Richard Dedekind, in the 1880 s [7], naturally arrived at the following arithmetic function. Let ((x)) be the sawtooth function defined by

Transcription

1 DEDEKIND CARLITZ POLYNOMIALS AS LATTICE-POINT ENUMERATORS IN RATIONAL POLYHEDRA MATTHIAS BECK, CHRISTIAN HAASE, AND ASIA R MATTHEWS Astrat We study higher-dimensional analogs of the Dedekind Carlitz polynomials X u, v; a, : u ka v k, where u and v are indeterminates and a and are positive integers polynomials satisfy the reiproity law k v u, v; a, + u v, u;, a u a v, Carlitz proved that these from whih one easily dedues many lassial reiproity theorems for the Dedekind sum and its generalizations We illustrate that Dedekind Carlitz polynomials appear naturally in generating funtions of rational ones and use this fat to give geometri proofs of the Carlitz reiproity law and various extensions of it Our approah gives rise to new reiproity theorems and omputational omplexity results for Dedekind Carlitz polynomials, a haraterization of Dedekind Carlitz polynomials in terms of generating funtions of lattie points in triangles, and a multivariate generalization of the Mordell-Pommersheim theorem on the appearane of Dedekind sums in Ehrhart polynomials of 3-dimensional lattie polytopes Introdution While studying the transformation properties of ηz : e πiz/2 n e 2πinz under SL 2 Z, Rihard Dedekind, in the 880 s [7], naturally arrived at the following arithmeti funtion Let x e the sawtooth funtion defined y { {x} x : 2 if x / Z, 0 if x Z Here {x} x x denotes the frational part of x For positive integers a and, we define the Dedekind sum as ka k s a, : k0 The Dedekind sum and its generalizations have sine intrigued mathematiians from various areas suh as analyti [, 8] and algerai numer theory [3, 20], topology [, 4], algerai [5, 6] and ominatorial geometry [3, 5], and algorithmi omplexity [2] Date: 3 Septemer Mathematis Sujet Classifiation Primary P2, L03; Seondary 05A5, 52C07 Key words and phrases Dedekind sum, Carlitz polynomial, reiproity, lattie point, rational polyhedron, polytope, generating funtion, Ehrhart polynomial Researh of Haase supported y DFG Emmy Noether fellowship HA 4383/

2 2 MATTHIAS BECK, CHRISTIAN HAASE, AND ASIA R MATTHEWS Almost a entury after the appearane of Dedekind sums, Leonard Carlitz introdued the following polynomial generalization, whih we will all a Dedekind Carlitz polynomial: u, v; a, : u ka v k Here u and v are indeterminates and a and are positive integers Undoutedly the most important asi property for any Dedekind-like sum is reiproity For the Dedekind Carlitz polynomials, this takes on the following form [6] Theorem Carlitz If u and v are indeterminates, and a and are relatively prime positive integers, then v u, v; a, + u v, u;, a u a v Carlitz s reiproity theorem generalizes that of Dedekind [7], whih states that for relatively prime positive integers a and, s a, + s, a 4 + a 2 + a + a Dedekind reiproity follows from Theorem y applying the operators u u twie and v v one to Carlitz s reiproity identity and onverting the greatest-integer funtions into frational parts Our motivation for this paper stems from the appearane of Dedekind sums in lattie-point enumerators for rational polyhedra The first suh instane was disovered y Mordell [5] the ase t in the following theorem and vastly generalized y Pommersheim [6], who laid the foundation for the appearane of Dedekind-like sums in Ehrhart polynomials L P t : # tp Z d ; here P is a lattie d-polytope ie, P has integral verties and t denotes a positive integer Theorem 2 Mordell Pommersheim Let T e the onvex hull of a, 0, 0, 0,, 0, 0, 0,, and 0, 0, 0, where a, and are pairwise relatively prime positive integers Then the Ehrhart polynomial of T is L T t a 6 t3 + + a + a a t 2 + k a + a + a + a s, a s a, s a, t + There are natural geometri interpretations for three of the four oeffiients appearing in the Ehrhart polynomial of the Mordell Pommersheim tetrahedron T : the leading oeffiient is the volume of T, the seond leading oeffiient equals half of the sum of the areas of the faes of T, and the onstant term is the Euler harateristi of T In fat, similar interpretations hold for any d-dimensional lattie polytope [3, 9] However, aside from tori varieties attahed to T [6], a geometri reason for the appearane of Dedekind sums in the linear term of the Ehrhart polynomial in Theorem 2 has so far eluded mathematiians In this paper, we study Dedekind Carlitz sums through the generating funtion attahed to the lattie points of a rational polyhedron P R d, namely, the integer-point transform σ P z σ P z, z 2, z d : z m, m P Z d where z m z m zm 2 2 z m d d Our goals in this paper are as follows: We show that Dedekind Carlitz polynomials appear naturally in integer-point transforms of rational ones Setion 2

