lattice points of a rational polytope Local Euler-Maclaurin expansion

Summing a function over

lattice points of a rational polytope Local Euler-Maclaurin expansion

Nicole Berline

Ecole polytechnique, France

Report on joint work with Mich`ele Vergne arxiv math.CO/0507256

Snowbird. June 2006

Theorem (B. and V. 2005):

F (p) = set of faces of rational convex polytope p ⊂ R ^d Choose a scalar product on R ^d

X

x∈p∩ Z ^d

h(x) = X

f∈F (p)

Z

f

D ( p , f ) · h

For each face f, D(p, f) differential operator of infinite order,

constant coefficients on R ^d , derivatives normal to f

Theorem (B. and V. 2005):

F (p) = set of faces of rational convex polytope p ⊂ R ^d Choose a scalar product on R ^d

X

x∈p∩ Z ^d

h(x) = X

f∈F (p)

Z

f

D ( p , f ) · h

For each face f, D(p, f) differential operator of infinite order, constant coefficients on R ^d , derivatives normal to f

D (p, f) local

Theorem (B. and V. 2005):

F (p) = set of faces of rational convex polytope p ⊂ R ^d Choose a scalar product on R ^d

X

x∈p∩ Z ^d

h(x) = X

f∈F (p)

Z

f

D ( p , f ) · h

For each face f, D(p, f) differential operator of infinite order, constant coefficients on R ^d , derivatives normal to f

D (p, f) local

D (p, f) canonical when scalar product is fixed

Theorem (B. and V. 2005):

F (p) = set of faces of rational convex polytope p ⊂ R ^d Choose a scalar product on R ^d

X

x∈p∩ Z ^d

h(x) = X

f∈F (p)

Z

f

D ( p , f ) · h

For each face f, D(p, f) differential operator of infinite order, constant coefficients on R ^d , derivatives normal to f

D (p, f) local

D (p, f) canonical when scalar product is fixed

D (p, f) computable

P

x ∈ p ∩ Z

h(x) =

_{f ∈F (p ) f} D(p , f ) · h

Historical Euler-Maclaurin summation formula

k ₁ k ₂

3 faces

k ₂

X

k ₁

h(x) =

Z k ₂ k ₁

h(x)dx − X

n ≥1

b(n)

n! h ⁽ⁿ⁻¹⁾ (k ₁ ) + X

n ≥1

(−1) ⁿ b(n)

n! h ⁽ⁿ⁻¹⁾ (k ₂ ) b(n) = signed Bernoulli number _e ^Xe X −1 ^X = P _∞

0

b(n)

n! X ⁿ .

P

x ∈ p ∩ Z

h ( x ) =

P

f ∈F (p ) f D ( p , f ) · h

Non integer endpoints

k ₁ k ₂

a ₁ a ₂

k ₂

X

k ₁

h(x) =

Z a ₂ a ₁

h(x)dx − X

n≥1

b(n, k ₁ − a ₁ )

n! h ⁽ⁿ⁻¹⁾ (a ₁ )

+ X

n≥1

(−1) ⁿ b(n, a ₂ − k ₂ )

n! h ⁽ⁿ⁻¹⁾ (a ₂ ) b(n, s ) = Bernoulli polynomial

Xe s ^X

e ^X −1 = P ∞ 0

b(n, s ⁾

n! X ⁿ .

P

x ∈ p ∩ Z

h ( x ) =

P

f ∈F (p ) f D ( p , f ) · h

Triangle [0, 0],[1, 0],[0, 1]

h(x) = x ²⁰ ₁ x ₂

As expected, sum of 7 faces contributions = 0

P

x ∈ p ∩ Z

h ( x ) =

P

f ∈F (p ) f D ( p , f ) · h

Triangle [0, 0],[1, 0],[0, 1]

h(x) = x ²⁰ ₁ x ₂

As expected, sum of 7 faces contributions = 0 ( Doctor Cosinus!)

