Multivariate Splines and Polytopes

Multiv ariate Splines and P olytop es ∗ Zhiqiang Xu † LSEC, Institute of Compu t ational Mathematics, Academy of Mathematics and System Sciences, Chinese Academ y of Sciences, Beijing, 100091, Chin a Abstract In this pap er, we use m ultiv a riate splines to inv estigate the vol ume of p olytop es. W e first present an explicit form ula for th e multiv ari ate truncated pow er, which can b e considered as a d ual version of th e fa- mous Brion’s formula for the volume of p olytopes. W e also prov e that the integration of p olynomials ove r p olyt op es can b e dealt with by the multiv ariate truncated p ow er. Moreov er, we show that the volume of the cub e slicing can b e considered as the maxim um v alue of th e box spline. Based on this connection, we give a simple p roof for Go o d’s conjecture, whic h has b een settled b efore b y p robabilit y metho ds. Keywor d s: Box Splines; Multiv ariate T runcated Pow ers; P olytop es; Unit C ub es 1 In tro duction Box splines a nd m ultiv a r iate truncated p ow ers were first introduced in [5] and [8], respectively . They have a wide rang e of importa nt and v aried applications in nu merical analysis a nd approximation theory . F rom the p oint of view of discrete geo metry , b ox splines and multiv ar iate tr uncated p owers ar e closely related to the volume o f cub e slicing and the v olume of p olytop es, re sp e ctively . How ev er, people working in the disc rete g eometry do not seem to b e fully aw are of the means of m ultiv ariate splines. The aim of this paper is to recast some problems related to the computatio n of volumes of p olytop es and s olve them by multiv ariate splines. W e b elieve that the means o f multiv a riate splines sheds some light on pro blems concerning p olyto p es . ∗ The work of this author was supp orted i n part by NSFC gran t (10871196) and National Basic Researc h Program of China (973 Program 2010CB832702), and was p erfor med i n part at T ec hn ische Universit¨ at Berl in. † Email: x uzq@lsec.cc.ac.cn. 1 The main results in this pap er ar e as follows. Some o f the main results concern the v olumes of po ly top es. The exa c t computation o f the volume o f a po lytop e P is an imp orta nt a nd difficult pr oblem which has close ties to v arious mathematical a reas. Brion’s form ula in contin uous form (se e [2, 7]), which is well known in discrete geometry , giv es an explicit formula for the volume of po lytop es. How ev er, t he formula lyings on the vertex repr esentation of polytop es and requires the ge ner ators for each vertex co ne. Based on the multiv ar iate exp onential truncated p ow er [1 9], we give a n e xplicit formula for the m ultiv ariate truncated power, which can b e rega rded as a dual version of Brio n’s formula. As another effort, La sserre [13] gav e a r ecursive for mula for the volume of po lytop es, which has b ecome a po pular metho d to day . W e re- prov e the for m ula by an iterative for m ula for the multiv aria te truncated p ower [16], which was pr esented by Micchelli. Another problem a ddressed is the in tegratio n of co ntin uous functions over po lytop es. This problem ha s imp or tant applications. F or exa mple, in most finite element in tegration metho ds, the domain o f integration is deco mp os ed int o p olytop es . Hence the integration of r e a l functions ov er poly top es is al- wa ys r e quired. In [12] an exact formula for the integration of p olynomials over simplices is pres e nted. An iter a tive formula for computing the integration over po lytop es is given in [14], and the formula is e x tended in [22] by F-s plines. W e shall show that integration of po