3 DEDEKIND CARLITZ POLYNOMIALS AS LATTICE-POINT ENUMERATORS IN RATIONAL POLYHEDRA 3 We give novel geometri proofs of Theorem, some of its generalizations, and some new reiproity theorems Setions 3 and 5 We show that our geometri setup immediately implies that higher-dimensional Dedekind Carlitz polynomials an e omputed in polynomial time Setion 4 We realize the equivalene of Dedekind Carlitz polynomials and the integer-point transform of a two-dimensional analogue of the Mordell Pommersheim tetrahedron Setion 6 We give an intrinsi geometri reason why Dedekind sums appear in Theorem 2 y applying Brion s deomposition theorem [5] to the Mordell Pommersheim tetrahedron Setion 7 2 Polyhedral Cones Give Rise to Dedekind Carlitz Polynomials We start y deomposing the first quadrant R 2 0 into two ones, namely K {λ 0, + λ 2 a, : λ, λ 2 0}, K 2 {λ, 0 + λ 2 a, : λ 0, λ 2 > 0} Thus K is losed, K 2 is half-open, and R 2 0 is the disjoint union of K and K 2 Let s ompute their integer-point transforms By a simple tiling argument see, for example, [3, Chapter 3], σ K u, v σ Π u, v v j u ka v k σ Π u, v v u a v, j 0 k 0 where Π {λ 0, + λ 2 a, : 0 λ, λ 2 < } is the fundamental parallelogram of the one K Analogously, we an write σ K2 u, v σ Π2 u, v u u a v, where Π 2 {λ, 0 + λ 2 a, : 0 λ <, 0 < λ 2 } Note that we need to inlude different sides of the half-open parallelograms Π and Π 2 Figure The fundamental parallelogram Π Now we list the integer points in the half-open parallelepiped Π We may assume that a and are relatively prime Then, sine Π has height, { } k Π Z 2 0, 0, k, + : k a, k Z, a

4 4 MATTHIAS BECK, CHRISTIAN HAASE, AND ASIA R MATTHEWS whene σ K u, v + a k uk v k a + v u a v + uv v, u;, a v u a v We otain the integer-point transform for K 2 in the same way, arefully adjusting our sums for the half-open parallelepiped Π 2 : σ K2 u, v u + k vk u ka + u u a v u + uv u, v; a, u u a v 3 Carlitz Reiproity The reiproity theorem for Dedekind Carlitz sums now follows almost instantly from our geometri setup Proof of Theorem We have onstruted two ones K and K 2 suh that R 2 0 K K 2 as a disjoint union In the language of integer-point transforms, this means 2 σ R 2 0 u, v σ K u, v + σ K2 u, v We just omputed the rational generating funtions on the right-hand side, and the integer-point transform for the first quadrant is simple: σ R 2 u, v 0 u v Thus 2 eomes + uv v, u;, a u + uv u, v; a, u v v u a v + u u a v, whih yields the identity of Theorem after learing denominators Our new, geometri proof of Carlitz s reiproity theorem has a natural generalization to higher dimensions, whih yields the reiproity identity for the higher-dimensional Dedekind Carlitz polynomials u, u 2,, u n ; a, a 2,, a n : a n k u j k ka an u j k ka2 an 2 u j kan k an n u k n, where u, u 2,, u n are indeterminates and a, a 2,, a n are positive integers The reiproity theorem for these polynomials is due to Berndt and Dieter [4], and we give a novel proof using essentially the same geometri piture as in our previous proof Theorem 3 Berndt Dieter If a, a 2,, a n are pairwise relatively prime positive integers, then u n u, u 2,, u n ; a, a 2,, a n + u n u n, u,, u n 2, u n ; a n, a,, a n 2, a n + + u u 2, u 3,, u n, u ; a 2, a 3,, a n, a u a u a 2 2 u an n Proof Analogous to our proof of Theorem, we onstrut a single ray in n-dimensional spae, and then we deompose the non-negative orthant into n ones as follows Let a : a, a 2,, a n R n,

