Asymptotics for the Ehrhart polynomial of a polytope and intermediate Todd classes

Asymptotics for the Ehrhart polynomial of a polytope and intermediate Todd classes

May 2011 in Mittag-leffler

Let P be a polytope in R

with rational vertices. The Ehrhart polynomial Eh(P )(k) computes the number of integral points in the dilated polytope kP . In other words, if M is a toric variety of dimension n with ample line bundle L, Eh(P )(k) = dim(H

(M, L

)). The coefficient of k

in Eh(P )(k) is thus the symplectic volume of the manifold M , with respect to the symplectic form determined by L. Inspired by results of Barvinok on the complexity of the Ehrhart polynomial (arXiv 050444), I will introduce intermediate Todd classes T

(M ). One application (see arXiv 1011.1602) is the efficient com- putation of the top coefficients k

, i = 0, 1, 2, ..., c when M is a weighted projective space (c being fixed). This is also known as the knapsack problem.

However, the geometric interpretation of these Todd classes is unclear to me.

History and Motivations

Consider a system of inequalities f

(x) <= a

in R

with integral coefficients.

How to see if there is a points x ∈ Z

satisfying these inequations ?. For example in R, the inequalities 2x ≥ 1, 2x ≤ 3 do not have solutions. But this can be more hidden: Example: x

≥ 1/4, x

≥ 1/4, 1000 ∗ x

+ x

≤ 1000, interior piece of a large area triangle with no interior points.

H. Lenstra showed in 1983 that there is, for a fixed dimension n, a poly- nomial time algorithm checking if a rational polytope P contains an integral point.

Then, A. Barvinok showed in 1994 that there is, for a fixed dimension n,

a polynomial time algorithm giving the number |P ∩ Z

| in function of the

input: the rational vertices of P .

Barvinok algorithm is of exponential complexity with respect to the di- mension of P .

LATTE Program implement (with improvements) the Barvinok algo- rithm, to count the number of integral points on polytopes, and other ques- tions on counting on polytopes. As expected, the dimension of P needs to be small.

I will concentrate here on the Ehrhart polynomial of a polytope P with integral vertices to simplify. I am not a specialist of complexity, but my pur- pose is to show that some questions on complexity motivates some interesting questions in index theory.

The Ehrhart polynomial of a polytope P is by definition

Eh(P )(k) = |kP Z

| = k

vol(P ) + 1/2 ∗ k

vol

delta(P ) + ....

THEOREM (BARVINOK) The Ehrhart polynomial Eh(P )(k) can be computed in polynomial times when the dimension is fixed.

Barvinok algorithm uses:

•: the Atiyah-Bott fixed point formula for the corresponding toric vari- ety. Or more exactly the corresponding Atiyah-Bott-Baum formula for any singular toric variety. Each fixed point gives a local term.

• Barvinok gives a very striking algorithm to determine each local term of the fixed point formula, when the toric variety is not smooth.

Equivariant Riemann Roch: the smooth case

Let us first recall Riemann-Roch theorem and Atiyah-Bott formula for the smooth case and a general manifold with a torus action (not necessarily 1/2 dimensional).

Let M be a smooth projective manifold of dimension n with corresponding line bundle L; Consider

E(k) := X

(−1)

dim(H

(M, O(L

)).

Riemann Roch formula gives E(k) =

Z

M

e

T odd(M ).

This is a polynomial in k with highest term

k

Z

M

c

(l)

n! =

Z

M

ω

/n!

that is the symplectic volume of M (with the symplectic form ω determined by the projective embedding).

If there is an action of a torus T on M ⊂ P (V ) with a number of isolated fixed points p, then we can compute by Atiyah-Bott fixed point formula:

T r(expX) = X

p

e

Q

i

(1 − exp < g

, X >) .

There the g

are the characters of T in the tangent bundle at the fixed points p, µ(p)(X) =

<p,p>

is the moment of the action of X at p.

It is worth to mention the Duistermaat-Heckman (Berline-Vergne) fixed point formula for the volume of P (the equivariant volume of the toric man- ifold):

We have

Z

M

e

ω

= X

p

e e

Q

i

< g

, X > .

Then the number of points or the volume is obtained from this fixed point formula by a limit procedure when X = 0.

Return to the toric case, where everything can be translated in terms of the polytope.

If P ⊂ R

with integral vertices is such that the primitive edges g

from each vertex s form a basis of Z

, then the corresponding manifold M

is smooth, and the translation of the equivariant Riemann-Roch theorem is

X

λ∈kP∩Zⁿ

e

= X

s

e

Q

i

(1 − e

) Z

kP

e

e

= X

s

e

Q

i

< g

, X > . I take a simple example:

Triangle with vertices [0, 0], [1, 0], [0, 1].

