Lower bounds on the coefficients of Ehrhart polynomials

LO WER BOUNDS ON THE COEFFIC IENTS OF EHRHAR T POL YNOMIALS MAR TIN HENK AND MAKOTO T AGAMI Abstra ct. W e present lo w er b oun ds for the coefficients of Ehrhart p olyn o- mials of conv ex lattice p olytop es in terms of their vol ume. Concerning the coefficients of the Ehrhart series of a lattice p olytop e we sho w that Hibi’s lo w er b ound is not true for lattice p olytop es without interio r lattice p oints. The counterexample is based on a formula of the Ehrhart series of the join of tw o lattice p olytop e. W e also present a formula for calculating th e Ehrhart series of integra l dilates of a polytop e. 1. Introduction Let P d b e the set of a ll con vex d -dimensional lat tice p olytop es in the d - dimensional Euclidean space R d with resp ect to the standard lattice Z d , i.e., all ve rtices of P ∈ P d ha v e inte gral co ord in ates and dim( P ) = d . The lattice p oin t enumerato r of a set S ⊂ R d , denoted by G( S ), count s t he n umb er of lattice (inte gral) p oint s in S , i.e., G( S ) = #( S ∩ Z d ). In 1962, Eug´ ene Ehr hart (see e.g. [3, Chapter 3], [7 ]) s ho w ed that for k ∈ N the lattice p oint en umerator G( k P ), P ∈ P d , is a p olynomial of degree d in k where the co efficien ts g i ( P ), 0 ≤ i ≤ d , dep end only on P : (1.1) G( k P ) = d X i =0 g i ( P ) k i . The p olynomial on the r igh t hand side is called th e Ehrhart p olynomial, and regarded as a formal p olynomial in a complex v ariable z ∈ C it is denoted b y G P ( z ). Tw o o f the d + 1 co efficien ts g i ( P ) are almost ob vious , namely , g 0 ( P ) = 1, the Euler c haracteristic of P , and g d ( P ) = vo l( P ), where v ol() denotes the v olume, i.e., t he d -dimensional Leb esgue measure on R d . It was sho wn b y Ehrhart (see e.g. [3, Theorem 5.6], [8]) that also the second leading co efficien t admits a simple geometric in terpretation as latti ce su rface area of P (1.2) g d − 1 ( P ) = 1 2 X F f acet of P v ol d − 1 ( F ) det(aff F ∩ Z d ) . Here vo l d − 1 ( · ) d enotes the ( d − 1)-dimensional vo lume and det(aff F ∩ Z d ) denotes the d etermin ant of th e ( d − 1 )-dimensional sublattice cont ained in the affine h ull of F . All other coefficien ts g i ( P ), 1 ≤ i ≤ d − 2, h a ve n o suc h kno wn explicit 2000 Mathemat ics Subje ct Classific ation. 52C07, 52B 20, 11H06. Key wor ds and phr ases. La ttice p olytop es, Ehrhart p olynomial. The second author w as supp orted by R esearc h F ello wships of the Japan So ciety for the Promotion of Science for Y oun g Scien tists. 1 2 MAR TIN HENK A ND MAKOTO T AG AMI geometric meaning, except for sp ecial classes of p olytop es. F or this and as a general reference on the theory of lattice p olytop es w e refer to the recen t b o ok of Matthias Bec k and Sin ai Robins [3] and the referen ces within. F or more information regarding lattices and the role of the lattic e p oin t en u merator in con vexit y see [9]. In [4, Theorem 6 ] Ulrich Betk e and Pete r McMullen prov ed the follo wing upp er b ounds on the co efficien ts g i ( P ) in terms of the v olume: g i ( P ) ≤ ( − 1) d − i stirl( d, i )v ol ( P ) + ( − 1) d − i − 1 stirl( d, i + 1) ( d − 1)! , i = 1 , . . . , d − 1 . Here stirl( d, i ) denote the Stirling n u m b ers of the first kind w h ic h can b e defined via the id en tit y Q d − 1 i =0 ( z − i ) = P d i =1 stirl( d, i ) z i . In order to p r esen t our lo wer b ounds on g i ( P ) in terms of the volume we need some n otatio n. F or an integ er i and a v ariable z w e consider the p olynomial ( z + i )( z + i − 1) · . . . · ( z + i − ( d − 1)) = d !  z + i d  , and w e denote its r -th co efficien t by C d r,i , 0 ≤ r ≤ d . F or instance, it is C d d,i = 1, and f or 0 ≤ i ≤ d − 1 we ha v e C d 0 ,i = 0. F or d ≥ 3 we are interested in (1.3) M r,d = min { C d r,i : 1 ≤ i ≤ d − 2 } . Ob viously , we hav e M 0 ,d = 0, M d,d = 1 and it is also easy to s ee that (cf. Prop o- sition 2.1 iii)) (1.4) M d − 1 ,d = C d d − 1 , 1 = − d ( d − 3) 2 . With th e help of these n umbers M r,d w e obtain the follo w ing lo w er b ound s. Theorem 1.1. L et P ∈ P d , d ≥ 3 . Then for i = 1 , . . . , d − 1 we have g i ( P ) ≥ 1 d ! n ( − 1) d − i stirl( d + 1 , i + 1) + ( d ! vo l( P ) − 1) M i,d o . W e remark th at the co efficien ts g i ( P ), 1 ≤ i ≤ d − 2, might b e negativ e and th us also the lo w er b ound s giv en abov e. In general, the b ound s of T heorem 1.1 are not b est p ossible. F or instance, in the case i = d − 1 we get together with (1.4) the b ound g d − 1 ( P ) ≥ 1 ( d − 1)!  