Dedekind sums: a combinatorial-geometric viewpoint

Transcription

1 DIMACS Series in Discrete Mathematics and Theoretical Computer Science Dedekind sums: a cominatorial-geometric viewpoint Matthias Beck and Sinai Roins Astract The literature on Dedekind sums is vast In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational polytopes In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the uilding locks of the numer of partitions of an integer from a finite set of positive integers This prolem also goes y the name of the coin exchange prolem Dedekind sums have enjoyed a resurgence of interest recently, from such diverse fields as topology, numer theory, and cominatorial geometry The Fourier-Dedekind sums we study here include as special cases generalized Dedekind sums studied y Berndt, Carlitz, Grosswald, Knuth, Rademacher, and Zagier Our interest in these sums stems from the appearance of Dedekind s and Zagier s sums in lattice point count formulas for polytopes Using some simple generating functions, we show that generalized Dedekind sums are natural ingredients for such formulas As immediate geometric corollaries to our formulas, we otain and generalize reciprocity laws of Dedekind, Zagier, and Gessel Finally, we prove a polynomial-time complexity result for Zagier s higher-dimensional Dedekind sums 1 Introduction In recent years, Dedekind sums and their various silings have enjoyed a new renaissance Historically, they appeared in analytic numer theory (Dedekind s η- function [De]), algeraic numer theory (class numer formulae [Me]), topology (signature defects of manifolds [HZ]), cominatorial geometry (lattice point enumeration [Mo]), and algorithmic complexity (pseudo random numer generators [K]) In this expository paper, we define some road generalizations of Dedekind sums, which are in fact finite Fourier series We show that they appear naturally in the enumeration of lattice points in polytopes, and prove reciprocity laws for them In cominatorial numer theory, one is interested in partitions of an integer n from a finite set That is, one writes n as a nonnegative integer linear comination of a given finite set of positive integers We showed in [BDR] that the numer of such partitions of n from a finite set is a quasipolynomial in n, whose coefficients are uilt up from the following generalization of Dedekind sums 2000 Mathematics Suject Classification Primary 05A15, 11L03; Secondary 11P21, 52C07 Key words and phrases Dedekind sums, rational polytopes, lattice points, partition function The second author is supported y the NSA Young Investigator Grant MSPR-OOY c 2002 American Mathematical Society

2 2 MATTHIAS BECK AND SINAI ROBINS as Definition 11 For,, a d, n Z, we define the Fourier-Dedekind sum σ n (a 1,, a d ; ) := 1 λ =1 λ n (1 λ a1 ) (1 λ a d ) Here the sum is taken over all a th 0 roots of unity for which the summand is not singular In [G], Gessel systematically studied sums of the form R(λ), λ a =1 where R is a rational function, and the sum is taken over all a th roots of unity for which R is not singular He called them generalized Dedekind sums, since his definition includes various generalizations of the Dedekind sum as special cases Hence we study Gessel s sums where the poles of R are restricted to e roots of unity In Section 2, we give a rief history on those generalizations of the classical Dedekind sum (due to Rademacher [R], and Zagier [Z]) which can e written as Fourier-Dedekind sums Our interest in these sums stems from the appearance of Dedekind s and Zagier s sums in lattice point enumeration formulas for polytopes [Mo, P, BV, DR] Using generating functions, we show in Section 3 that generalized Dedekind sums are natural ingredients for such formulas, which also apply to the theory of partition functions In Section 4 we otain and generalize reciprocity laws of Dedekind [De], Zagier [Z], and Gessel [G] as geometric corollaries to our formulas Finally, in Section 5, we prove that Zagier s higher-dimensional Dedekind sums are in fact polynomial-time computale in fixed dimension For Dedekind sums in 2 dimensions, this fact follows easily from their reciprocity law; ut for higher dimensional Dedekind sums the polynomial-time complexity does not seem to follow so easily, and we therefore invoke some recent work of [BP] and [DR] 2 Classical Dedekind sums and generalizations According to Riemann s will, it was his wish that Dedekind should get Riemann s unpulished notes and manuscripts [RG] Among these was a discussion of the important function ( η(z) = e πiz 12 1 e 2πinz ), n 1 which Dedekind took up and eventually pulished in Riemann s collected works [De] Definition 21 Let ((x)) e the sawtooth function defined y { {x} 1 ((x)) := 2 if x Z 0 if x Z Here {x} = x [x] denotes the fractional part of x For two integers a and, we define the Dedekind sum as s(a, ) := (( )) (( )) ka k k mod Here the sum is over a complete residue system modulo