lynomials o ver p olyto p es can be dealt with via the m ultiv a riate truncated pow er leading to an explicit formula for the int egra l of p olynomials over p o lytop es. As contin uous functions on a compact set can be uniformly approximated by po lynomials, this re sult provides an appr oximate formula for in tegrating contin uous functions ov er polyto pe s . Moreover, this r e- sult als o shows tha t one can c ompute the integrals of p olynomials ov er polytop es by calcula ting the volumes of p oly top es. The volume o f c ub e slicing is another active resear ch topic in discr e te geom- etry (see [2 3]). Suppo se that Q n := [0 , 1) n is the unit cub e in R n and H is a n 2 ( n − 1)-dimensional hyper plane of R n through o f its c ent er. According to [11], Go o d conjectured that vol( H ∩ Q n ) ≥ 1 . Hensley (unexpectedly) int ro duced a probabilistic method fo r the study of this conjecture and finally solved it [11]. In fact, we sho w that the conjecture can b e reformulated as the follo wing box spline problem: max x B ( x | ( a 1 , . . . , a n )) ≥ 1 p P n i =1 a 2 i , (1.1) where B ( x | ( a 1 , · · · , a n )) is a univ ariate box spline (cf. Section 3), a nd a i are po sitive r eal num b ers fo r 1 ≤ i ≤ n . Based on the explicit for mu la for the F ourier transform o f the b ox spline, we g ive a simple pr o of o f (1.1 ). Hence w e present a spline method fo r pro ving the conjecture. The pro ble m o f computing the volume of the int ersection o f Q n and an ( n − 1)-hyperpla ne is also in teresting a nd it can b e traced ba ck to P´ olya’s thesis. In [18], the authors derived a formula for the volumes of suc h domains using combinatorial metho ds. Using b ox splines , we g ive an explicit formula for the volume of conv ex bo dies obta ine d b y intersecting Q n and a j -hyperplane, where j is a p ositive in teg e r < n . The form ula in [18] may be considered as a s pe c ial case of ours. The pap er is org anized as follows. After recalling some definitions and nota- tions in Section 2, we s how (Section 3) the connectio n betw een the m ultiv ariate truncated power a nd the volume of polyto p es . In Section 4, we transform the int egratio n of p olyno mials over p olytop es to a problem co ncerning the m ulti- v ariate trunca ted p ower. In Section 5, we inv es tig ate the volume of cub e slicing using box splines. Finally , Section 6 illustrates the a pplication of the formulas given in this pa pe r with some examples. 2 Definitions and Notatio ns A con vex po lytop e P is the con vex h ull of a finite set of p oints in R d . Throughout this pap er, we shall omit the qua lifier “conv ex” since we c onfine our discus sion to such p olytop es. W e also assume that the affine hull of P contains the origin. 3 Moreov er, we us e d -p olytop e to mean a d -dimensio nal p olytop e. When the po lytop e P is de fined as the conv ex hull of a finite set of p oints in R d , the finite s et is called as a vertex represe n tation or simply V -re presentation of P , while, if P is defined as { x ∈ R n + | M x = b } for so me s × n matrix M and s -vector b , then the pair ( M , b ) is ca lled a ha lf space repr esentation or simply H -repr esentation. F or a vertex v of P , w e define the vertex cone of v as the smallest c one with vertex v that co ntains P . If P ⊂ R n is a d -p o lytop e, then let vol n ( P ) denote the d -dimensiona l volume of P in R n . F or a rational polytop e P , i.e., a polytop e whose v er tices hav e ra tional co o rdinates, let R P denote the space that is spanned by the vertex vectors of P . The lattice p oints in R P form an Ab elian group o f rank d , i.e., R P ∩ Z d is isomor phic to Z d . Hence