d − 1 − d − 3 2 d ! v ol( P )  . On the other h and, sin ce the lattice su rface area of any facet is at least 1 / ( d − 1)! w e h av e the trivial inequalit y (cf. (1.2)) (1.5) g d − 1 ( P ) ≥ 1 2 d + 1 ( d − 1)! . Hence the lo w er b ound on g d − 1 ( P ) given in Theorem 1.1 is only b est p ossible if v ol ( P ) = 1 /d !. In the cases i ∈ { 1 , 2 , d − 2 } , ho w ever, Theorem 1.1 giv es b est p ossible b ounds for any v olume. LOW ER BOUNDS ON THE COEFFICIENTS OF EHRHAR T POL YNOMIALS 3 Corollary 1.2. L et P ∈ P d , d ≥ 3 . Th en i) g 1 ( P ) ≥ 1 + 1 2 + · · · + 1 d − 2 + 2 d − 1 − ( d − 2)! v ol( P ) , ii) g 2 ( P ) ≥ ( − 1) d d ! × n stirl( d + 1 , 3) +  ( − 1) d ( d − 2)! + stirl( d − 1 , 2)  ( d ! v ol( P ) − 1) o , iii) g d − 2 ( P ) ≥    1 d ! ( d − 1) d ( d +1) 24 { 3( d + 1) − d ! vol( P ) } : if d o dd , 1 d ! ( d − 1) d 24 { 3 d ( d + 2) − ( d − 2) d ! v ol( P ) } : if d ev en . And the b ounds ar e b e st p ossible for any volume. F or some recent inequalities in v olving more co efficien ts of Ehr hart p olynomi- als w e refer to [2]. Next we come to another family of coefficient s of a p olynomial asso ciated to lattice p olytop es. The generating function of the lattice p oin t enumerato r, i.e., the formal p o w er series Ehr P ( z ) = X k ≥ 0 G P ( k ) z k , is called the Eh rhart series of P . It is well kno w n that it can b e expressed as a rational function of the form Ehr P ( z ) = a 0 ( P ) + a 1 ( P ) z + · · · + a d ( P ) z d (1 − z ) d +1 . The p olynomial in the numerator is calle d the h ⋆ -p olynomial. Its degree is also called the d egree of the p olytop e [1] and it is denoted by deg ( P ). Concerning the co efficien ts a i ( P ) it is known that they are integ ral and that a 0 ( P ) = 1 , a 1 ( P ) = G( P ) − ( d + 1) , a d ( P ) = G(int( P )) , where int( · ) denotes the in terior. Moreo ver, due to Stanley’s famous n on - negativit y theorem (see e.g. [3, Theorem 3.12], [17]) w e also kno w that a i ( P ) is non-negativ e, i.e., for these co efficien ts we ha v e the lo wer b oun d s a i ( P ) ≥ 0. In the case G(int( P )) > 0, i.e., deg( P ) = d , these b oun ds w ere improv ed by T ak a yuki Hibi [13] to (1.6) a i ( P ) ≥ a 1 ( P ) , 1 ≤ i ≤ deg ( P ) − 1 . In this cont ext it w as a quite natural question wh ether the assump tion deg( P ) = d can b e w eak en (see e.g. [15]), i.e., wh ether these low er b oun ds (1.6 ) are also v alid for p olytop es of degree less than d . As we sho w in Example 1.4 the ans wer is already n egativ e f or p olytop es ha ving d egree 3. The problem in order to stud y suc h a q u estion is th at only very f ew geometric constructions of p olytop es are kno wn for whic h we can explicitly calculate the Ehrh art series. In [3, Th eorem 2.4, T heorem 2.6] th e Ehrh art s er ies of sp ecial pyramids and doub le pyramids o ver a b asis Q are determined in terms of the Ehr hart series of Q . In a recen t pap er Braun [6] ga ve a v ery nice p ro duct form ula for the Ehr h art series o f the free s u m of t w o lattice polytop es, where one of the p olytop es has to b e reflexiv e. Here we consider a related constru ction, kn o w n as the join of tw o 4 MAR TIN HENK A ND MAKOTO T AG AMI p olytop es [11]. As we learned b y Matthias Bec k the Ehr hart series of suc h a join is already describ ed as E x er cise 3.32 in the b o ok [3] and it w as p ersonally comm u nicated to the authors of the b o ok by Kevin W o o ds. F or completeness’ sak e we present its sh ort p ro of in Section 3. Lemma 1.3. F or P ∈ P p and Q ∈ P q let P ⋆ Q b e the join of P and Q , i.e., P ⋆ Q = con v { ( x, 0 q , 0) ⊺ , (0 p , y , 1) ⊺ : x ∈ P , y ∈ Q } ∈ P p + q +1 , wher e 0 p and 0 q denote the p - and q -dimensional 0 -ve ctor, r esp e ctively. Then