3 DEDEKIND SUMS: A COMBINATORIAL-GEOMETRIC VIEWPOINT 3 Through the study of the transformation properties of η under SL 2 (Z), Dedekind naturally arrived at s(a, ) The classic introduction to the arithmetic properties of the Dedekind sum is [RG] The most important of these, already proved y Dedekind [De], is the famous reciprocity law: Theorem 22 (Dedekind) If a and are relatively prime then s(a, ) + s(, a) = ( a a + ) a This reciprocity law is easily seen to e equivalent to the transformation law of the η-function [De] Due to the periodicity of ((x)), we can reduce a modulo in the Dedekind sum: s(a, ) = s(a mod, ) Therefore, Theorem 22 allows us to compute s(a, ) in polynomial time, similar in spirit to the Euclidean algorithm The Dedekind sum s(a, ) has various generalizations, two of which we introduce here The first one is due to Rademacher [R], who generalized sums introduced y Meyer [Me] and Dieter [D]: Definition 23 For a, Z, x, y R, the Dedekind-Rademacher sum is defined y s(a, ; x, y) := (( )) (( )) (k + y)a k + y + x k mod This sum posesses again a reciprocity law: Theorem 24 (Rademacher) If a and are relatively prime and x and y are not oth integers, then s(a, ; x, y) + s(, a; y, x) = ((x))((y)) + 1 ( a 2 B 2(y) + 1 a B 2(ay + x) + ) a B 2(x) Here B 2 (x) := (x [x]) 2 (x [x]) is the periodized second Bernoulli polynomial If x and y are oth integers, the Dedekind-Rademacher sum is simply the classical Dedekind sum, whose reciprocity law we already stated As with the reciprocity law for the classical Dedekind sum, Theorem 24 can e used to compute s(a, ; x, y) in polynomial time The second generalization of the Dedekind sum we mention here is due to Zagier [Z] From topological considerations, he arrived naturally at expressions of the following kind: Definition 25 Let a 1,, a d e integers relatively prime to N Define the higher-dimensional Dedekind sum as s( ; a 1,, a d ) := ( 1)d/2 a 0 1 cot πka 1 cot πka d This sum vanishes if d is odd It is not hard to see that this indeed generalizes the classical Dedekind sum: the latter can e written in terms of cotangents [RG], which yields s(a, ) = 1 4 k mod cot πka Again, there exists a reciprocity law for Zagier s sums: cot πk = 1 s(; a, 1) 4

4 4 MATTHIAS BECK AND SINAI ROBINS Theorem 26 (Zagier) If,, a d N are pairwise relatively prime then d s(a j ;,, â j,, a d ) = φ(,, a d ) Here φ is a rational function in,, a d, which can e expressed in terms of Hirzeruch L-functions [Z] It should e mentioned that a version of the higher-dimensional Dedekind sums had already een introduced y Carlitz [C]: (( )) (( )) (( )) a1 k a d k d a1 ad k 1,,k d mod Berndt [B] noticed that these sums are, up to trivial factor, Zagier s higher-dimensional Dedekind sums If we write the higher-dimensional Dedekind sum as a sum over roots of unity, s( ; a 1,, a d ) = 1 λ =1 λ λ a1 + 1 λ a λad λ a, d 1 it ecomes clear that it suffices to study sums of the form 1 1 (λ a1 1) (λ a d 1) λ =1 λ Zagier s Dedekind sum can e expressed as a sum of expressions of this kind On the other hand, we consider special cases of the Dedekind-Rademacher sum, namely, for n Z, s (a, ; n ), 0 = k mod (( ka + n )) (( )) k Knuth [K] discovered that these generalized Dedekind sums descrie the statistics of pseudo random numer generators In [BR], we used the convolution theorem for finite Fourier series to show that, if a and are relatively prime, (21) s (a, ; n ), 0 = 1 λ =1 λ λ n (1 λ a )(1 λ) 1 2 { n } Here {x} = x [x] denotes the fractional part of x Comparing this with the representation we otained for Zagier s Dedekind sums motivates the study of the Fourier-Dedekind sum σ n (a 1,, a d ; ) = 1 λ n (1 λ a1 ) (1 λ a, d ) λ =1 a finite Fourier series in n Gessel [G] gave a new reciprocity law for a special case of Fourier-Dedekind sums: Theorem 27 (Gessel) Let p and q e relatively prime and suppose that 1 n p + q Then 1 p λ p =1 λ λ n (1 λ q ) (1 λ) + 1 q ( 1 p + 1 q + 1 pq = n2 2pq + n 2 λ q =1 λ ) 1 4 λ n (1 λ p ) (1 λ) ( 1 p + 1 ) q ( p q + 1 pq + q ) p

5 DEDEKIND SUMS: A COMBINATORIAL-GEOMETRIC VIEWPOINT 5 It is easy to see that the reciprocity law for classical Dedekind sums (Theorem 22) is a special case of Gessel s theorem We can rephrase the statement of Gessel s theorem in terms of Dedekind-Rademacher sums y means of (21): for p and q relatively prime, and 1 n p + q, ( q, p; n ) ( p, 0 + s p, q; n ) q, 0 p 1 (( )) (( )) def qk n k q 1 (( )) (( )) pk n k = + p p q q k=0 k=0 = n2 2pq n ( 1 2 p + 1 q + 1 ) + 1 pq ( p 12 q + 1 pq + q ) 1 p 2 s { } t 1 p 2 { } t q We will now view the Fourier-Dedekind sum from a generating-function point of view, which will allow us to otain and extend geometric proofs of Dedekind s, Zagier s and Gessel s reciprocity laws 3 A new cominatorial identity for partitions from a finite set The form of the Fourier-Dedekind sum σ n (a 1,, a d ; ) = 1 suggests the use of a generating function f(z) := λ =1 λ λ n (1 λ a1 ) (1 λ a d ) 1 z n 1 z a0 (1 z a1 ) (1 z a d ) In fact, let s expand this generating function into partial fractions: suppose, for simplicity, that n > 0, and,, a d are pairwise relatively prime Then we can write f(z) = λ =1 λ A λ z λ + + λ a d =1 λ d+1 A λ z λ + n B k (z 1) k + C k z k The coefficient A λ for, say, a nontrivial th root of unity λ can e derived easily: A λ = lim (z λ)f(z) = λ λ n z λ (1 λ a1 ) (1 λ an ) Hence we otain the Fourier-Dedekind sums if we consider the constant coefficient of f (in the Laurent series aout z = 0): (31) const(f) = λ =1 λ A λ λ + + λ a d =1 λ d+1 A λ λ + ( 1) k B k d+1 = σ n (a 1,, a d ; ) + + σ n (,, a d 1 ; a n ) + ( 1) k B k The coefficients B k are simply the coefficients of the Laurent series of f aout z = 1, and are easily computed, y hand or using mathematics software such as

6 6 MATTHIAS BECK AND SINAI ROBINS Maple or Mathematica It is not hard to see that they are polynomials in n whose coefficients are rational functions of the,, a d 1 To simplify notation, define d+1 (32) q(,, a d, n) := ( 1) k B k On the other hand, we can compute the constant coefficient of f y rute force: By expanding f(z) = z k0a0 z k da d z n, k0 0 kd 0 we can see that const(f) enumerates the ways of writing n as a linear comination of the,, a d with nonnegative coefficients: (33) const(f) = # { (k 0,, k d ) Z d+1 : k j 0, k k d a d = n } = p {a0,,a d }(n) This defines the partition function with parts in the finite set A := {,, a n } Geometrically, p A (n) enumerates the integer points in n-dilates of the rational polytope P := { (x 0,, x d ) R d+1 : x j 0, x x d a d = 1 } This geometric interpretation allows us to use the machinery of Ehrhart theory, which will e advantageous in the following section We next give an explicit formula for the famous coin-exchange prolem that is, the numer of ways to form n cents from a finite set of coins with given denominations,, a d : comparing (31) with (33) yields our central result [BDR] Theorem 31 Suppose,, a d are pairwise relatively prime, positive integers We recall that the numer of partitions of an integer n from the finite set of a i s is defined y Then p {a0,,a d }(n) := { (k 0,, k d ) Z d+1 : k j 0, k k d a d = n } p {a0,,a d }(n) = q(,, a d, n) + where q(,, a d, n) is given y (32) d σ n (,, â j,, a d ; a j ), [BGK] 1 After this paper was sumitted, general formulas for these polynomials were discovered in

