[PPT] - Dedekind sums ingredients Dedekind sums Fourier- Dedekind sums PowerPoint Presentation

SLIDE 1

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Dedekind sums

Mirco Kraenz

Seminar Discrete Convex Geometry Seminar Chair: Prof. Dr. Martin Henk Technical University Berlin

February 7, 2017

SLIDE 2

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Outline

1 The ingredients

Dedekind sums Fourier-Dedekind sums Restricted partition function

2 Reciprocity laws 3 How to compute Dedekind sums 4 Mordell-Pommersheim tetrahedron

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Dedekind sums - Definition

Let a, b relatively prime integers, b > 0. Define the Dedekind sums s(a, b) =

b−1

k=1

ka b k b

,

where ((x)) denotes the saw-tooth function.

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Dedekind sum - Properties

s(a, b) =

b−1

k=1

ka b k b

periodic in a with period b

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Dedekind sum - Properties

s(a, b) =

b−1

k=1

ka b k b

periodic in a with period b

special values s(1, k) = − 1

4 + 1 6k + k 12

SLIDE 6

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Dedekind sum - Properties

s(a, b) =

b−1

k=1

ka b k b

periodic in a with period b

special values s(1, k) = − 1

4 + 1 6k + k 12

ur first goal: s(a, b) + s(b, a) = some simple function

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Fourier-Dedekind sums - Definition

Let a1, a2, . . . , ad pairwise relatively prime, positive integers. Let n some integer. Define the Fourier-Dedekind sums sn(a1, . . . , ad; b) = 1 b

b−1

k=1

ξnk

b

Πd

i=1(1 − ξkai b )

, where ξb = e2πi/b denotes the b-th root of unity.

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Fourier-Dedekind sums - Properties

sn(a1, . . . , ad; b) = 1 b

b−1

k=1

ξnk

b

Πd

i=1(1 − ξkai b )

commutative in ai arguments

SLIDE 9

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Fourier-Dedekind sums - Properties

sn(a1, . . . , ad; b) = 1 b

b−1

k=1

ξnk

b

Πd

i=1(1 − ξkai b )

commutative in ai arguments → write sn(A; b) with A = {a1, a2 . . . ad}

SLIDE 10

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Fourier-Dedekind sums - Properties

sn(a1, . . . , ad; b) = 1 b

b−1

k=1

ξnk

b

Πd

i=1(1 − ξkai b )

commutative in ai arguments → write sn(A; b) with A = {a1, a2 . . . ad} periodic in each ai with period b

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Fourier-Dedekind sums - Properties

sn(a1, . . . , ad; b) = 1 b

b−1

k=1

ξnk

b

Πd

i=1(1 − ξkai b )

commutative in ai arguments → write sn(A; b) with A = {a1, a2 . . . ad} periodic in each ai with period b s0(a, 1; b) = −s(a, b) + b−1

4b

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Restricted partition function

Let A = {a1, a2 . . . ad} ⊂ Nd, where the ai are pairwise relatively prime. For a positive integer n, the restricted partition function is defined to be pA(n) = #{(k1 . . . kd) ∈ Zd

≥0| d

i=1

kiai = n}.

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Polynomial part of pA(n)

pA(n) =

d

i=1

(−1)iBi +

d

i=1

s−n( ˆ Ai; ai) with Bi coefficients at z = 1 of the partial fraction expansion of the shifted generating function of pA(n). Bi are polynomials in

n. Here ˆ

Ai = A \ {ai}.

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Polynomial part of pA(n)

pA(n) =

d

i=1

(−1)iBi +

d

i=1

s−n( ˆ Ai; ai) with Bi coefficients at z = 1 of the partial fraction expansion of the shifted generating function of pA(n). Bi are polynomials in

n. Here ˆ

Ai = A \ {ai}.Therefore, define polyA(n) =

d

i=1

(−1)iBi the polynomial part of the restricted partition function.

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Zagier reciprocity

Theorem 8.4 (Zagier reciprocity) For A = {a1, a2 . . . ad} ⊂ Nd, with the ai are pairwise relatively prime,

d

i=1

s0( ˆ Ai; ai) = 1 − polyA(0), where ˆ Ai = A \ {ai}.

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Dedekind’s reciprocity law

Corollary 8.5 (Dedekind’s reciprocity law) For all a, b relatively prime, positive integers we have s(a, b) + s(b, a) = 1 12(a b + b a + 1 ab) − 1 4.