Ehr P ⋆Q ( z ) = Ehr P ( z ) · Ehr Q ( z ) . In order to apply this lemma we consider t wo families of lattice simplices. F or an intege r m ∈ N let T ( m ) d = con v { o, e 1 , e 1 + e 2 , e 2 + e 3 , . . . , e d − 2 + e d − 1 , e d − 1 + m e d } , S ( m ) d = con v { o, e 1 , e 2 , e 3 , . . . , e d − 1 , m e d } , where e i denotes the i -th un it v ector. It was sho wn in [4] that (1.7) Ehr T ( m ) d ( z ) = 1 + ( m − 1) z ⌈ d 2 ⌉ (1 − z ) d +1 and E h r S ( m ) d ( z ) = 1 + ( m − 1) z (1 − z ) d +1 . Actually , in [4] the formula for T ( m ) d w as only pro v ed for o dd dimensions, bu t the ev en case can b e treated completely analogously . Example 1.4. F or q ∈ N o dd and l , m ∈ N we have Ehr T ( l +1) q ⋆S ( m +1) p ( z ) = 1 + m z + l z q +1 2 + m l z q +3 2 (1 − z ) p + q +2 . In p articular, for q ≥ 3 and l < m this shows that (1.6) is, in gener al, false for lattic e p olytop es without i nterior lattic e p oints. Another f orm ula for calculating the Ehr h art Series from a give n one concerns dilates. Here we will sho w Lemma 1.5. L et P ∈ P d , k ∈ N and let ζ b e a primitive k -th r o ot of u nity. Then Ehr k P ( z ) = 1 k k − 1 X i =0 Ehr P ( ζ i z 1 k ) . The lemma can b e used, for instance, to calculate the Ehr hart series of the cub e C d = { x ∈ R d : | x i | ≤ 1 , 1 ≤ i ≤ d } . Example 1.6. F or two inte gers j, d , 0 ≤ j ≤ d , let A ( d, j ) = j X k =0 ( − 1) k  d + 1 k  ( j − k ) d LOW ER BOUNDS ON THE COEFFICIENTS OF EHRHAR T POL YNOMIALS 5 b e the Eulerian numb ers (se e e.g . [3, pp. 28] ). F urthermor e, we set A ( d, j ) = 0 if j / ∈ { 0 , . . . , d } . Then, for 0 ≤ i ≤ d , we have a i ( C d ) = d +1 X j = 0  d + 1 j  A ( d, 2 i + 1 − j ) . Of course, the cub e C d ma y b e also r egarded as a prism o v er a ( d − 1)-cub e, and as a coun terp art to the bip yramid construction in [3] w e calculate here also the Ehrhart ser ies of some sp ecial prism. Example 1 .7. L et Q ∈ P d − 1 , m ∈ N , and let P = { ( x, x d ) ⊺ : x ∈ Q, x d ∈ [0 , m ] } b e the prism of height m over Q . Then a i ( P ) = ( m i + 1)a i ( Q ) + ( m ( d − i + 1) − 1) a i − 1 ( Q ) , 0 ≤ i ≤ d, wher e we set a d ( Q ) = a − 1 ( Q ) = 0 . It seems to b e quite likel y that for the class of 0-symmetric lattic e p olytop es P d o the lo w er b ound s on a i ( P ) can consider ab ly b e impro v ed. In [5] it was conjectured that for P ∈ P d o a i ( P ) + a d − i ( P ) ≥  d i  (a d ( P ) + 1) , where equalit y holds for instance for the cross-p olytop es C ⋆ d (2 l − 1) = con v {± l e 1 , ± e i : 2 ≤ i ≤ d } , l ∈ N , with 2 l − 1 inte rior lattice p oint s. It is also conjec- tured that these cross-p olytop es ha v e m in imal v olume among all 0-symmetric lattice p olytop es with a give n num b er of in terior lattice p oints. The maximal v olume of those p olytop es is k n o w n b y the w ork of Blic hfeldt and v an der Cor- put (cf. [9, p. 51]) and, f or in stance, th e maximum is attained b y the b o xes Q d (2 l − 1) = {| x 1 | ≤ l , | x i | ≤ 1 , 2 ≤ i ≤ d } with 2 l − 1 interior p oint s. By the Examples 1.6 and 1.7 we can easily calculate the Ehrh art series of these b o xes. Example 1.8. L et l ∈ N . Then, for 0 ≤ i ≤ d , a i ( Q d (2 l − 1)) = (2 l i + 1) a i ( C d − 1 ) + (2 l ( d − i + 1) − 1) a i − 1 ( C d − 1 ) . It is quite tempting to conjecture th at the b o x Q d (2 l − 1) maximizes a i ( P ) + a d − i ( P ) for 0-symmetric p olytop e with 2 l − 1 in terior lattice p oin ts. In the 2-dimensional case this follo w s easily from a result of P aul Scott [16] which implies th at a 1 ( P ) ≤ 6 l = a 1 ( Q 2 (2 l − 1)) for an y 0-symmetric conv ex lattice p olygon with 2 l − 1 in terior lattice p oints. In fact, the result of S cott wa s recen tly generalized b y Jaron T r eutlein [19] to all degree 2 p olytop es. Theorem 1.9 (T reutlein) . L et P ∈ P d of de g r e e 2 and let a i = a i ( P ) . Th en (1.8) a 1 ≤ ( 7 , if a 2 = 1 , 3 a 2 + 3 , if a 2 ≥ 2 . In Section 3 w e will show that these conditions indeed classify all h ⋆ -p olynomials of degree 2. Prop osition 1.10. L et f ( z ) = a 2 z 2 + a 1 z + 1 , a i ∈ N , satisfying the ine qualities in (1.8) . Then f is the h ⋆ -p