7 DEDEKIND SUMS: A COMBINATORIAL-GEOMETRIC VIEWPOINT 7 The first few expressions for q(,, a d, n) are q(, n) = 1 q(, a 1, n) = q(, a 1, a 2, n) = n + 1 ( 1 a a 1 n 2 + n ( a 1 a 2 2 a 1 a 2 ( a 1 + a 2 a 1 a 2 a 1 a 2 a 2 a 1 n 3 ) a 1 a 2 ) q(, a 1, a 2, a 3, n) = 6 a 1 a 2 a ( 3 ) + n a 1 a 2 a 1 a 3 a 2 a 3 a 1 a 2 a 3 + n ( ) 4 a 1 a 2 a 3 a 1 a 2 a 1 a 3 a 2 a 3 + n ( a0 + a 1 + a 2 + a ) 3 12 a 1 a 2 a 3 a 2 a 3 a 1 a 3 a 1 a ( a a 1 + a 1 + a 1 24 a 1 a 2 a 1 a 3 a 2 a 3 a 2 a 3 a 2 a 3 + a 2 + a 2 + a 2 + a 3 + a 3 + a ) 3 a 1 a 3 a 1 a 3 a 1 a 2 a 1 a ( ) 8 a 1 a 2 a 3 ) 4 Reciprocity laws We will now use Theorem 31 to prove and extend some of the reciprocity theorems stated earlier We will make use of two results due to Ehrhart for rational polytopes, that is, polytopes whose vertices are rational Ehrhart [E] initiated the study of the numer of integer points ( lattice points ) in integer dilates of such polytopes: Definition 41 Let P R d e a rational polytope, and n a positive integer We denote the numer of lattice points in the dilates of the closure of P and its interior y respectively L(P, n) := # ( np Z d) and L(P, n) := # ( np Z d), Ehrhart proved that L(P, n) and L(P, n) are quasipolynomials in the integer variale n, that is, expressions of the form c d (n) n d + + c 1 (n) n + c 0 (n), where c 0,, c d are periodic functions in n Ehrhart conjectured the following fundamental theorem, which estalishes an algeraic connection etween our two lattice-point-count operators Its original proof is due to Macdonald [Ma]

8 8 MATTHIAS BECK AND SINAI ROBINS Theorem 42 (Ehrhart-Macdonald reciprocity law) Suppose the rational polytope P is homeomorphic to a d-manifold Then L(P, n) = ( 1) d L(P, n) This enales us to rephrase Theorem 31 for the quantity p {,,a d } (n) := # { (k 0,, k d ) Z d+1 : k j > 0, k k d a d = n } By Theorem 42, we have the following result Corollary 43 Suppose,, a d N are pairwise relatively prime Then d (n) = ( 1)d q(,, a d, n) + σ n (,, â j,, a d ; a j ), p {,,a d } where q(,, a d, n) is given y (32) We note that we could have derived this identity from scratch in a similar way as Theorem 31, without using Ehrhart-Macdonald reciprocity The reason for switching to p {,,a d }(n) is that p {,,a d } (n) = 0 for 0 < n < + + a d, y the very definition of p {,,a d }(n) This yields a reciprocity law: Theorem 44 Let,, a d e pairwise relatively prime integers and 0 < n < + + a d Then d σ n (,, â j,, a d ; a j ) = q(,, a d, n), where q(,, a d, n) is given y (32) For d = 2, = p, a 1 = q, a 2 = 1, this is the statement of Gessel s Theorem 27, which, in turn, implies Dedekind s reciprocity law Theorem 22 To prove Zagier s Theorem 26 in the language of Fourier-Dedekind sums, we make use of another result of Ehrhart [E] on lattice polytopes, that is, polytopes whose vertices have integer coordinates Recall that the reduced Euler characteristic of a polytope P can e defined as χ(p) := σ ( 1) dim σ, where the sum is over all su-simplices of P Theorem 45 (Ehrhart) Let P e a lattice polytope Then L(P, n) is a polynomial in n whose constant term is χ(p) We note that the polytope P corresponding to p {a0,,a d }(n) is convex and hence has Euler characteristic 1 If we now dilate P only y multiples of a d, say n = a d m, we otain the dilates of a lattice polytope Theorem 31 simplifies for these n to d p {a0,,a d }( a d m) = q(,, a d, a d m) + σ 0 (,, â j,, a d ; a j ),