0 − 28224572717107 / 66853011456 5131761430387 / 12155092992

− 1 / 252

287696501 / 133706022912 0

1 / 10626

p = triangle [ ¹⁶¹⁰⁵¹ ₃ , ¹⁶¹⁰⁵¹ ₅ ], [ ^2576816 ₃ , ¹⁶¹⁰⁵¹ ₇ ], [ ^5958887 ₅ , ^14816692 ₇ ] Computation time 3617 sec.

approx. 10 ¹² points (844497845921)

X

(x,y) ∈ p ∩ Z

x ⁴⁸ y ⁴⁸ =

55969247458735493271268368615238071121335974262337882261418363621

89704055956429496253759473056373507451253522021344188115187647607

84555431172202923756940824265247663088847763429436570335188702325

06644969965841257822711805056447218921550669146263582661876630783

21357671611262065293901983868557252464459832189159990869820527095

53646871654914800005753059422066576204781923454823934475242960034

42199041253798398004263030681714027295470241663946228744550160085

43856624239377702107746492579014275563017167813144052693763385569

75239252588060279466314599314734680953729093269435217987689840619

0740089242444014302.

P

x ∈ p ∩ Z

h(x) =

_{f ∈F (p ) f} D(p , f ) · h

D (p, f) local : depends only on class

(mod projected lattice translations) of normal cone to p

along the face f.

Choose a rational scalar product on R ^d .

Normal cone n(p, f) := projected cone in vector subspace

< f > ^⊥ orthogonal to affine span < f >

local

n(p, f) is a solid affine cone in < f > ^⊥

Vertex = projection of < f > on < f > ^⊥

h(x) = 1 Card(p ∩ Z ^d ) = X

f ∈F (p )

ν ₀ (p, f) vol(f) ν ₀ (p, f) = constant term of D(p, f)

depends only on class (mod projected lattice translations) of normal cone n(p, f)

Morelli 1993, McMullen 1993

Pommersheim and Thomas 2004

Riemann-Roch on toric varieties

Card( p ∩ Z ^d ) =

_{f ∈F (p)} ν ₀ ( p , f ) vol( f )

ν ₀ (p, f) local

89133678169939 / 66088208614500 − 4281800310619 / 2106396270216

− 401172431621091 / 457987274773000

1 / 210

− 1 / 210 1 / 1050

34187 / 1050

Triangle [ ¹ ₃ , ¹ ₅ ], [ ¹⁶ ₃ , ¹ ₇ ], [ ³⁷ ₅ , ⁹² ₇ ] (31 points)

Card(p ∩ Z ^d ) =

_{f ∈F (p)} ν ₀ (p , f) vol(f)

ν ₀ (p, f) local

210849514883 / 127956322980 − 4281800310619 / 2106396270216

− 179008247 / 706816180

− 4382929 / 6869864

1 / 210

− 1 / 210 11 / 35

1 / 30 699 / 14

Quadrangle with extra vertex [3, 10] (49 points)

P

x ∈ p ∩ Z

h(x) =

_{f ∈F (p ) f} D( p , f ) · h

Computation of operator D(p, f) uses Barvinok’s signed

decomposition of a cone into unimodular cones

P

x ∈ p ∩ Z

h(x) =

_{f ∈F (p ) f} D(p , f ) · h

Differential operator D (p, f) defined through its symbol D.e ^hξ,xi = µ(ξ )e ^hξ,xi

D = µ( ∂

∂x )

Function µ must depend only on normal cone n(p, f) µ(n(p, f))(ξ)

actually a function of ξ | _{<f >} ^⊥ , hence normal derivatives

a rational affine cone

in rational subspace with projected lattice Λ.

Function µ(a)(ξ ) defined recursively w.r.to dim a

a rational affine cone

in rational subspace with projected lattice Λ.

Function µ(a)(ξ ) defined recursively w.r.to dim a a not acute or a has no lattice points ⇒ µ(a) = 0 In dim 0, set µ({s}) = 1 if s is a lattice point

µ({s}) = 0 otherwise

a rational affine cone

in rational subspace with projected lattice Λ.