olynomial of a lattic e p olytop e. 6 MAR TIN HENK A ND MAKOTO T AG AMI Concerning lo wer b ou n ds on the coefficients g i ( P ) for 0-symmetric p olytop es P w e only know, except the trivial case i = d , a lo w er b ou n d on g d − 1 ( P ) (cf. (1.5 )). Namely g d − 1 ( P ) ≥ g d − 1 ( C ⋆ d ) = 2 d − 1 ( d − 1)! , where C ⋆ d = con v {± e i : 1 ≤ i ≤ d } denotes the regular cross-p olytop e. This follo ws immediately from a result of Ric hard P . Stanley [18, T heorem 3.1] on the h -v ector of ”sym m etric” Cohen-Macaula y simp licial complex. Motiv ated by a problem in [12] w e study in the last section also the related question to b ound the surface area F( P ) of a lattice p olytop e P . In con trast to the g i ( P )’s the surface area is not in v arian t under u nimo dular transformations. In ord er to d escrib e our r esult we denote by T d the s tand ard simplex T d = con v { 0 , e 1 , . . . , e d } . Prop osition 1.11. L et P ∈ P d . Then F( P ) ≥    F( C ⋆ d ) = 2 d d ! d 3 2 , if P = − P , F( T d ) = d + √ d ( d − 1)! , otherw ise . The pap er is organized as follo ws. In the next section we giv e the pro of of our main Th eorem 1.1. Then, in S ection 3, we prov e the Lemmas 1.3 and 1.5 and s h o w how the Ehrhart s eries in the Examples 1.4 and 1.6 can b e d educed. Moreo ver, w e will giv e th e pro of of Pr op osition 1.10. Finally , in the last section w e provide a p ro of of Prop osition 1.11 whic h in the symmetric cases is based on a isop erimetric inequalit y for cross-p olytop es (cf. Lemma 4.1). 2. Lower b ounds on g i ( P ) In the follo wing w e denote for an in tege r r and a p olynomial f ( x ) th e r -th co efficien t of f ( x ), i.e. the coefficient of x r , b y f ( x ) | r . Before pro ving Theorem 1.1 w e need some b asic prop erties of the n um b ers C d r,i and M r,d defined in the in tro duction (see (1.3) ). W e b egin with some sp ecial cases. Prop osition 2.1. L et d ≥ 3 . Th en M 0 ,d = 0 , M d,d = 1 and i) M 1 ,d = C d 1 ,d − 2 = − ( d − 2)! , ii) M 2 ,d = C d 2 ,d − 2 = ( d − 2)! + ( − 1) d stirl( d − 1 , 2) , iii) M d − 1 ,d = C d d − 1 , 1 = − d ( d − 3) 2 , iv) M d − 2 ,d =    C d d − 2 , d − 1 2 = − 1 4  d +1 3  , if d o dd , C d d − 2 , d 2 = − 1 4  d 3  , if d even . Pr o of. The cases M 0 ,d and M d,d are trivial. Since C d r,l is the ( d − r )-th elemen tary symmetric function of { l, l − 1 , . . . , l − ( d − 1) } w e ha ve C d 1 ,i = ( − 1) d − i − 1 i ! ( d − LOW ER BOUNDS ON THE COEFFICIENTS OF EHRHAR T POL YNOMIALS 7 i − 1)! and M 1 ,d = min { C d 1 ,i : 1 ≤ i ≤ d − 2 } = C d 1 ,d − 2 = − ( d − 2)! In th e case r = 2 we obtain b y element ary calculations that C d 2 ,i = i ! stirl( d − i, 2) + ( − 1) d ( d − i − 1)! stirl( i + 1 , 2) = i ! ( d − i − 1)! ( − 1) d − i d − i − 1 X k =1 1 k − i X k =1 1 k ! , from w hic h we conclude M 2 ,d = C d 2 ,d − 2 = ( d − 2)! + ( − 1) d stirl( d − 1 , 2). F or iii) we note that C d d − 1 ,i = i X j = i − ( d − 1) j = − d 2 ( d − 1 − 2 i ) , and s o M d − 1 ,d = C d d − 1 , 1 . Finally , for th e v alue of M d − 2 ,d w e first observe that C d d − 2 ,i − C d d − 2 ,i − 1 = ( z + i ) ( z + i − 1) · . . . · ( z + i − ( d − 1))   d − 2 − ( z + i − 1) · . . . ( z + i − ( d − 1)) ( z + i − d )   d − 2 = i − 1 X j = − d + i +1 j ( i − ( − d + i )) = d i − 1 X j = − d + i +1 j = d ( d − 1)( − d + 2 i ) 2 . Th us the fu nction C d d − 2 ,i is decreasing in 0 ≤ i ≤ ⌊ d/ 2 ⌋ and increasing in ⌊ d/ 2 ⌋ ≤ i ≤ d . So it take s its minimum at i = ⌊ d/ 2 ⌋ . First let u s assume that d is o dd . Then M d − 2 ,d = C d d − 2 , d − 1 2 = d !  z + ( d − 1) / 2 d       d − 2 = z ( z 2 − 1) ( z 2 − 4) · . . . · ( z 2 − (( d − 1) / 2) 2 )     d − 2 = − ( d − 1) / 2 X i =0 i 2 = − 1 4  d + 1 3  . The even case can b e treated similarly .  