- Common work with Nicole Berline. Preprint in arkiv; CO/ Thanks to Charles Cochet for drawings (the good ones).

Transcription

1 Euler-Maclaurin formula for polytopes.. p.1 - Common work with Nicole Berline. Preprint in arkiv; CO/ Thanks to Charles Cochet for drawings (the good ones).

2 Euler-Maclaurin formula in dimension 1.. p.2 Consider the interval [0, k] and the function x x m on R. Compute the sum of x m over the first k + 1 integers, and compare it to the integral k 0 xm dx = km+1 m k 0 = k k = k(k + 1) k 2 = k3 3 + k2 2 + k 6.

3 Bernoulli polynomials. p.3 Theorem: (Jakob BERNOULLI -( )) For any non negative integer m, the sum S m (k) = k a=0 a m of the m th -powers of the k + 1 first integers is given by a polynomial in k.

4 . p.4 IMPORTANT for computational purposes: Fix m. Then the polynomial behaviour in k of S m (k) = km+1 m+1 + implies that to compute the value of S m (k) (a sum of k numbers), we need only a number of operations only less or equal to C Log(k) m+1. In other words, we can compute this sum in "polynomial time" with respect to k.

5 Bernoulli numbers. p.5 Define the analytic function Ber(u) = 1 (1 exp(u)) + 1 u = 1 2 u 12 + = j=0 B j u j (j + 1)! B j Bernoulli numbers. B j = 0 if j is odd, except B 1 = 1/2.

6 Euler-Maclaurin formula for the interval. p.6 Let φ be a polynomial function over R, A and B two integers. B S(φ, A, B) = φ(a), a=a a runs over integers between A,B. Then S(φ, A, B) = B A φ(x)dx + R R sum of derivatives of φ at boundary of interval R := 1 2 (φ(a) + φ(b)) + j=1 B 2j 2j! (φ(2j 1) (A) φ (2j 1) (B))

7 Euler-Maclaurin formula in dimension 1. p.7 More condensed formula: S(φ, A, B) = B A φ(x)dx+(ber( x )φ)(b)+(ber( x )φ)(a).

8 . p.8 We want to generalize this formula Let P be a polytope in R d, with vertices in Z d, φ a polynomial function: compute φ(a) = (D F φ(x))dx a P L Ffaces de P with D F différential operators with constant coefficients. We will impose conditions on D F that I will describe later. F

9 Idea of the proof of Euler-Maclaurin formula. p.9 Do it for cones and decompose a polytope in cones.

10 . p.10 A B A B A B A B

11 Important formulae (1 e u ) e ku = 1 Continuous version a=0 (1 e u ) e ku = 0 a= u u x=0 x= e xu dx = 1 e xu dx = 0. p.11

12 Thus explicit formula for summing exponentials u négatif. Thus S(u) := I(u) := n=0 0 e nu = 1 1 e u e ux dx = 1 u B n=a e nu = ebu 1 e u + eau 1 e u AS n= e nu = 0. B A e ux dx = eau u + ebu u AS e ux dx = 0.. p.12

13 Comparison between sums and integrals for cones. p.13 We want to write n=0 in function of the integral and of a sum of derivatives of e xu at the boundary. Compare S(u) = 1 1 e et I(u) = 1 u u.??: e nu S(u) = I(u) + Ber(u) with Ber(u) = 1 1 e u + 1 u ANALYTIQUE. S(u) = I(u) + (Ber( x )e ux ) x=0.

14 Any dimension.. p.14 Polytope P R d with vertices in Z d. φ polynomial function on R d : Sum(φ, P) = a Z d φ(a). Theorem: Ehrhart (generalizes d Bernoulli) E(k) = φ(a) a kp Z d is a polynomial in k = φ(x)dx + kp

15 Ehrhart. p.15 Example: φ = 1. The number card(kp Z d ) of integral points in kp is a polynômial in k: Ehrhart polynomial. Example: standard simplex in R d. Polynôme d Ehrhart (k+1)(k+2) (k+d) d!. No formula in general, even for simplices. Mordell example: simplex in R 3 with vertices [0, 0, 0] [a, 0, 0], [0, b, 0], [0, 0, c].

16 . p.16 k o a k o b volume(k ) = k2 2, (k Z2 ) = (k+1)(k+2) 2

17 Euler-MacLaurin formula in any dimension:. p.17 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 φ(a) = (D F φ(x))dx a P L Ffaces de P with D F differential operators with constant coefficients. Here dx is the canonical measure on faces determined by the lattice. WITH Conditions satisfied by D F : F

18 Normal cone. p.18 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

19 . p.19 Conditions satisfied by D F Locality: Operators D F depends only of the normal cone. Invariants: Invariants modulo translation by an element of the lattice. : Explicit formula for D F, computable in polynomial time up to some order when the dimension and the order of D F are fixed. Calculable: Function 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 F.