5 DEDEKIND CARLITZ POLYNOMIALS AS LATTICE-POINT ENUMERATORS IN RATIONAL POLYHEDRA 5 denote the j th unit vetor y e j, and define K {λ 2 e 2 + λ 3 e λ n e n + λa : λ 2,, λ n, λ 0}, K 2 {λ e + λ 3 e λ n e n + λa : λ > 0, λ 3,, λ n, λ 0}, K j { λ e + + λ j e j + λ j+ e j+ + + λ n e n + λa : λ,, λ j > 0, λ j+,, λ n, λ 0 }, K n {λ e + λ 2 e λ n e n + λa : λ,, λ n > 0, λ 0} The fundamental parallelepiped of K j is { λ e Π j + + λ j e j + λ j+ e j+ + + λ n e n + λa : 0 < λ,, λ j, 0 λ j+,, λ n, λ < } Thus a point in Π j will look like λ + λa,, λ j + λ, λ, λ j+ + λ+,, λ n + λa n, and a slie of this parallelepiped at x j k where k will ontain ka kaj kaj+ kan +,, +, k, +,, + as the only integer point Note also that the integer point e + + e j is in Π j Putting it all together just like in Setion 2 yields the integer-point transform σ Πj u σ Kj u u a u u j u j+ u n ka + kaj + u u 2 u j + k u uj u k j u j+ u u a u u j u j+ u n kaj+ + kan + n u u 2 u j + u u 2 u n u,, u j, u j+,, u n, u j ; a,,, +,, a n, u a u u j u j+ u n Sine n j K j R n 0 is a disjoint union of the nonnegative n-dimensional orthant, σ K u + σ K2 u + + σ Kn u σ R n 0 u u u n Theorem 3 follows upon learing denominators in this identity To virtually every theorem in this paper, there exist translate ompanions, ie, we an shift the ones involved in our proofs y a fixed vetor This gives rise to shifts in the greatest-integer funtions, and the resulting Carlitz sums are polynomial analogues of Dedekind Rademaher sums [9] For the sake of larity of exposition, we only give integer-vertex versions of our ones, ut the reader should keep in mind that aritrary verties do, in prinipal, not ause any additional prolems

6 6 MATTHIAS BECK, CHRISTIAN HAASE, AND ASIA R MATTHEWS 4 Computational Complexity Dedekind s reiproity law together with the identity sa, sa mod, allows us to ompute the lassial Dedekind sum in an Eulidean-algorithm style and thus very effiiently, namely in linear time Computational omplexity is measured in terms of the input length, eg, in this ase log a + log We do not know how to apply a similar reasoning to the Dedekind Carlitz polynomials; however, the following entral theorem of Barvinok [2] allows us to dedue that higher-dimensional Dedekind Carlitz polynomials an e omputed effiiently Theorem 4 Barvinok In fixed dimension, the integer-point transform σ P z, z 2,, z d of a rational polyhedron P an e omputed as a sum of rational funtions in z, z 2,, z d in time polynomial in the input size of P Note that Barvinok s theorem says, in partiular, that the rational funtions whose sum represent σ P z are short, ie, the set of data needed to output this sum of rational funtions is of size that is polynomial in the input size of P The appliation of Barvinok s theorem to any of the ones appearing in the proof of Theorem 3 immediately yields the following novel omplexity result Theorem 5 For fixed n, the higher-dimensional Dedekind Carlitz polynomial u, u 2,, u n ; a, a 2,, a n an e omputed in time polynomial in the size of a, a 2,, a n In partiular, this result says that there is a more eonomial way to write the long polynomial u, u 2,, u n ; a, a 2,, a n as a short sum of rational funtions Theorem 5 implies that any Dedekind-like sum that an e derived from Dedekind Carlitz polynomials eg, y applying differential operators an also e omputed effiiently 5 Variations on a Theme The geometry we have used so far is very simple: it is ased on one ray in spae Naturally, this an e extended in numerous ways To illustrate one of them, onsider two rays through the points a, and, d in the first quadrant of the plane, where a,,, d are positive integers This onstrution deomposes the first quadrant into three rational ones as shown in Figure 2 Figure 2 Two-ray deomposition of the first quadrant The integer-point transform of eah of the two exterior ones, K and K 3, is easily omputed in the same manner as in Setion 2 However, the one in the middle, K 2, is ounded on either side y a non-unit vetor, and therefore the interior points of the fundamental parallelogram of this one are not trivial to list We will use unimodular transformations to ompute σ K2 This onstrution