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Equalities for Dedekind sums

periodicity s(a, b) = s(a mod b, b) Dedekind’s reciprocity law s(a, b) + s(b, a) = 1 12(a b + b a + 1 ab) − 1 4. special values s(1, k) = −1 4 + 1 6k + k 12

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Mordell-Pommersheim tetrahedron

For positive integers a, b, c call P =

(x, y, z) ∈ R3

≥0|x

a + y b + z c ≤ 1

the Mordell-Pommersheim tetrahedron.

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Mordell-Pommersheim tetrahedron

For positive integers a, b, c call tP =

(x, y, z) ∈ R3

≥0|x

a + y b + z c ≤ t

the t-th dilate of the Mordell-Pommersheim tetrahedron.

SLIDE 20

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Mordell-Pommersheim tetrahedron

For positive integers a, b, c call tP =

(x, y, z) ∈ R3

≥0|x

a + y b + z c ≤ t

the t-th dilate of the Mordell-Pommersheim tetrahedron.

Its Lattice-point enumerator is LP(t) = #

(k, l, m) ∈ Z3

≥0|k

a + l b + m c ≤ t

SLIDE 21

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Mordell-Pommersheim tetrahedron

For positive integers a, b, c call tP =

(x, y, z) ∈ R3

≥0|x

a + y b + z c ≤ t

the t-th dilate of the Mordell-Pommersheim tetrahedron.

Its Lattice-point enumerator is LP(t) = #

(k, l, m) ∈ Z3

≥0|k

a + l b + m c ≤ t

= #
(k, l, m) ∈ Z3

≥0|kbc + lac + mab ≤ abct

SLIDE 22

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Mordell-Pommersheim tetrahedron

For positive integers a, b, c call tP =

(x, y, z) ∈ R3

≥0|x

a + y b + z c ≤ t

the t-th dilate of the Mordell-Pommersheim tetrahedron.

Its Lattice-point enumerator is LP(t) = #

(k, l, m) ∈ Z3

≥0|k

a + l b + m c ≤ t

= #
(k, l, m) ∈ Z3

≥0|kbc + lac + mab ≤ abct

= #
(k, l, m, n) ∈ Z4

≥0|kbc + lac + mab + n = abct

SLIDE 23

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Mordell-Pommersheim tetrahedron

For positive integers a, b, c call tP =

(x, y, z) ∈ R3

≥0|x

a + y b + z c ≤ t

the t-th dilate of the Mordell-Pommersheim tetrahedron.

Its Lattice-point enumerator is LP(t) = #

(k, l, m) ∈ Z3

≥0|k

a + l b + m c ≤ t

= #
(k, l, m) ∈ Z3

≥0|kbc + lac + mab ≤ abct

= #
(k, l, m, n) ∈ Z4

≥0|kbc + lac + mab + n = abct

= p{bc,ac,ab,1}(abct).

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Lattice-point enumerator of M-P tetrahedron

LP(t) = p{bc,ac,ab,1}(abct)

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Lattice-point enumerator of M-P tetrahedron

LP(t) = p{bc,ac,ab,1}(abct) = const( 1 1 − zbc 1 1 − zac 1 1 − zab 1 1 − z 1 zabct )

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Lattice-point enumerator of M-P tetrahedron

LP(t) = p{bc,ac,ab,1}(abct) = const( 1 1 − zbc 1 1 − zac 1 1 − zab 1 1 − z 1 zabct ) = const( z−abct − 1 (1 − zbc)(1 − zac)(1 − zab)(1 − z)) + 1

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Partial fraction expansion

Theorem 1.1 (Partial fraction expansion) Given a rational function f (z) = p(z) q(z) = p(z) Πm

k=1(z − ak)αk

with deg q = m

k=1 αk ≥ deg p, there exist unique coefficients

ck,i ∈ C with f (z) =

m

k=1

( ck,1 z − ak + ck,2 (z − ak)2 + · · · ck,αk (z − ak)αk ).

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Assuming a, b, c pairwise relatively prime, the coefficient at ξk

a

is − t a(1 − ξkbc

a

)(1 − ξk

a ).

SLIDE 29

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Assuming a, b, c pairwise relatively prime, the coefficient at ξk

a

is − t a(1 − ξkbc

a

)(1 − ξk

a ).