20 What was known before??.. p.20 Mac-Mullen: proof of the existence of the operators D F satisfying locality and invariance. But no construction. Cappell-Shaneson, Brion-Vergne: Existence and explicit construction of operators D F, 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) = F µ(f)vol(f)

21 Theorem: Berline-Vergne. p.21 We give ourselves a rational scalar product on V. then φ(a) = (D F φ(x))dx a P L Ffaces de P with D F differential operators with constant coefficients : -Local,invariant, rational, derivation with respect to normal directions to the face. -explicit construction, computable in polynomial time. F

22 Example for polygones in R 2.. p.22 Pick theorem: Let P be a polygone in R 2, with vertices with integral coordinates. then Number of points in Z 2 P = AreaP lenght Z (a) + 1 a=edges Pommersheim-Berline-Vergne: = AreaP lenght Z (a) + a=edges s=vertices µ(s).

23 Example of constants µ(s) adding to 1. Otherwise, formulae with Dedekind sums (computable by. p The middle drawing is the constants found by Cappell-Shaneson (CRAS). Clearly not local constants. Berline-Vergne (If P is good): CS BV 1 4 µ(v) = α, β 12 ( 1 α, α + 1 β, β ).

24 Another example µ([3, 4]) = 3 50 µ([4, 4]) = µ([3, 1]) = µ([0, 0]) = 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 2 P = 9;. p.24

25 . p.25 D C A B a (P Z 2 ) φ(a) = P φ(x)dx + B A (D ABφ)(x)dx + C B (D BCφ)(x)dx + D C (D CDφ)(x)dx + A D (D DAφ)(x)dx +(D A φ)(a) + (D B φ)(b) + (D C φ)(c) + (D D φ)(d). Remarque: opérateurs correspondant aux arêtes déjà connus, car cône normal de dimension 1. Exemple: D AB

26 0pérateurs différentiels D F s additionnant à 1. p.26 D [0,0] = x 5 24 y + D [3,1] = x 1 12 y + D [4,4] = x y + D [3,4] = x y +

27 A function on cones.. p.27 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 ξ C L e <ξ,u> = FfacesofC µ(normal(c, F))(u) F e <x,u> dx.

28 Properties of µ. p.28 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.

29 Euler-MacLaurin pour les cones. p.29 Transformer µ en opérateur différentiel: e <ξ,u> = µ(normal(c, F))( x )e <x,u> dx. ξ C L FfacesdeC Propriétés de µ: -Invariante par translations. -Additive sur les cônes duaux: -Calculable en temps polynomial. F

30 translation: equivariant homology of toric varieties. p.30 Let T be a torus with Lie algebra t. Lattice t Z avec T = t/2iπt Z. 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 σ of M(Ev) invariant by T C. This open set contains a unique closed orbit O σ of T C.

31 P 1 (C). p.31 Example: T = {e iu } = R/2πZ Ev = {R + } {0} {R } M(Ev) = P 1 (C) U R + = C with fixed point 0 = N U R = C with fixed point: 0 = S. U 0 = C with orbit C. P 1 (C) = C C guled over C 3 orbites de C.

32 P 1 (C) glued with affines open set. p.32 N N et et S S C C 3 invariant cycles [P 1 (C)], N et S. C \ {0}

33 Equivariant Homology. p.33 We can define the equivariant homology H T (M) of an algebraic variety M provided with an algebraic action of a torus T = (C ) d. This is a module under the action of S(t ). Equivariant homology is generated by the cycles invariants by T with relations. N N et et S S C C C \ {0} Relations u[p 1 (C)] = [N] [S] in equivariant homology, au lieu de N = S (u = 0).

34 Equivariant Todd class. p.34 The equivariant Todd class of M belongs to Ŝ(t )H T (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 Todd(M(Ev)) = σ Ev µ(σ )[O σ ].

35 Euler-Maclaurin in equivariant homology. p.35 Return to P 1 (C). In homology: 1 Todd(P 1 (C)) = [P 1 (C)]+( 1 e 1 u u )[N]+( 1 1 e + 1 u u )[S].

36 Question. p.36 If we write Todd(P 1 (C)) in a basis, we only need two orbits: for example Schubert cells [P 1 ], and N, as we have the relation u[p 1 ] = [S] [N]. Todd(P 1 ) = u 1 e u[p 1] e u[n]. Not so nice. Is there a beautiful invariant formula for Todd class of flag varieties??