7 DEDEKIND CARLITZ POLYNOMIALS AS LATTICE-POINT ENUMERATORS IN RATIONAL POLYHEDRA 7 leads to a new reiproity identity Theorem 7 elow We start y omputing the integer-point transform for a general rational one in R 2, ie, K 2 in Figure 2 Lemma 6 Suppose a,,, d are positive integers suh that ad > and gda, gd, d, and let x, y Z suh that ax + y Then the integer-point transform of the one K : {λa, + µ, d : λ, µ 0} is σ K u, v + ua y v +x u a v, u y v x ; x + dy, ad u a v u v d Proof To ompute the generating funtion σ K, note that the linear transformation given y x y M : a maps K to the one MK generated y, 0 and x + dy, ad, whose integer-point transform we know how to ompute from Setion 2 Sine M a y x we otain σ K u, v m,n K Z 2 u m v n m,n MK Z 2 u am yn v m+xn + ua y v +x u a v, u y v x ; x + dy, ad u a v u v d m,n MK Z 2 u a v m u y v x n Theorem 7 Suppose a,,, d are positive integers suh that ad > and gda, gd, d, and let x, y Z suh that ax + y Then uvu u a v v, u; d, + uvv u v d u, v; a, + u a y v +x u v u a v, u y v x ; x + dy, ad u a+ v +d u a v uv v + u v d uv u + + uv Proof Again, we prove the theorem geometrially Let K {λ 2 e 2 + λ d, d : λ 2 > 0, λ d 0}, K 2 {λ a a, + λ d, d : λ a, λ d 0}, K 3 {λ e + λ a a, : λ > 0, λ a 0}, so that K 2 is losed and K and K 3 are half-open, and K K 2 K 3 R 2 0 is a disjoint union of the first quadrant With the methods introdued in Setion 2, the integer-point transforms of K and K 3 are σ K u, v v + k uk v kd + v u v d σ K3 u, v u + k vk u ka + u u a v v + uv v, u; d, v u v d, u + uv u, v; a, u u a v, and the integer-point transform of K 2 was omputed in Lemma 6 By our onstrution, σ K u, v + σ K2 u, v + σ K3 u, v σ R 2 u, v 0 u v, from whih Theorem 7 follows upon learing denominators