there exists a n invertible affine linear transformatio n T : R P → R d satisfying T ( R P ∩ Z n ) = Z d . The r elative volume of P , denoted as v ol( P ), is just the d -dimensional volume of the ima ge T ( P ) ⊂ R d . F or mor e detaile d infor ma tion ab out the relative v olume, the rea der is referred to [20]. Throughout this paper, Z + and R + denote the no n-negative in teger and non-negative r eal sets, res pec tively . Given a set D , let χ D ( x ) = 1 if x ∈ D , otherwise let χ D ( x ) = 0. Elements o f R s can b e regar ded as row or column s -vectors a ccording to circumstances. Let M b e an s × n matrix. Then M can be considered as a multiset of its columns. The co ne spanned by M , denoted by cone( M ), is the set { P m ∈ M a m m | a m ≥ 0 for a ll m } . Mor eov er , we set [[ M )) := { P m ∈ M a m m | 0 ≤ a m < 1 , fo r all m ∈ M } . F urther more, we use # A to denote the cardina lit y of the finite set A . “ · ” stands for scalar pro duct and k · k for the E uclidean norm. E s × s denotes the s × s identit y matrix. As a final piece of notation, A − T := ( A − 1 ) T when A is an in vertible matrix. 4 3 Multiv ariate truncated p o w ers and the v ol- ume of p olytop es Let M b e an s × n rea l matrix with r ank( M ) = s . Recall that M is also viewed as the mult iset o f its column vectors. Throughout this section we a lwa ys assume that the interior of the conv ex h ull of M do es not contain the origin. The multivariate t runc ate d p ower T ( ·| M ) asso cia ted with M , fir st in tr o duced by Dahmen [8 ], is the distribution given by the rule Z R s T ( x | M ) φ ( x ) dx = Z R n + φ ( M u ) du, φ ∈ D ( R s ) , (3.1) where D ( R s ) is the space of test functions on R s . If we define P := { y ∈ R n + | M y = x } , then (see [6]) T ( x | M ) = vol n ( P ) p det( M M T ) . (3.2) Note that vol n ( P ) vol ( P ) = p det( M M T ) # { [[ M T )) ∩ Z s } provided M is a n in teger matrix (see [1]). Hence w e have T ( x | M ) = vol ( P ) # { [[ M T )) ∩ Z s } (3.3) provided M is an int eger matrix. In par ticular, if E s × s ⊂ M then T ( x | M ) = vol ( P ), since # { [[ M T )) ∩ Z s } = 1. So, the relativ e volume of polytop es P can be obtained b y computing T ( x | M ). In the following, we s ha ll g ive an explicit formula for T ( x | M ). W e firs t int ro duce the multivariate exp onential trunc ate d p ower E c ( x | M ) asso ciated with a complex vector c = ( c 1 , . . . , c n ) ∈ C n and a matrix M . The function E c ( x | M ) is the distribution g iven by the rule (see [19]): Z R s E c ( x | M ) φ ( x ) dx = Z R n + exp ( − c · u ) φ ( M u ) du, φ ( x ) ∈ D ( R s ) . (3.4) It is conv enient for us to index the constants c 1 , . . . , c n by an element m i ∈ M , that is, we set c m i := c i for i = 1 , . . . , n . F or the submatrix M ′ = ( m i 1 , . . . , m i k ) in M we set c M ′ := ( c i 1 , . . . , c i k ) and M ′ /c M ′ := ( m i 1 /c i 1 , . . . , m i k /c i k ). 5 W e recall a n ex plicit formula for E c ( ·| M ). In this formula, we de no te, given a square inv er tible Y ⊂ M , θ Y := Y − T c Y and α Y := Q y ∈ M \ Y ( θ Y · y − c y ) − 1 . Lemma 3.1. ([19]) E c ( x | M ) = X Y ⊂ M # Y =rank( Y )= s α Y E c Y ( x | Y ) , (3.5) for al l c ∈ C n such that t he deno minators in α Y do n ot vanish. Using Lemma 3.1 we can now give an explicit formula for T ( x | M ). Theorem 3.1. T ( x | M ) = 1 ( n − s )! X Y ⊂ M # Y =rank( Y )= s α Y | det Y | − 1 ( − θ Y · x ) n − s χ cone( Y ) ( x ) , for al l c ∈ C n such t hat the denominators on the right-hand side do not vanish, wher e b oth α Y and θ Y ar e define d in L emma 3.1. Pr o of. F or an inv ertible Y ⊂ M , one has E c Y ( x | Y ) = 1 | det Y | exp ( − θ Y · x ) χ cone( Y ) ( x ) . According to Lemma 3.1, for ρ ∈ R \ { 0 } , we have E ρc ( x | M ) = ρ − n + s X Y ⊂ M # Y =rank( Y )= s α Y E ρc Y ( x | Y ) . Then the T aylor ex pansion of E ρc ( x | M ) ab out 0 in the v ariable ρ is E ρc ( x | M ) = ∞ X l =0 ρ l − n + s p l ( x ) , (3.6) where p l ( x ) = 1 l ! X Y ⊂ M # Y =rank( Y )= s α Y | det Y | − 1 ( − θ Y · x ) l χ cone( Y ) ( x ) . The definition of T ( x | M ) implies that it is the c onstant term in (3.6). Hence, we hav e T ( x | M ) = p n − s ( x ). The theor em follows. Brion’s formula, which is obtained b y Brion’s Theorem, is useful for com- puting the rela tive volumes o f poly to pe s . W e state it her e. 6 Theorem 3.2. ([7]) Supp ose that P is a simple ra tional c onvex d -p olytop e. F or a vert ex c one K v of P , fix a set of gener ators w 1 ( v ) , w 2 ( v ) , . . . , w d ( v ) ∈ Z d . Then vol ( P ) = ( − 1) d d ! X v a v ertex of P ( v · c ) d | det( w 1 ( v ) , . . . , w d ( v )) | Q d k =1 ( w k ( v ) · c ) for al l c ∈ C d such that the denominators on the right-hand side do not vanish. Remark 3.1. Brion ’s formula r e quir es the V -r epr esentation and t he gener ators for e ach vertex c one, while t he formula pr esente d in The or em 3.1 r e quir es the H -r epr esentation. H en c e, the formula in The or em 3.1 c an b e c onsider e d as a dual versio n of Brion ’s. W e next turn to another for m ula for computing the relative volume of poly- top es. W e first in tro duce an iterative form ula fo r calculating the multiv ar iate truncated p ow er. Theorem 3.3. ([16]) L et M b e an s × n matrix with c olumns m 1 , . . . , m n ∈ R s \ { 0 } such that the origin is not c ontaine d in t he interior of conv( M ) . Su pp ose that n > s + 1 . F or any λ 1 , . . . , λ n ∈ R and x = P n j =1 λ j m j , we have T ( x | M ) = 1 n − s n X j =1 λ j T ( x | M \ m j ) . (3.7) W e next int ro duce Lasserr e’s formula for the volume of p olyto pes , which has bec ome well-known today (see [13, 3 ]). Cons ide r the co nv ex p oly top e defined by D ( b ) := { x ∈ R n | Ax ≤ b } , where A is an s × d matrix and b an s -vector. The i th face of D ( b ) is defined b y D i ( b ) := { x ∈ R d | a i · x = b i , Ax ≤ b } , where a i is the i th row of A . W e s et V ( d, A, b ) := vol d ( D ( b )) and set V i ( d − 1 , A, b ) := v o l d ( D i ( b )). Now we can describ e Lasserr e’s formula and show that it can b e proved by (3.7). 7 Theorem 3.4. ([13]) If V ( d, A, b ) is differ entiable at b , then V ( d, A, b ) = 1 d s X i =1 b i k a i k V i ( d − 1 , A, b ) . (3.8) Pr o of. Without los s of generality , we can s uppo se that all p oints in D ( b ) ar e non-negative, i.e., D ( b ) := { x ∈ R d + | Ax ≤ b } . W e fir st consider the case where each entry in A is an integer. B y (3.2) and (3.3), when A is an in teger matrix, T ( b | M ) = vol ( P ) = V ( d, A, b ) , where P := { x ∈ R d + s + | M x = b } and M := ( A, E s × s ). Let e i be the s -vector with 1 at the i th p os ition and 0 for j 6 = i . Using (3.7), w e o btain T ( b | M ) = 1 d s X i =1 b i T ( b | M \ e i ) . (3.9) Note that the ( d − 1)-p olytop e D i ( b ) lies in a hyper plane { x ∈ R d | a i · x = b i } and that the ( n − 1)-dimensio nal volume o f the unit parallelo gram in the hyper pla ne is k a i k gcd( a i 1 ,...,a id ) (cf. [1]), where a ij is the j th entry in the vector a i . So, one has vol d ( D i ( b )) vol ( D i ( b )) = k a i k gcd( a i 1 , . . . , a id ) . (3.10) By (3.3) and (3.1 0), we hav e T ( b | M \ e i ) = vol ( D i ( b )) # { [[ M \ e i )) ∩ Z s } = vol ( D i ( b )) gcd( a i 1 , . . . , a id ) = vol d ( D i ( b )) k a i k = V i ( d − 1 , A, b ) k a i k . Substituting T ( b | M ) = V ( d, A, b ) and T ( b | M \ e i ) = V i ( d − 1 ,A,b ) k a i k int o (3.9), we get (3.8). By taking limit, (3.8) ho lds for any ma tr ix A . 