêtes déjà connus, car cône normal de dimension 1 .

Exemple: D

. – p.25

0pérateurs différentiels D

s’additionnant à 1

D

= 83

200 − 1

4 ∂

− 5

24 ∂

+ · · · D

= 3

10 + 1

12 ∂

− 1

12 ∂

+ · · · D

= 9

40 + 1

12 ∂

+ 1

8 ∂

+ · · · D

= 3

50 + 1

12 ∂

+ 1

6 ∂

+ · · ·

A function on cones.

Let V be a vector space with a lattice L and scalar product.

Let Cones the set of affine rational cones in V

There exists a unique function µ on Cones with values in analytic function on V

(defined near 0 ) such that

1): µ({s}) = 1 or 0 according to the fact that s in integer or not.

2) For any rational cone C in V and −u in the dual cone in V

X

ξ∈C∩L

e

= X

F f acesof C

µ(N ormal(C, F ))(u) Z

F

e

dx.

. – p.27

Properties of µ

1) µ(C ) = 0 if the cone C is not acute.

2) µ is invariant by translation by an integral vector.

2) µ is additive when restricted to cones with fixed vertice

s .

Euler-MacLaurin pour les cones

Transformer µ en opérateur différentiel:

X

ξ∈C∩L

e

= X

F f acesdeC

Z

F

µ(N ormal(C, F ))(∂

)e

dx.

Propriétés de µ :

-Invariante par translations.

-Additive sur les cônes duaux:

-Calculable en temps polynomial.

. – p.29

translation: equivariant homology of toric varieties Let T be a torus with Lie algebra t .

Lattice t

avec T = t /2iπ t

.

Ev a fan: decomposition of t in rational cones.

Corresponds to a variety M (Ev ) : each cone σ of the fan

indexe an open affine set U

of M (Ev ) invariant by T

.

This open set contains a unique closed orbit O

of T

.

P

( C )

Example: T = {e

} = R /2π Z

Ev = { R

} ∪ {0} ∪ { R

} M (Ev ) = P

( C )

U

= C with fixed point 0 = N U

= C with fixed point: 0 = S . U

= C

with orbit C

.

P

( C ) = C ∪ C guled over C

3 orbites de C

.

. – p.31

P

( C ) glued with affines open set

S N

S

N

C C C \ {0}

et et

3 invariant cycles [P

( C )] , N et S .

Equivariant Homology

We can define the equivariant homology H

(M ) of an

algebraic variety M provided with an algebraic action of a torus T = ( C

)

. This is a module under the action of

S ( t

) . Equivariant homology is generated by the cycles invariants by T with relations.

S N

S

N

C C C \ {0}

et et

Relations u[P

( C )] = [N ] − [S ] in equivariant homology, au lieu de N = S ( u = 0 ).

. – p.33

Equivariant Todd class

The equivariant Todd class of M belongs to S ˆ ( t

)H

(M ) Recall that we have associated to any rational cone in V a fonction on V

.

Theorem: Let Ev be a fan and M (Ev ) the corresponding variety Then the equivariant Todd class of M (Ev ) is equal to

T odd(M (Ev )) = X

σ∈Ev

µ(σ

)[O

].

Euler-Maclaurin in equivariant homology

Return to P

( C ) . In homology:

T odd(P

( C )) = [P

( C )] + ( 1

1 − e

− 1

u )[N ] + ( 1

1 − e

+ 1

u )[S ].

. – p.35

Question

If we write T odd(P

( C )) in a basis, we only need two

orbits: for example Schubert cells [P

] , and N , as we have the relation u[P

] = [S ] − [N ] .

T odd(P

) = u

1 − e

[P

] + 1

1 − e

[N ].

Not so nice.

Is there a beautiful invariant formula for Todd class of flag

varieties ??