This corresponds to the projective space. We write the equivariant in-

dex for the usual action [(e

, e

, 1)] on [z

, z

, z

]C

. of the torus, with

fixed points the 3 lines [1, 0, 0], [0, 1, 0], [0, 0, 1]. Then we obtain the Lefschetz formula

T r(e

, e

) = 1

(1 − e

)(1 − e

) + e

(1 − e

)(1 − e

) + e

(1 − e

)(1 − e

) which indeed gives 1 + e

+ e

after simplifications.

The number of points is then evaluated (via a limit formula) at θ

= θ

= 0. We obtain 3 points!!

We could have enumerated them in this case... but if the polytope is large and complicated, impossible to enumerate all points inside, as Lenstra result shows..

If we dilate [0, 0], [k, 0], [0, k], the Lefshetz formula stays ”the same”. Just move the vertex in the formula:

Remak that the denominators stay the same.

1 Difficulties for the singular case

Retrurn to the preceding example:

At the fixed point [0, 0], the local term is 1

(1 − e

)(1 − e

) = X

n1≥0,n2≥0

e

e

the series P

λ∈C(s,P)∩Zⁿ

e

for λ in the tangent cone at s to P .

If M is not smooth the local term in the Atiyah-Bott-Baumformula is given by the computation of the series: P

λ∈C(s,P)∩Zⁿ

e

That is if P is a polytope, s a vertex of P , then we consider the cone C(P, s) generated by the edges of P . The local term in the Atiyah-Bott fixed point formula is the series P

λ∈C(P,s)∩Zⁿ

e

. But to compute this series is very difficult in general: It is only easy to compute this series if the generators of the cone is a basis of Z

.

Remarkable algorithm of Barvinok to decompose a cone in a signed sum unimodular cones...

Do THE DRAWING in R

. with q = 1000.

Not 1000 terms, but 3.

This is the remarkable discovery of Barvinok: FOR EACH FIXED di- mension n, there is an algorithm with input: a rational cone L

output: the corresponding local term (a quotient of two regular functions on the torus).

2 The simplex

Assume that P is a simplex (with integral vertices). If the dimension of P is not fixed, Barvinok algorithm for the computation of the number of integral points in P has exponential complexity in the inputs.

We now consider the dilation kP of P Then

(kP ) ∩ Z

= vol(P )k

+ e

(P )k

+ ...

Now the vol(P ) is a determinant. This is computed in polynomial time.

Barvinok proved in 2005, that if cod is fixed, then there is a polynomial time algorithm that given a list L of (n + 1) integral points in Z

returns the cod top coefficients of the Ehrhart polynomial. Quite nice, it uses the c-equal arrangement, and its Moebius function computed by Lovasz and Bjorner.

His method is to introduce intermediates between number of points and volume. He consider an explicit family F

of subspaces of codimension less than cod stable by addition. For each F in the family he slices the simplex in F

:= (x + F ) ∩ P , computes the volume, and add all P

F

µ(F ) P

x

vol(F

) when µ(F ) is the Moebius function of the family arrangement.

We produced a simpler algorithm which uses intermediated Todd classes.

It raises some questions even in the smooth case. So I want to discuss them here.

Consider n variables and the symmetric function T :=

Y

i=1

x

(1 − e

) Define the elementary symmetric functions c

by

Y

i

(1 + x

) = X

i

c

Thus T = P (c

, c

, . . . , c

).

If M is a manifold, the Todd class is computed by replacing the c

by the Chern classes of M .

When computing the highest coefficients of R

M

e

T odd(M ) we need only to compute the lowest term of the polynomial P (c

, c

, . . . , c

) up to degree cod.

Thus we introduced an easier class which has the same lowest terms...

Let us consider ”partial Todd functions”

let T (u) =

Y

i=1

(u+ x

(1 − e

) = u

+u

X

i

x

1 − e

+u

X

i<j

x

1 − e

x

1 − e

+· · · The highest coefficient in u is 1. It can be expanded in elementary sym- metric functions

T (u) = X

i=0

u

T T

Each T T

gives a characteristic class. i = 0 gives the Todd class while i = n gives 1.

Here is our theorem (same in the singular case) with appropriate refor- mulation

Define

IT

(M) = X

j=0

(−1)

binomial(n − j − 1, n − c − 1)T T

(M ) Then this characteristic class is ”easier” and copincides with T T (M ) up to degree cod.

To say that is is easier, has no meaning here, but it has in the singular case. The local term is computed in terms of linear combination of local terms for the transverse cones to faces of codimension less than cod (thus computable by Barvinok algoritm as these transverses cones are of small dimensions).

I have no idea of (even for an arbitrary smooth manifold) of what com-

putes Z

M

e

T T

(M )

For i = 0 this is the alernate dimension of the cohomology groups, if

i = n, the volume..

In between ???

Using this technic we can for example calculate for a random weighted projective space up to dimension 18, the three top coefficients of the Ehrhart quasi polynomial... this is with a basic maple implementation, and we hope to do it with our collaborators, in a faster way soon.

(The complete Ehrhart polynomial for random weighted projective space

can be computed with Latte up to dimension ??)