8 8 MATTHIAS BECK, CHRISTIAN HAASE, AND ASIA R MATTHEWS Theorem 7 is the polynomial analogue of the following result due to Pommersheim [6, Theorem 7]: Theorem 8 Pommersheim Suppose a,,, d are positive integers suh that gda, gd, d, and let x, y Z suh that ax + y Then sa, + s, d sx dy, ad dad + + d ad + + ad + d This, in turn, generalizes Rademaher s three-term reiproity theorem [7]: Corollary 9 Rademaher If a,, are pairwise oprime positive integers then s a, + s a, + s, a 4 + a 2 + a +, a where a a mod, mod, and mod a The ase n 3 of Berndt Dieter s Theorem 3 is the polynomial analogue of this three-term law Clearly, Dedekind s reiproity theorem is implied y Corollary 9, and Girstmair showed [0] that implies Theorem 8, so that, in fat, Dedekind s, Rademaher s, and Pommersheim s reiproity theorems an e derived from eah other Suh an equivalene an learly not hold on the polynomial level, ut one ould ask whether Theorems and 7 an e derived from eah other Theorem 7 ould e generalized in several ways, eg, to higher dimensions or to more than three ones in dimension 2 this yields a Carlitz polynomial analogue of [6, Theorem 8], ut we digress Theorem 7 simplifies when ad : then the third Dedekind Carlitz polynomial disappears Geometrially, this stems from the fat that the one K 2 is unimodular, ie, its fundamental parallelogram ontains only the origin Corollary 0 If a,,, d are positive integers suh that ad, then u u a v v, u; d, + v u v d u, v; a, u a+ v +d u a v u v d + u a v + u v d u a v u v d + This is the polynomial analogue of the following result due to Rademaher [8] Corollary Rademaher If a,,, d are positive integers suh that ad, then sa, + sd, 2 + a 2 + a + d + d 6 Dedekind Carlitz Polynomials as Integer-Point Transforms of Triangles We have shown that Dedekind Carlitz polynomials appear as natural ingredients of integerpoint transforms of ones In this light, it should ome as no surprise that the following oni deomposition theorem of Brion [5] will prove useful We remind the reader that a rational polytope is the onvex hull of finitely many rational points in R d If v is a vertex of the polytope P, then the vertex one K v is the smallest one with apex v that ontains P Theorem 2 Brion Suppose P is a rational onvex polytope Then we have the following identity of rational funtions: σ P z σ Kv z, v where the sum is over all verties of P Brion s theorem allows us to give a novel expression for the Dedekind Carlitz polynomial as the integer-point transform of a ertain triangle

9 DEDEKIND CARLITZ POLYNOMIALS AS LATTICE-POINT ENUMERATORS IN RATIONAL POLYHEDRA 9 Theorem 3 Let a and e relatively prime positive integers and u and v e indeterminates If is the triangle with verties 0, 0, a, 0, and 0,, then the Dedekind Carlitz polynomial u, v; a, and the integer-point transform of are related in the following manner: u σ u, v u a v, v; a, u Proof The triangle omes with the vertex ones K {j, 0 + k0, : j, k 0}, + u u a + v v+ v K 2 {a, 0 + j, 0 + k a, : j, k 0}, K 3 {0, + ja, + k0, : j, k 0} One more we apply the methods of Setion 2 and otain the integer-point transforms σ K u, v u v, σ K2 u, v u a + k vk ka u ua+ u u a v + u a v u, v; a, u u a v σ K3 u, v v + a k uk k v a v+ v u a v + uv v, u;, a v u a v Now Brion s Theorem 2 gives σ u, v u v ua+ + u a v u, v; a, u u a v v+ + uv v, u;, a v u a v or, after some manipulations, 3 u v u a v σ u, v u a v + u 2a+ v v 2+ u + u 2a vv, v; a, uv 2 u, u;, a u v The two Dedekind Carlitz polynomials in this expression are atually related y reiproity: after a simple hange of variales, Carlitz s Theorem implies uv u, u;, a u a vv, v; a, u a v uv, v u and sustituting this into 3 finishes the proof We remark that we ould also deal with triangles with rational verties The setup is exatly the same, however, the formulas eome quite a it messier, 7 The Mordell Pommersheim Tetrahedron In this final setion we derive a generating-funtion analogue to Theorem 2 It is natural to extend the appliation of Brion s theorem in the last setion to polytopes in higher dimensions, whih we will do with the tetrahedron T Sine our ultimate goal is a geometri proof of Theorem 2, we will apply Brion s theorem to the dilate tt for any positive integral t As we will see, the following higher-dimensional extensions of Dedekind Carlitz polynomials will appear naturally in