In addition to the previous p rop osition we also need Lemma 2.2. i) C d r,i = ( − 1) d − r C d r,d − 1 − i for 0 ≤ i ≤ d − 1 . ii) L et d ≥ 3 . Then M r,d ≤ 0 for 1 ≤ r ≤ d − 1 , and M r,d = 0 only in the c ase d = 3 and r = 2 . Pr o of. The fir st statemen t is just a consequen ce of the fact that C d r,l is th e ( d − r )-th elemen tary symmetric f unction of { l, l − 1 , . . . , l − ( d − 1) } . F or ii) w e first observ e th at the case d = 3 follo ws dir ectly from Prop osition 2.1. Hence it 8 MAR TIN HENK A ND MAKOTO T AG AMI remains to sho w that M r,d < 0 for d ≥ 4 and 1 ≤ r ≤ d − 1. On accoun t of i) it suffices to pro v e this when d − r is ev en and w e will pro ceed b y ind uction on d . The case d = 4 is cov ered by Pr op osition 2.1. So let d ≥ 5. By Prop osition 2.1 i) we also ma y assu me r ≥ 2. It is easy to see that (2.1) C d r,i = ( i − d + 1) C d − 1 r,i + C d − 1 r − 1 ,i , and by induction w e ma y assum e that there exists a j ∈ { 1 , . . . , d − 3 } with C d − 1 r − 1 ,j < 0. Observe that d − 1 − ( r − 1) is ev en. If C d − 1 r,j ≥ 0 we obtain b y (2.1) that C d r,j < 0 and we are d one. So let C d − 1 r,j < 0. By part i) w e know that C d − 1 r,j = ( − 1) d − 1 − r C d − 1 r,d − 2 − j and C d − 1 r − 1 ,j = ( − 1) d − r C d − 1 r − 1 ,d − 2 − j . Since d − r is ev en w e conclude C d − 1 r,d − 2 − j > 0 and C d − 1 r − 1 ,d − 2 − j < 0. Hence, on accoun t of (2.1) we get C d r,d − 2 − j < 0 and so M r,d < 0.  No w w e are able to give the pro of of our m ain Theorem. Pr o of of The or em 1.1. W e follo w th e ap p roac h of Betk e and McMullen used in [4, Th eorem 6]. By expandin g the E h rhart series at z = 0 one gets (see e.g. [3, Lemma 3.14]) (2.2) G P ( z ) = d X i =0 a i ( P )  z + d − i d  . In p articular, we ha v e (2.3) 1 d ! d X i =0 a i ( P ) = g d ( P ) = vol( P ) . F or short, w e will write a i instead of a i ( P ) and g i instead of g i ( P ). With this notation w e ha v e d ! g r = d ! G P ( z ) | r = d ! d X i =0 a i  z + d − i d       r = C d r,d + (a 1 C d r,d − 1 + a d C d r, 0 ) + d − 1 X i =2 a i C d r,d − i . (2.4) Since C d r,d − 1 ≥ 0 we get with Lemma 2.2 i) that C d r,d − 1 = | C d r, 0 | . T oget her with a 1 = G( P ) − ( d + 1) ≥ G(in t( P )) = a d and C d r,d = ( − 1) d − r stirl( d + 1 , r + 1) w e find d ! g r ≥ ( − 1) d − r stirl( d + 1 , r + 1) + d − 1 X i =2 a i C d r,d − i = ( − 1) d − r stirl( d + 1 , r + 1) + d − 1 X i =2 a i  C d r,d − i − M r,d  + d X i =1 a i M r,d − (a 1 + a d ) M r,d ≥ ( − 1) d − r stirl( d + 1 , r + 1) + ( d ! v ol( P ) − 1) M r,d , (2.5) LOW ER BOUNDS ON THE COEFFICIENTS OF EHRHAR T POL YNOMIALS 9 where the last inequalit y follo ws from the definition of M r,d and the n on - p ositivit y of M r,d (cf. Pr op osition 2.1 and L emm a 2.2 ii)).  W e remark that for d ≥ 3, r ∈ { 1 , . . . , d − 1 } and ( r , d ) 6 = (2 , 3) w e can sligh tly improv e the inequalities in T h eorem 1.1, b ecause in these cases we ha ve M r,d < 0 (cf. Lemma 2.2 ii)), and since C d r,d − 1 is th e ( d − r )-th elemen tary symmetric function of { 0 , . . . , d − 1 } w e also kno w C d r,d − 1 > 0 for 1 ≤ r ≤ d − 1. Hence we get (cf. (2.4) and (2.5)) d ! g r = C d r,d + d X i =1 a i C d r,d − i = C d r,d + a 1  C d r,d − 1 − M r,d  + d X i =2  C d r,d − i − M r,d  + d X i =1 a i M r,d ≥ ( − 1) d − r stirl( d + 1 , r + 1) + 2 a 1 ( P ) + ( d ! vo l( P ) − 1) M r,d = ( − 1) d − r stirl( d + 1 , r + 1) − 2( d + 1) + 2G( P ) + ( d ! vol( P ) − 1) M r,d . Corollary 1.2 is an immediate consequence of Theorem 1.1 and Prop osition 2.1. Pr o of of Cor ol lary 1.2. Th e inequ alities just f ollo w b y in serting the v alue of M r,d giv en in Prop osition 2.1 in th e general inequalit y of Theorem 1.1. Here w e also hav e used the iden tities stirl( d + 1 , 2) = ( − 1) d +1 d ! d X i =1 1 i and stirl( d + 1 , d − 1) = 3 d + 2 4  d + 1 3  . It remains to show that the in equalities are b est p ossible for any vo lume. F or r = d − 2 we consid er the simp lex T ( m ) d (cf. (1.7)) with a 0 ( T ( m ) d ) = 1, a ⌈ d/ 2 ⌉ ( T ( m ) d ) = ( m − 1) and a i ( T ( m ) d ) = 0 for i / ∈ { 0 , ⌈ d/ 2 ⌉} . Then vo l( T ( m ) d ) = m/d ! and on accoun t of Prop osition 2.1 we ha v e equ alit y in (2.4) and (2.5). F or r = 1 , 2 and d ≥ 4 we consid er the ( d − 4)-fol d pyramid ˜ T ( m ) d o ver T ( m ) 4 giv en by ˜ T ( m ) d = con v { T ( m ) 4 , e 5 , . . . , e d } . Th en v ol( ˜ T ( m ) d ) = m/d ! and in view of (1.7) and [3, Theorem 2.4] w e obtain a 0 ( ˜ T ( m ) d ) = 1 , a 2 ( ˜ T ( m ) d ) = m − 1 and a i ( ˜ T ( m ) d ) = 0 , i / ∈ { 0 , 2 } . Again, by Prop osition 2.1 we ha ve equalit y in (2.4) and (2.5).  