4 In tegration of p olynomial s o v er p olytop es In this section, we co nsider the problem of in tegrating of p olynomials ov er po lytop es. Since each p o ly nomial can b e wr itten as the sum of monomials, we only consider the mo nomial cas e. F or e very k = ( k 1 , . . . , k n ) ∈ Z n + and M = ( m 1 , . . . , m n ) we set M k := ( m 1 , . . . , m 1 | {z } k 1 +1 , m 2 , . . . , m 2 | {z } k 2 +1 , . . . , m n , . . . , m n | {z } k n +1 ) . 8 The following theorem shows that the integration of monomia ls can b e handled by the multiv ar iate truncated p ow er. Theorem 4.1. Su pp ose k = ( k 1 , . . . , k n ) ∈ Z n + and f ( u ) = Q n j =1 u k j j . Set P := { u ∈ R n + | M u = x } . Then T ( x | M k ) = 1 k ! · p det( M M T ) Z P f ( u ) du, wher e k ! := k 1 ! · · · k n ! . Pr o of. W e set T k ( x | M ) := 1 p det( M M T ) Z P f ( u ) du and consider its Laplace tra nsform, i.e., b T k ( ω | M ) := Z R s exp ( − ω · x ) T k ( x | M ) dx. F or every u in P := { u ∈ R n + | M u = x } , we can write u in a unique wa y as u = z + y wher e z ∈ ker ( M ) and y ⊥ ker ( M ). Note that x = M y a nd hence p det( M M T ) dy = dx (see [6]). The n w e have Z R s exp ( − ω · x ) T k ( x | M ) dx = 1 p det( M M T ) Z R s exp ( − ω · x ) Z { u ∈ R n + | M u = x } f ( u ) dudx = Z y ⊥ ker M exp ( − ω · M y ) Z z ∈ ker M χ R n + ( z + y ) f ( z + y ) dz dy = Z y ⊥ ker M Z z ∈ ker M χ R n + ( z + y ) f ( z + y ) exp ( − ω · M ( z + y )) dz dy = Z R n + f ( u ) ex p ( − ω · M u ) du = k ! · n Y j =1 1 ( ω · m j ) k j +1 . Also, note that b T ( ω | M k ) = n Y j =1 1 ( ω · m j ) k j +1 . Hence the theorem follows from the inv ers e theo r em for Laplace transform. 9 5 Multiv ariate b o x splines and the v olume of cub e slicing The m u ltivariate b ox spli ne B ( ·| M ) asso ciated with M is the distribution given by the rule (see [4, 5]) Z R s B ( x | M ) φ ( x ) dx = Z [0 , 1) n φ ( M u ) du, φ ∈ D ( R s ) . (5.1) According to [6], one has B ( x | M ) = vol n ( P ∩ [0 , 1) n ) p det( M M T ) , (5.2) where P := { y ∈ R n + | M y = x } . The form ula (5.2) sho ws the connection betw een the b ox spline and the volume of cube slicing. Bas ed on this connection, we can study s ome interesting problems concerning the unit cub e. Recall that Q n is the unit cub e in R n . W e assume that H is an ( n − 1)- dimensional hyperplane of R n through the center of Q n , i.e., H := { y ∈ R n | a 1 y 1 + · · · + a m y m = m X i =1 a i / 2 } , where 1 ≤ m ≤ n , a i is a real num b er for 1 ≤ i ≤ m . Bas e d on the symmetry of Q n , w e may assume that a i > 0 for 1 ≤ i ≤ m . W e set A := ( a 1 , . . . , a m ). Then from (5.2) we ha ve vol n ( H ∩ Q n ) = q det( AA T ) B (( a 1 + · · · + a m ) / 2 | A ) . By the symmetry of the box spline, B ( x | A ) achiev es its maximum v a lue at ( a 1 + · · · + a m ) / 2. So, as allow ed to be fore, Go o d’s co njecture is equiv alent to max x B ( x | A ) ≥ 1 p P m i =1 a 2 i , (5.3) where 1 ≤ m ≤ n . W e next presen t a spline metho d for pr oving (5.3). Theorem 5.1. max x B ( x | ( a 1 , . . . , a m )) ≥ 1 p P m i =1 a 2 i , wher e a i ar e p ositive r e al numb ers. The e qu ality holds if and only if m = 1 . 