10 0 MATTHIAS BECK, CHRISTIAN HAASE, AND ASIA R MATTHEWS the integer-point transforms of the vertex ones of T For positive integers a,,, and indeterminates u, v, w, we define the Dedekind Rademaher Carlitz DRC sum dru, v, w; a,, : u ja + ka v j w k The DRC sums are the main ingredients for the integer-point transform of T Theorem 4 Let T e the onvex hull of a, 0, 0, 0,, 0, 0, 0,, and 0, 0, 0, where a, and are pairwise relatively prime positive integers Then u v w u a v u a w v w σ tt u, v, w u t+2a v w v w u + dr u, v, w; a,, v t+2 u w u a w v + dr v, u, w;, a, + w t+2 u w u a v w + dr w, u, v;, a, u a v u a w v w Proof The tetrahedron tt has the vertex ones K 0 {λ, 0, 0 + λ 2 0,, 0 + λ 3 0, 0, : λ, λ 2, λ 3 0}, K {ta, 0, 0 + λ, 0, 0 + λ 2 a,, 0 + λ 3 a, 0, : λ, λ 2, λ 3 0}, K 2 {0, t, 0 + λ a,, 0 + λ 2 0,, 0 + λ 3 0,, : λ, λ 2, λ 3 0}, K 3 {0, 0, t + λ a, 0, + λ 2 0,, + λ 3 0, 0, : λ, λ 2, λ 3 0}, and we will ultimately apply Brion s Theorem 2 to otain 4 σ tt z σ K0 z + σ K z + σ K2 z + σ K3 z, whih will yield Theorem 4 The omputation of the integer-point transforms of the vertex ones should e routine y now; we will derive only one of them in detail The vertex one K has the fundamental parallelepiped Π ta, 0, 0 + {λ, 0, 0 + λ 2 a,, 0 + λ 3 a, 0, : 0 λ, λ 2, λ 3 < } There exists exatly one integer point in Π at eah integer j along the y-axis and k along the z-axis In other words, for j, k Z 0, x, j, k Π is written as x, j, k λ λ 2 a λ 3 a, λ 2, λ 3 and hene λ 2 j and λ 3 k Thus and Π Z 3 { ja ka }, j, k : j, k 0,,, ja σ K u, v, w u ta u ka v j w k u u a v u a w u ta + u ja + ka v j w k u u 2a u u a v u a w [ ut+2a u + dr u, v, w; a,, ] u u a v u a w

11 DEDEKIND CARLITZ POLYNOMIALS AS LATTICE-POINT ENUMERATORS IN RATIONAL POLYHEDRA Similarly, we derive and [ σ K2 u, v, w vt+2 v + dr v, u, w;, a, ] v u a v v w, [ σ K3 u, v, w wt+2 w + dr w, u, v;, a, ] w u a w v w, σ K0 u, v, w u v w Sustituting these rational funtions into 4 and learing denominators gives the theorem Proof of Theorem 2 Theorem 4 gives σ tt u, v, w as a rational funtion with numerator u t+2a v w v w u + dr u, v, w; a,, v t+2 u w u a w v + dr v, u, w;, a, + w t+2 u w u a v w + dr w, u, v;, a, u a v u a w v w and denominator u v w u a v u a w v w We otain the Ehrhart polynomial L T t σ tt,, y taking the limit of this rational funtion as u, v, w We need to use L Hospital s rule and take partial derivatives with respet to u one, v twie, and w three times in oth numerator and denominator efore sustituting u v w This is easily done for the denominator with the result 2 2 The numerator of σ tt u, v, w is not handled as easily After differentiating and setting u v w it eomes 5 2a 2 3 t 3 6a 2 3 t t 2 4a 2 3 t 8 2 t j + 6t a + k j 6 2 t a + k 2t t 6t j k a + k j a + k j a + k 6t t 2 j + a + k j a + k j a + k plus numerous terms that do not depend on t Fortunately, eause the onstant term of L T is see, for example, [3, Corollary 35], we are not onerned with these extra terms As efore, we