3. Ehrhar t se ries of so me special pol yt opes W e start w ith the short pro of of L emma 1.3. Pr o of of L emma 1.3. Since Ehr P ( z ) Eh r Q ( z ) = X k ≥ 0 X m + l = k G P ( m )G Q ( l ) ! z k , 10 MAR TIN HENK A ND MAKOTO T AG AMI it suffices to pro v e that the Ehrhart p olynomial G P ⋆Q ( k ) of the lattice p olytop e P ⋆ Q ∈ P p + q +1 is giv en b y G P ⋆Q ( k ) = X m + l = k G P ( m )G Q ( l ) . This, h o wev er, follo ws im m ediately f rom the definition since k ( P ⋆ Q ) = { λ ( x, o q , 0) ⊺ + ( k − λ ) ( o p , y , 1) ⊺ : x ∈ P , y ∈ Q, 0 ≤ λ ≤ k } .  Example 1.4 in the in tro d uction sho w s an application of this construction. F or Example 1.6 we need Lemma 1.5. Pr o of of L emma 1.5. With w = z 1 k w e may write 1 k k − 1 X i =0 Ehr P ( ζ i w ) = 1 k k − 1 X i =0 X m ≥ 0 G P ( m )( ζ i w ) m = 1 k X m ≥ 0 G P ( m ) w m k − 1 X i =0 ζ i m . Since ζ is a k -th ro ot of unity th e sum P k − 1 i =0 ζ i m is equal to k if m is a multiple of k and otherwise it is 0. T h us we obtain 1 k k − 1 X i =0 Ehr P ( ζ i w ) = X m ≥ 0 G P ( m k ) w m k = X m ≥ 0 G k P ( m ) z m = Ehr k P ( z ) .  As an application of Lemma 1.5 we calculate the Ehrhart series of the cub e C d (cf. Example 1.6). Instead of C d w e consider the translated cub e 2 ˜ C d , where ˜ C d = { x ∈ R d : 0 ≤ x i ≤ 1 , 1 ≤ i ≤ d } . In [3, Th eorem 2.1] it was shown that a i ( ˜ C d ) = A ( d, i + 1) where A ( d, i ) d enotes the Eulerian num b ers. Setting w = √ z Lemma 1.5 leads to Ehr C d ( z ) = 1 2  Ehr ˜ C d ( w ) + Ehr ˜ C d ( − w )  = 1 2 P d i =1 A ( d, i ) w i − 1 (1 − w ) d +1 + P d i =1 A ( d, i ) ( − w ) i − 1 (1 + w ) d +1 ! = 1 2 1 (1 − z ) d +1 d X i =1 A ( d, i ) w i − 1 (1 + w ) d +1 + d X i =1 A ( d, i ) ( − w ) i − 1 (1 − w ) d +1 ! = 1 (1 − z ) d +1   d X i =1 A ( d, i ) d +1 X j = 0 , i + j − 1 ev en  d + 1 j  w i + j − 1   LOW ER BOUNDS ON THE COEFFICIENTS OF EHRHAR T POL YNOMIALS 11 Substituting 2 l = i + j − 1 giv es Ehr C d ( z ) = 1 (1 − z ) d +1 d X l =0 2 l +1 X i =2 l − d  d + 1 2 l + 1 − i  A ( d, i ) w 2 l ! = 1 (1 − z ) d +1   d X l =0 z l d +1 X j = 0  d + 1 j  A ( d, 2 l + 1 − j )   , whic h explains th e formula in Example 1.6 . In order to calculat e in general the Ehrhart series of the prism P = { ( x, x d ) ⊺ : x ∈ Q, x d ∈ [0 , m ] } where Q ∈ P d − 1 , m ∈ N (cf. Example 1.7), we use the differen tial op erator T d efined by z d dz . Considered as an op erator on the ring of formal p o wer series we ha v e (cf. e.g. [3, p. 28]) (3.1) X k ≥ 0 f ( k ) z k = f ( T ) 1 1 − z for any p olynomial f . S ince G P ( k ) = ( m k + 1) G Q ( k ) we dedu ce from (3.1) Ehr P ( z ) = ( m T + 1)Ehr Q ( z ) = mz d dz Ehr Q ( z ) + Ehr Q ( z ) . Th us Ehr P ( z ) = m z P d − 1 i =0 i a i ( Q ) z i − 1 (1 − z ) + P d − 1 i =0 d a i ( Q ) z i (1 − z ) d +1 + P d − 1 i =0 a i ( Q ) z i (1 − z ) d = P d − 1 i =0 ( m i + 1)a i ( Q ) z i (1 − z ) + P d − 1 i =0 m d a i ( Q ) z i +1 (1 − z ) d +1 = 1 (1 − z ) d +1 d X i =1 (( m i + 1)a i ( Q ) + ( m ( d − i + 1) − 1) a i − 1 ( Q )) z i , whic h is the formula in Example 1.7. Finally , we come to the classification of h ⋆ -p olynomials of degree 2. Pr o of of Pr op osition 1.10. W e recall that a 1 ( P ) = G( P ) − ( d + 1) and a d ( P ) = G(in t( P )) for P ∈ P d . In the case a 2 = 1, a 1 = 7 the triangle con v { 0 , 3 e 1 , 3 e 2 } has the d esired h ⋆ -p olynomial. Next we distinguish t w o cases: i) a 2 < a 1 ≤ 3 a 2 + 3. F or in tegers k, l, m with 0 ≤ l, k ≤ m + 1 let P ∈ P 2 giv en by P = con v { 0 , l e 1 , e 2 + ( m + 1) e 1 , 2 e 2 , 2 e 2 + k e 1 } . Then it is easy to see that a 2 ( P ) = m and P has k + l + 4 lattice p oin ts on the b ound ary . Thus a 1 ( P ) = k + l + m + 1. ii) a 1 ≤ a 2 . F or inte gers l , m with 0 ≤ l ≤ m let P ∈ P 3 giv en b y P = con v { 0 , e 1 , e 2 , − l e 3 , e 1 + e 2 + ( m + 1) e 3 } . The only lattice p oin ts con tained in P are the v ertices and the lattice p oint s on the edge con v { 0 , − l e 3 } . Thus a 3 ( P ) = 0 and a 1 ( P ) = l . On the other hand , since ( l + m + 1) / 6 = vol( P ) = ( P 3 i =0 a i ( P )) / 6 (cf. (2.3)) it is a 2 ( P ) = m .  