Function µ(a)(ξ ) defined recursively w.r.to dim a a not acute or a has no lattice points ⇒ µ(a) = 0 In dim 0, set µ({s}) = 1 if s is a lattice point

µ({s}) = 0 otherwise

For a with vertex s define µ(a)(ξ)e ^hξ,si = X

x∈a ∩Λ

e ^hξ,xi − X

dim f >0

µ(n(a, f))(ξ) Z

f

e ^hξ,xi

Dimension one

If < a > contains lattice points, mod lattice translation we may assume

a = (s + R ₊ )v v primitive lattice vector, s ∈ Q

µ(a)(ξ) = e ^{(n−s)hξ,vi}

1 − e ^hξ,vi + 1 hξ, v i . where n ∈ Z s.t. n − 1 < s ≤ n

hence Bernoulli polynomials in dim 1 Euler-Maclaurin

Theorem:

• Fix rational vertex s.

c 7→ µ(s + c)(ξ ) is a valuation

on the set of rational cones c with values in the space of (germs at 0 of) holomorphic functions.

µ(a ₁ ∪ a ₂ ) = µ(a ₁ ) + µ(a ₂ ) − µ(a ₁ ∩ a ₂ )

Theorem:

• Fix rational vertex s.

c 7→ µ(s + c)(ξ ) is a valuation

on the set of rational cones c with values in the space of (germs at 0 of) holomorphic functions.

µ(a ₁ ∪ a ₂ ) = µ(a ₁ ) + µ(a ₂ ) − µ(a ₁ ∩ a ₂ )

• c 7→ µ(s + c)(0) is a valuation with values in Q

Theorem:

• Fix rational vertex s.

c 7→ µ(s + c)(ξ ) is a valuation

on the set of rational cones c with values in the space of (germs at 0 of) holomorphic functions.

µ(a ₁ ∪ a ₂ ) = µ(a ₁ ) + µ(a ₂ ) − µ(a ₁ ∩ a ₂ )

• c 7→ µ(s + c)(0) is a valuation with values in Q

• µ(x + a) = µ(a) for any lattice point x

• a 7→ µ(a) equivariant w.r. to lattice preserving

isometries.

µ(a)(ξ )e ^{h ξ,s i} = X

x ∈ a ∩Λ

e ^{h ξ,x i} − X

dim f >0

µ(n(a, f))(ξ ) Z

f

e ^{h ξ,x i}

µ(a)(ξ )e ^{h ξ,s i} = X

x ∈ a ∩Λ

e ^{h ξ,x i} − X

dim f >0

µ(n(a, f))(ξ ) Z

f

e ^{h ξ,x i}

Vertex s with other faces (dim f > 0), get Euler-Maclaurin expansion of P

x∈a ∩Λ e ^{h ξ,x i} : X

x∈a ∩Λ

e ^hξ,xi = X

f ∈F ( a )

µ(n(a, f))(ξ) Z

f

e ^hξ,xi

µ(a)(ξ )e ^{h ξ,s i} = X

x ∈ a ∩Λ

e ^{h ξ,x i} − X

dim f >0

µ(n(a, f))(ξ ) Z

f

e ^{h ξ,x i}

Vertex s with other faces (dim f > 0), get Euler-Maclaurin expansion of P

x∈a ∩Λ e ^{h ξ,x i} : X

x∈a ∩Λ

e ^hξ,xi = X

f ∈F ( a )

µ(n(a, f))(ξ) Z

f

e ^hξ,xi

Compare with leitmotiv X

x∈p ∩ Z ^d

h(x) = X

f ∈F (p )

Z

f

D(p, f) · h

use Brion’ theorem

P

x∈p ∩ Z ^d h(x) = P

f∈F (p )

R

f (D(p, f) · h)(x) D(p, f) local

⇒ Ehrhart quasipolynomial for p and h X

x∈ t ^{p ∩} Z ^d

h(x) =

d+deg X h

m=0

E m (p, h, t)t ^m

Ehrhart coefficients are periodic w.r.to t ∈ N

P

x∈p ∩ Z ^d h(x) = P

f∈F (p )