Summing over all powers of a-th roots of unity yields − t a

a−1

k=1

1 (1 − ξkbc

a

)(1 − ξk

a )

SLIDE 30

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Assuming a, b, c pairwise relatively prime, the coefficient at ξk

a

is − t a(1 − ξkbc

a

)(1 − ξk

a ).

Summing over all powers of a-th roots of unity yields − t a

a−1

k=1

1 (1 − ξkbc

a

)(1 − ξk

a ) = −ts0(bc, 1; a)

SLIDE 31

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Lattice-point enumerator of M-P tetrahedron

LP(t) = p{bc,ac,ab,1}(abct) = const( 1 1 − zbc 1 1 − zac 1 1 − zab 1 1 − z 1 zabct ) = const( z−abct − 1 (1 − zbc)(1 − zac)(1 − zab)(1 − z)) + 1

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Back to lattice-point enumeration

LP(t) =abc 6 t3 + ab + ac + bc + 1 4 t2 + 1 4

a + b + c + 1

a + 1 b + 1 c + 1 3(bc a + ac b + ab c )

t

+ (s0(bc, 1; a) + s0(ac, 1; b) + s0(ab, 1, c))t + 1

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Ehrhart quasipolynomial of M-P tetrahedron

Theorem 8.11 Let P be the Mordell-Pommersheim tetrahedron with parameters a, b, c pairwise relatively prime. Then its Ehrhart quasipolynomial is given by the formula LP(t) =abc 6 t3 + ab + ac + bc + 1 4 t2 + 1 4

3 + a + b + c + 1

3(bc a + ac b + ab c + 1 abc )

t

− (s(bc, a) + s(ac, b) + s(ab, c))t + 1

SLIDE 34

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Ehrhart series of M-P tetrahedron

Corollary 8.12 Let P be the Mordell-Pommersheim tetrahedron with parameters a, b, c pairwise relatively prime. Then its Ehrhart series is given by EhrP(z) = h∗

3z3 + h∗ 2z2 + h∗ 1z + 1

(1 − z)4 , where

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Ehrhart series of M-P tetrahedron

Corollary 8.12 Let P be the Mordell-Pommersheim tetrahedron with parameters a, b, c pairwise relatively prime. Then its Ehrhart series is given by EhrP(z) = h∗

3z3 + h∗ 2z2 + h∗ 1z + 1

(1 − z)4 , where h∗

3 =abc

6 − ab + ac + bc + a + b + c 4 − 1 2 + 1 12 bc a + ac b + ab c

− (s(bc, a) + s(ca, b) + s(ab, c))

SLIDE 36

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Ehrhart series of M-P tetrahedron

Corollary 8.12 Let P be the Mordell-Pommersheim tetrahedron with parameters a, b, c pairwise relatively prime. Then its Ehrhart series is given by EhrP(z) = h∗

3z3 + h∗ 2z2 + h∗ 1z + 1

(1 − z)4 , where h∗

3 =abc

6 − ab + ac + bc + a + b + c 4 − 1 2 + 1 12 bc a + ac b + ab c

− (s(bc, a) + s(ca, b) + s(ab, c))

and h∗

2 and h∗ 1 are given by similar formulas as h∗ 3.

SLIDE 37

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Summary

Dedekind sums s(a, b) Fourier-Dedekind sums sn(a1, . . . ad; b) Zagier reciprocity and Dedekind’s reciprocity law how to compute Dedekind sums explicit formulas for Mordell-Pommersheim tetrahedron geometric interpretation of coefficients of Ehrhart quasipolynomial

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

References

Matthias Beck, Christian Haase, and Asia R. Matthews. Dedekind–carlitz polynomials as lattice-point enumerators in rational polyhedra. Mathematische Annalen, 341(4):945–961, 2008.

M. Beck and S. Robins.

Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Springer New York, 2015. James E. Pommersheim. Toric varieties, lattice points and dedekind sums.

Math. Ann., 295(1):1–24, 1993.

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Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Extra - Dedekind-Rademacher sums

rn(a, b) =

b−1

k=1

ka + n b k b

SLIDE 40

Dedekind sums Mirco Kraenz The ingredients

Dedekind sums Fourier- Dedekind sums Restricted partition function

Reciprocity laws How to compute Dedekind sums Mordell- Pommersheim tetrahedron

Extra - Dedekind-Rademacher sums

rn(a, b) =

b−1

k=1

ka + n b k b

Of course, r0(a, b) = s(a, b).