12 2 MATTHIAS BECK, CHRISTIAN HAASE, AND ASIA R MATTHEWS replae all greatest-integer funtions with frational-part funtions to modify 5 to 6 2a 2 3 t t 2 3a 3 t t a t 6a3 t 3 3 t a 3 t + 6a 2 2 t + 6a 2 t 8 2 t 5a 2 t { j 6t a + k } { j t a + k } + 2t + 2t + 22 a a t a + k } j a + k } 24 2 t t Now we use three elementary identities: a + k } They allow us to simplify 6 to a 2 a + k k a + k }, } 6 2 t 2 j a + k } a sa, +, 4 a + k } 2 a 2a 6a 2a 2 3 t t 2 3a 3 t t t { j k a + k } a t 33 t a 3 t t 3a 2 t 9 2 t a 2 t a t + 22 sa, + sa, + s, a t a + k } a + k } 2 We divide y the denominator 2 2 and add the onstant term to arrive at the desired formula for L T t Referenes [] Gert Almkvist, Asymptoti formulas and generalized Dedekind sums, Experiment Math 7 998, no 4, [2] Alexander I Barvinok, A polynomial time algorithm for ounting integral points in polyhedra when the dimension is fixed, Math Oper Res 9 994, no 4, [3] Matthias Bek and Sinai Roins, Computing the ontinuous disretely: Integer-point enumeration in polyhedra, Undergraduate Texts in Mathematis, Springer-Verlag, New York, 2007 [4] Brue C Berndt and Ulrih Dieter, Sums involving the greatest integer funtion and Riemann-Stieltjes integration, J Reine Angew Math , [5] Mihel Brion, Points entiers dans les polyèdres onvexes, Ann Si Éole Norm Sup , no 4, [6] Leonard Carlitz, Some polynomials assoiated with Dedekind sums, Ata Math Aad Si Hungar , no 3-4, 3 39 [7] Rihard Dedekind, Erläuterungen zu den Fragmenten xxviii, Colleted Works of Bernhard Riemann, Dover Pul, New York, 953, pp [8] Ulrih Dieter, Das Verhalten der Kleinshen Funktionen log σ g,h ω, ω 2 gegenüer Modultransformationen und verallgemeinerte Dedekindshe Summen, J Reine Angew Math , 37 70

13 DEDEKIND CARLITZ POLYNOMIALS AS LATTICE-POINT ENUMERATORS IN RATIONAL POLYHEDRA 3 [9] Eugène Ehrhart, Sur les polyèdres rationnels homothétiques à n dimensions, C R Aad Si Paris , [0] Kurt Girstmair, Some remarks on Rademaher s three-term relation, Arh Math Basel , no 3, [] Friedrih Hirzeruh and Don Zagier, The Atiyah-Singer Theorem and Elementary Numer Theory, Pulish or Perish In, Boston, Mass, 974 [2] Donald E Knuth, The Art of Computer Programming Vol 2, seond ed, Addison-Wesley Pulishing Co, Reading, Mass, 98 [3] Curt Meyer, Üer einige Anwendungen Dedekindsher Summen, J Reine Angew Math , [4] Werner Meyer and Roert Szeh, Üer eine topologishe und zahlentheoretishe Anwendung von Hirzeruhs Spitzenauflösung, Math Ann , no, [5] Louis J Mordell, Lattie points in a tetrahedron and generalized Dedekind sums, J Indian Math So NS 5 95, 4 46 [6] James E Pommersheim, Tori varieties, lattie points and Dedekind sums, Math Ann , no, 24 [7] Hans Rademaher, Generalization of the reiproity formula for Dedekind sums, Duke Math J 2 954, [8], Zur Theorie der Dedekindshen Summen, Math Z , [9], Some remarks on ertain generalized Dedekind sums, Ata Arith 9 964, [20] David Solomon, Algerai properties of Shintani s generating funtions: Dedekind sums and oyles on PGL 2Q, Compositio Math 2 998, no 3, Department of Mathematis, San Franiso State University, San Franiso, CA 9432, USA address: ek@mathsfsuedu URL: Fahereih Mathematik & Informatik, Freie Universität Berlin, 495 Berlin, Germany address: hristianhaase@mathfu-erlinde URL: Department of Mathematis & Statistis, Queen s University, Kingston, ON, Canada, K7L 3N6 address: asiarosem@yahooom