10 Pr o of. Set C ( x | A ) := B ( x + m X i =1 a i / 2 | A ) . The F ourier transfo r m o f C ( x | A ) is b C ( ω | A ) = m Y i =1 sin( ω a i / 2) ω a i / 2 . W e ca n see that b C (0 | A ) = 1. According to the definition of F our ier tra nsform, we conclude that Z ∞ 0 t 2 C ( t | A ) dt = − 1 2 b C ′′ (0 | A ) = P m i =1 a 2 i 24 . (5.4) Put S ( t ) := Z t 0 C ( x | A ) dx. By  t max x C ( x | A )  2 ≥ S ( t ) 2 we hav e  max x C ( x | A )  2 Z ∞ 0 t 2 C ( t | A ) dt ≥ Z ∞ 0 S ( t ) 2 C ( t | A ) dt (5.5) = 1 3 Z ∞ 0 dS ( t ) 3 dt dt = 1 24 , where the last equality follows fro m Z ∞ 0 C ( x | A ) dx = b C (0 | A ) 2 = 1 2 . W e combine (5 .4) a nd (5.5) to obtain max x B ( x | A ) = max x C ( x | A ) ≥ 1 p P m i =1 a 2 i . F rom (5.5), we see that the equality holds if and only if m = 1. In the following and final theorem, w e pr esent an e x plicit formula for the volume o f j - slice of Q n . The form ula g iven in [1 8] is the ( j = n − 1)-case in the next theorem. 11 Theorem 5.2. S u pp ose that M is an ( n − j ) × n matrix with rank( M ) = n − j and let P := { y ∈ R n + | M y = x } . D enote Ξ = { 0 , 1 } n and | ε | = P n i =1 ε i . Then we hav e vol n ( P ∩ Q n ) = p det( M M T ) j ! X Y ⊂ M # Y =rank( Y )= s α Y | det( Y ) | − 1 X ε ∈ Ξ ( − 1) | ε | ( − θ Y · ( x − M ε )) j χ cone( Y ) ( x − M ε ) , for al l c such that the denominators on the right-hand side do n ot vanish, wher e b oth α Y and θ Y ar e define d in L emma 3.1 . Pr o of. By (5.2), o ne has vol n ( P ∩ Q n ) = q det( M M T ) B ( x | M ) . Now we present an explicit form ula for B ( x | M ). Recall that B ( x | M ) = ∇ M T ( x | M ) (see [6]), where ∇ M := Q n i =1 ∇ m i and ∇ m i T ( x | M ) := T ( x | M ) − T ( x − m i | M ). F rom this formula, one obtains (see [17]): B ( ·| M ) = X ε ∈ Ξ ( − 1) | ε | T ( · − M ε | M ) . (5.6) The theorem is proved by putting (3.6) in to (5.6). 6 Examples Example 6.1. Set D ( z ) := { y ∈ R 2 + | y 1 + y 2 ≤ z ; − 2 y 1 + 2 y 2 ≤ z ; 2 y 1 − y 2 ≤ z } . The volume of D ( z ) has b e en c alculate d in [15 ] using Cauchy’s R esidue the or em. Her e we c an obtain it dir e ctly. Base d on ( 3.3), we have vol( D ( z )) = T ( z | M ) wher e M =   1 1 1 0 0 − 2 2 0 1 0 2 − 1 0 0 1   , z = ( z , z , z ) T . We use m i to denote the i th c olumn in M . A s hort c alculation shows that the c ones sp anne d by the squar e matric es ( m 1 , m 2 , m 4 ) , ( m 1 , m 2 , m 5 ) , ( m 1 , m 3 , m 4 ) , 12 ( m 2 , m 3 , m 5 ) and ( m 3 , m 4 , m 5 ) c ontain z . We sele ct c = (1 , 1 , 1 , 1 , 1 / 2) in The- or em 3. 1 and obtain that T ( z | M ) = 17 48 z 2 which agr e es with the r esult pr esent e d in [15]. Example 6.2. Set Ω d := { y ∈ R d + | P d i =1 y i ≤ 1 } . We c ons ider the pr oblem of inte gr ating of monomial s over Ω d , i.e ., J d := Z Ω d y k 1 1 · · · y k d d dy 1 · · · dy d . The value of J d is also c alculate d in [10, 21]. Base d on The or em 4.1, we c an c ompute it e asily. W e set e d := (1 , . . . , 1) ∈ Z d . Using The or em 4.1, we have J d = k 1 ! · · · k d ! T (1 | e P d i =1 k i + d +1 ) . It is wel l known that T ( x | e d ) = x d − 1 + ( d − 1)! . Henc e, we have J d = k 1 ! · · · k d ! T (1 | e P d i =1 k i + d +1 ) = k 1 ! · · · k d ! ( P d i =1 k i + d )! . Ac knowledgmen ts . The author is grateful to the referees for their carefully reading o f the manuscript and for their helpful commen ts o n improving the final version of this pap er. The autho r also thanks J. Gag elman, O. Holtz and B. Sturmfels for helpful discussions. References [1] A. I. Barvinok, A p olynomial time algorithm for counting in tegral p oints in p olyhedra when the dimension is fixed, Math. Oper. Res. 19 (1994) 769-779. [2] M. Bec k and S . Robins, Computing the contin uous discretely: integer-point enumeratio n in p olyhedron, U n dergraduate T exts in Mathematics, Springer- V erlag, N ew Y ork, 2007. [3] B. Bueler, A . Enge and K. F ukuda, Exact vol ume computation for p olytop es: A practical study , In : Po lytop es-Combinatori cs and Computation, G. Kalai and G. M. 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