12 MAR TIN HENK A ND MAKOTO T AG AMI 4. 0 -symmetric l a ttice pol ytopes In order to study the surface area of 0-symmetric p olytop es we first pr ov e an isop erimetric inequalit y for the class of cross-p olytop es. Lemma 4.1. L et v 1 , . . . , v d ∈ R d b e line arly indep endent and let C = conv {± v i : 1 ≤ i ≤ d } . Then F( C ) d v ol( C ) d − 1 ≥ 2 d d ! d 3 2 d , and e quality holds if and only if C is a r e gular cr oss-p olytop e, i.e., v 1 , . . . , v d form an ortho gonal b asis of e qual length. Pr o of. Without loss of generalit y let vo l( C ) = 2 d /d !. Then we ha ve to sh ow (4.1) F( C ) ≥ 2 d d ! d 3 2 . By standard argumen ts fr om con v exity (see e.g. [10, Theorem 6.3] ) the set of all 0-symmetric cross-p olytop es with volume 2 d /d ! con tains a cross-p olytop e C ⋆ = con v {± w 1 , . . . , ± w d } , say , of min imal s urface area. Sup p ose th at some of the vec tors are not p airwise orthogonal, for instance, w 1 and w 2 . Then w e apply to C ⋆ a Steiner-Symmetrization (cf. e.g. [10, pp . 169]) with resp ect to the hyp erplane H = { x ∈ R d : w i x = 0 } . It is easy to c hec k that the Steiner-symmetral of C ⋆ is again a cross-p olytop e ˜ C ∗ , say , with v ol( ˜ C ⋆ ) = v ol( C ⋆ ) (cf. [10, Prop osition 9.1]). Since C ⋆ w as not symmetric with resp ect to the hyp erplane H w e also know that F( ˜ C ∗ ) < F( C ⋆ ) w hic h cont radicts the minimalit y of C ⋆ (cf. [10 , p. 171]). So we can assume that the vect ors w i are pairwise orthogonal. Next supp ose that k w 1 k > k w 2 k , wh ere k · k denotes the Euclidean norm. Then w e app ly Steiner-Symmetrization w ith resp ect to the hyp erplane H whic h is orthogonal to w 1 − w 2 and bisecting the edge conv { w 1 , w 2 } . As b efore we get a con tradiction to the min imalit y of C ⋆ . Th us we know th at w i are p airwise orthogonal and of same length. By our assumption on the v olume we get k w i k = 1, 1 ≤ i ≤ d , and it is easy to calculate that F( C ⋆ ) = (2 d /d !) d 3 / 2 . S o we ha v e F( C ) ≥ F( C ⋆ ) = 2 d d ! d 3 2 , and by the foregoing argumenta tion via Steiner-Sym metrizatio ns we also see that equalit y holds if and only C is a regular cross-p olytop e generated by ve ctors of unit-length.  The d etermination of the minimal surface area of 0-symmetric lattice p oly- top es is an immediate consequence of the lemma ab o v e, whereas the n on -sym - metric case do es not follo w from the corr esp onding isop erimetric inequalit y for simplices. Pr o of of Pr op osition 1.11. Let P ∈ P d with P = − P . T hen P con tains a 0- symmetric lattice cross-p olytop e C = conv {± v i : 1 ≤ i ≤ d } , sa y , and b y the LOW ER BOUNDS ON THE COEFFICIENTS OF EHRHAR T POL YNOMIALS 13 monotonicit y of th e su rface area and L emma 4.1 we get (4.2) F( P ) ≥ F( C ) ≥  2 d d !  1 d d 3 2 v ol( C ) d − 1 d . Since v i ∈ Z d , 1 ≤ i ≤ d , we ha ve v ol( C ) = (2 d /d !) | det( v 1 , . . . , v d ) | ≥ 2 d /d !, whic h sh o ws by (4.2) the 0-symmetric case. In the non-symmetric case we know that P conta ins a latt ice simplex T = { x ∈ R d : a i x ≤ b i , 1 ≤ i ≤ d + 1 } , sa y . Here w e ma y assume that a i ∈ Z n are primitiv e, i.e., con v { 0 , a i } ∩ Z n = { 0 , a i } , and that b i ∈ Z . F urth ermore, w e den ote the facet P ∩ { x ∈ R d : a i x = b i } by F i , 1 ≤ i ≤ d + 1. With these notations w e ha ve det(aff F i ∩ Z n ) = k a i k (cf. [14, Prop osition 1.2.9] ). Hence there exist intege rs k i ≥ 1 w ith (4.3) v ol d − 1 ( F i ) = k i k a i k ( d − 1)! , and s o we ma y wr ite F( P ) ≥ F( T ) = d +1 X i =1 v ol d − 1 ( F i ) ≥ 1 ( d − 1)! d +1 X i =1 k a i k . W e also ha v e P d +1 i =1 v ol d − 1 ( F i ) a i / k a i k = 0 (cf. e.g. [10 , T heorem 18.2]) and in view of (4.3 ) we obtain P d +1 i =1 k i a i = 0. Thus, since the d + 1 lattice v ectors a i are affin ely ind ep endent w e can find f or eac h index j ∈ { 1 , . . . , d } at least tw o v ecto rs a i 1 and a i 2 ha ving a non-trivial j -th co ordinate. Hence (4.4) d +1 X i =1 k a i k 2 ≥ 2 d. T ogether with the restrictions k a i k ≥ 1, 1 ≤ i ≤ d + 1, it is easy to argue that P d +1 i =1 k a i k is minimized if and only if d norms k a i k are equal to 1 and one is equal to √ d . F or instance, the in tersectio n of the cone { x ∈ R d +1 : x i ≥ 1 , 1 ≤ i ≤ d + 1 } with the h yp erplane H α = { x ∈ R d +1 : P d +1 i =1 x i = α } , α ≥ d + 1, is th e d -simplex T ( α ) with v ertices giv en by the p erm utations of the v ector (1 , . . . , 1 , α − d ) ⊺ of length p d + ( α − d ) 2 . Therefore, a vertex of that simp lex is con tained in { x ∈ R d +1 : P d +1 i =1 x 2 i ≥ 2 d } if α ≥ d + √ d . In other w ords, we alw ays ha v e d +1 X i =1 k a i k ≥ d + √ d, whic h give s the desired inequalit y in th e n on-symmetric case (cf. (4.3)).  