R

f (D(p, f) · h)(x) D(p, f) local

⇒ Ehrhart quasipolynomial for p and h X

x∈ t ^{p ∩} Z ^d

h(x) =

d+deg X h

m=0

E m (p, h, t)t ^m

Ehrhart coefficients are periodic w.r.to t ∈ N Ehrhart coefficient of highest degree (d + deg h) is

Z

p

h ^max (x)

Ehrhart coefficient = sum of contribution of faces X

x∈ t ^{p ∩} Z ^d

h(x) =

d+deg X h

m=0

X

dim f≤m

E _m (p, f, h, t) t ^m

Assume h homogeneous, deg h = n

Ehrhart coefficient = sum of contribution of faces X

x∈ t ^{p ∩} Z ^d

h(x) =

d+deg X h

m=0

X

dim f≤m

E _m (p, f, h, t) t ^m

Assume h homogeneous, deg h = n

E _d+n−k (p, h, t) involves only normal cones n(p, f) of dimension ≤ k (codim f ≤ k )

(face f contributes only to Ehrhart coefficients in degree m = dim f, . . . , dim f + n)

Face p particular (D(p, p) = 1), contributes only to highest

degree d + n

X

x ∈ t ^{p ∩} Z ^d

h(x) = X

f ∈F (p )

Z

t ^f D( t p, t f) · h(x)

D (tp, tf) = µ(n(tp, tf))(∂/∂x) D(tp, tf) = ν ₀ (p, f, t) +

X ∞

A,|A|=1

ν _A (p, f, t)∂ ^A

X

x ∈ t ^{p ∩} Z ^d

h(x) = X

f ∈F (p )

Z

t ^f D( t p, t f) · h(x)

D (tp, tf) = µ(n(tp, tf))(∂/∂x) D(tp, tf) = ν ₀ (p, f, t) +

X ∞

A,|A|=1

ν _A (p, f, t)∂ ^A Coefficients ν _A (p, f, t) depend only on class (mod

projected lattice translations in orthogonal space) of normal cone n(tp, tf)

⇒ ν _A (p, f, t) periodic (t ∈ N )

Period at most T _f s.t. T _f < f > contains lattice points.

P

x∈p ∩ Z ^d h(x) = P

f∈F (p )

R

f D (p, f) · h

Local D(p, f) related via Riemann-Roch to local coefficients for

Equivariant Todd genus of a toric variety

P

x∈p ∩ Z ^d h(x) = P

f∈F (p )

R

f D (p, f) · h

Local D(p, f) related via Riemann-Roch to local coefficients for

Equivariant Todd genus of a toric variety

Simplicial lattice polytope p → toric variety P with line bundle L

Riemann-Roch: X

x∈ p ∩Λ

e ^hξ,xi = hTodd(P )(ξ ), ch(L)(ξ)i

P

x∈p ∩ Z ^d h(x) = P

f∈F (p )

R

f D (p, f) · h

Local D(p, f) related via Riemann-Roch to local coefficients for

Equivariant Todd genus of a toric variety

Simplicial lattice polytope p → toric variety P with line bundle L

Riemann-Roch: X

x∈ p ∩Λ

e ^hξ,xi = hTodd(P )(ξ ), ch(L)(ξ)i Face f → subvariety P f → invariant homology class [P f ] Theorem: Todd(P )(ξ) = X

f ∈F ( p )

µ(n(p, f)(ξ))[P _f ]

General toric variety P ↔ fan E (collection of cones in Lie algebra of torus)

Each cone σ ∈ E → subvariety P _σ → homology class [P _σ ] ∈ H _∗ (P )

Theorem Todd(P )(ξ) = X

σ∈E

µ(ˇ σ)(ξ)[P _σ ]

where ˇ σ = projection of the dual cone σ ^∗ on

(dual of Lie algebra )/ < σ > ^⊥

ν (σ) := µ(ˇ σ)(0)

Danilov’s question 1978

Find ν (σ ) depending only on cone σ, not on whole fan E

s.t.

Todd(P ) = X

σ∈E

ν (σ)[P _σ ]

Answers by Morelli 1993, McMullen 1993 for polytope toric

varieties, Pommersheim and Thomas 2004 for general case .