9 DEDEKIND SUMS: A COMBINATORIAL-GEOMETRIC VIEWPOINT 9 y the periodicity of the Fourier-Dedekind sums However, χ(p) = 1, and Theorem 45 yields a result equivalent to Zagier s reciprocity law for his higher-dimensional Dedekind sums, Theorem 26: Theorem 46 For pairwise relatively prime integers a 1,, a d, d σ 0 (,, â j,, a d ; a j ) = 1 q(,, a d, 0), where q(,, a d, n) is given y (32) 5 The computational complexity of Zagier s higher-dimensional Dedekind sums In this section we give a proof of the polynomial-time complexity of Zagier s higher-dimensional Dedekind sums, in fixed dimension d In [BP], there is a nice theorem due to Barvinok which guarantees the polynomial-time computaility of the generating function attached to a rational polyhedron We will use his theorem for a cone First, we mention that a common way to enumerate lattice points in a cone K (and in polytopes) is to use the generating function f(k, x) := x m, m K Z d where we use the standard multivariate notation x m := x m1 1 x m d d It is an elementary fact that for rational cones these generating functions are always rational functions of the variale x Barvinok s theorem reads as follows: Theorem 51 (Barvinok) Let us fix the dimension d There exists a polynomial-time algorithm, which for a given rational polyhedron K R d, K = {x R d : c i, x β i, i = 1m}, where c i Z d and β i Q computes the generating function f(k, x) := in the form (a virtual decomposition) f(k, x) = i I ɛ i m K Z d x m x ai (1 x i1 ) (1 x id ), where ɛ i { 1, 1}, a i Z, and i1,, id is a asis of Z d for each i The computational complexity of the algorithm for finding this virtual decomposition is L O(d), where L is the input size of K In particular, the numer I of terms in the summand is L O(d) Thus Barvinok s algorithm finds the coefficients of the rational function f(k, x) in polynomial time In [DR], on the other hand, the generating function f(k, x) is given in terms of an average over a finite aelian group of a product of d cotangent functions, whose arguments are in terms of the coordinates of the vertices which generate the cone K (these are the extreme points of K whose convex hull is K) This is the main theorem in [DR] and we apply it elow to a special lattice cone which will give us the Zagier-Dedekind sums we want to study

10 10 MATTHIAS BECK AND SINAI ROBINS The following theorem is part of a igger project on the computaility of generalized Dedekind sums in all dimensions A slightly different proof is sketched in [BP] Theorem 52 For fixed dimension d, the higher-dimensional Dedekind sums s( ; a 1,, a d ) = ( 1)d/2 are polynomial-time computale a 0 1 cot πka 1 cot πka d Proof Let K R d+1 e the cone generated y the positive real span of the vectors v 1 = (1, 0,, 0, a 1 ) v 2 = (0, 1, 0,, 0, a 2 ) v d = (0,, 0, 1, a d ) v d+1 = (0,, 0, ) Then the right-hand side of the main theorem of [DR] is in this case a d ( 2 d coth π ) (s + ia j k) When we compute the coefficient of s 1 in this meromorphic function of s, we arrive at the following higher-dimensional Dedekind sum: a 0 1 cot πka 1 cot πka d plus other products of lower-dimensional Zagier-Dedekind sums By induction on the dimension, all of the lower-dimensional Zagier-Dedekind sums are polynomialtime computale, and since the left-hand side of the main theorem is polynomialtime computale y Barvinok s theorem, the aove Zagier-Dedekind sums in dimension d is now also polynomial-time computale References [BP] A I Barvinok, J E Pommersheim, An algorithmic theory of lattice points in polyhedra, in New perspectives in algeraic cominatorics, Berkeley, CA, ( ), Math Sci Res Inst Pul 38, Camridge Univ Press, Camridge (1999), [BDR] M Beck, R Diaz, S Roins, The Froenius prolem, rational polytopes, and Fourier- Dedekind sums, to appear in J Numer Th [BGK] M Beck, I M Gessel, T Komatsu, The polynomial part of a restricted partition function related to the Froenius prolem, Electronic J Comin 8, no 1 (2001), N 7 [BR] M Beck, S Roins, Explicit and efficient formulas for the lattice point count inside rational polygons, to appear in Discr Comp Geom [B] B Berndt Reciprocity theorems for Dedekind sums and generalizations, Adv in Math 23, no 3 (1977), [BV] M Brion, M Vergne An equivariant Riemann-Roch theorem for simplicial toric varieties, J reine angew Math 482 (1997), [C] L Carlitz, A note on generalized Dedekind sums, Duke Math J 21 (1954), [De] R Dedekind, Erläuterungen zu den Fragmenten XXVIII, in Collected works of Bernhard Riemann, Dover Pul, New York (1953),

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