W e remark that the pro of also sh o ws that equalit y in Prop osition 1.11 holds if and only if P is the o -symmetric cross-p olytop e C ⋆ d or the simplex T d (up to lattice translations). A cknow le dgement. The authors w ould like to thank Matthias Bec k, Benjamin Braun, Chr istian Haase and th e anon ymous referee for v aluable commen ts and suggestions. 14 MAR TIN HENK A ND MAKOTO T AG AMI Referen ces [1] V. V. Batyrev, L attic e p olytop es with a given h ∗ -p olynomial , Contemporary Mathematics, no. 423, AMS, 2007, pp . 1–10 . [2] M. Beck, J. De Loera, M. Develin, J. Pfeifle, and R.P . St an ley , Co efficients and r o ots of Ehrhart p olynomials , Contemp. Math. 374 (2005), 15–36. [3] M. Bec k and S. R obins, Computing the c ontinuous discr etely: Inte ger-p oint enumer ation in p olyhe dr a , Springer, 2007. [4] U. Betke and P . McMullen, L attic e p oints in lattic e p olytop es , Monatsh. Math. 99 (1985), no. 4, 253–265. [5] Ch. Bey , M. Henk, and J.M. Wills, Notes on the r o ots of Ehrhart p ol ynomials , Discrete Comput. Geom. 38 (2007), 81–98. [6] B. Braun, A n Ehrhart series f ormula for r eflexive p ol ytop es , Electron. J. Combinatorics 13 (2006), no. 1, Note 15. [7] E. Ehrhart, Sur les p oly` edr es r ationnels homoth ´ etiques ` a n di mensions , C. R. Acad. Sci., P aris, S´ er. A 254 (1962), 616–618. [8] , Sur un pr obl ` eme de g´ eom´ etrie di ophantienne lin´ eair e , J. R eine Angew. Math. 227 (1967), 25–49. [9] P . M. Gruber and C. G. Lekkerkerk er, Ge ometry of numb ers , second ed., vol. 37, North- Holland Publishing Co., Amsterd am, 198 7. [10] P .M. Grub er, Convex and di scr ete ge ometry , Grund lehren der mathematischen Wis- sensc haften, vol. 336, Springer-V erlag Berlin Heidelberg, 2007. [11] M. Henk and J. Rich ter-Gebert and G.M. Z iegler, Basic pr op erties of c onvex p olytop es , chapter 16 of the second edition of th e “CR C Handb o ok of Discrete and Computational Geometry”, edited by J.E. Goo dman and J. O’Rourke. , 2004, 355–382. [12] M. Henk and J.M. Wills, A Blichfeldt-typ e ine quality for the surfac e ar e a , to app ear in Mh. Math. (2007), Preprint a v ailable at http://arxiv.org /abs/0705. 2088 . [13] T. Hibi, A l ower b ound the or em for Ehrhart p olynomials of c onvex p olytop es , Adv . Math. 105 (1994), no. 2, 162–165. [14] J. Martinet, Perfe ct lattic es in Euclide an sp ac es , Grundlehren d er Mathematischen Wis- sensc haften [F u ndamental Principles of Mathematical S ciences], vol. 327, Sp ringer-V erlag, Berlin, 2003. [15] B. Nill, L attic e p olytop es having h ∗ -p olynomials wi th given de gr e e and line ar c o efficient , (2007), Preprint a v ailable at http://arxiv. org/abs/07 05.1082 . [16] P . R. Scott, On c onvex lattic e p olygons , Bull. A u stral. Math. Soc. 15 (1976), no. 3, 395– 399. [17] R.P . Stanley , De c omp ositions of r ational c onvex p olytop es , Ann. D iscrete Math. 6 (1980), 333–342 . [18] , On the numb er of fac es of c entr al ly-symmetric si mplicial p olytop es , Graphs and Com binatorics 3 (1987), 55–66. [19] J. T reutlein, L attic e p olytop es of de gr e e 2 , (2007), Preprint av ailable at http://arx iv.org/abs /0706.4178 . Mar tin H enk, Universit ¨ at Magdebur g, Institut f ¨ ur Algebra und Geometrie, Universit ¨ atspla tz 2, D-39106 Magdeburg, Ge rmany E-mail addr ess : henk@m ath.uni-ma gdeburg.de Mako to T a gami, Un iversit ¨ at Magdeburg, Institut f ¨ ur Algebra und Geometrie, Universit ¨ atspla tz 2, D-39106 Magdeburg, Ge rmany E-mail addr ess : tagami @kenroku.k anazawa-u. ac.jp