Journal of Al-Nahrain University Vol.11(1), April, 2008, pp Science 3 OF THE EHRHART POLYNOMIALS OF A POLYHEDRON IN

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1 ournl of Al-Nhrin Univrsity Vol( April 008 pp05-9 Scinc ON FINDING TH COFFICINTS c OF TH HRHART POLYNOMIALS OF A POLYHDRON IN Shth Ass Al-Nr Mnl N Al-Hrr n Vin A Al Al-Slhy Dprtnt of Appli Scinc Univrsity of Tchnology Bgh Irq Abstrct Coputing th volu n intgrl points of polyhron in is vry iportnt subct in iffrnt rs of thtics such s: nubr thory toric Hilbrt functions Kostnt's prtition function in rprsnttion thory hrhrt polynoil in cobintorics cryptogrphy intgr progring sttisticl contingncy n ss spctroscop nlysis Thrfor tho for fining th cofficints of this polynoil r to b list A progr in visul bsic lngug is for fining th gnrl iffrntition of th function tht us for fining th cofficints of th hrhrt polynoil which illustrt by flow chrt in Fig( Introuction Th hrhrt polynoil of convx lttic polytop counts th nubr of intgrl points in n intgrl ilt of th polytop(vry boun polyhron is si to b polytop hrhrt prov tht th function which counts th nubr of lttic points tht li insi th ilt polytop tp( Lt P b n intgrl polytop for positiv intgr t lt tp={tx : XP} not th ilt polytop P is polynoil in t n it is not by L(Pt which is th crinl of (tp whr is th intgr lttic in Fro th finitions of th hrhrt polynoil th ling cofficint is th volu of th polytop n th constnt tr is on; ths r tr s th trivil cofficint of th hrhrt polynoil th othr cofficints r nontrivil [] In this wor w prsnt tho for coputing th cofficints of th hrhrt polynoil tht pns on th concpts of Din su n rsiu thor in coplx nlysis Gnrl forul tht counts th rivtivs in th introuc tho is givn To th bst of our nowlg this tho ss to b nw Forultion of this tho Bfor w iscuss th tho th following thor whr n Thor ( []: Lt b lttic -polytop with th hrhrt polynoil L(Pt = i c Thn c is it i0 th volu of P whil th constnt tr is on which is qul to th ulr chrctristic of P Th othr cofficints of L(Pt r not sily ccssibl In fct tho of coputing ths cofficints ws unnown until quit rcntly [] [] n [] Counting intgrl points using Din sus In this sction w scrib th rltion btwn th Din su n th hrhrt polynoil of polytop n iscuss thor tht counts th nubr of intgrl points in polytop Rcll tht th Din su of two rltivly pri positiv intgrs n b not by S(b is fin s whr S (b (( x b i x 0 i i (( (( b b x x is th grtst intgr st of intgr nubrs if if x x x n is th 05

2 Shth Ass Al-Nr Rr (: Th iscrt Fourir xpnsions cn b us to rwrit th Din su in trs of th Din cotngnt su tht is for two rltivly pri positiv intgrs n b: b S (b cot( cot( b b b whr S(b is th Din su of n b [5 p 7] Counting intgrl points using th rsiu thor This sction is concrn with tho givn in [] to count th intgrl points of givn polytop by ns of th rsiu thor Thor ( []: Lt P b polytop fin s P (x x : x n x 0 ( with vrtics ( 00 0 (0 0 ( 0 00(00 whr r positiv intgrs A A â (whr â ns th fctor is oitt n A nots â â f t ( n Ω r fin s ta A A A f t ( ( n L(P t ( A { C \ {}: R s(f t ( ( } thn ( R s(f t ( Th hrhrt cofficints In this sction so tils for riving forul of hrhrt cofficints r givn For ch cofficint of th hrhrt polynoil t c0 L(Pt ct c forul for fining ths cofficints cn b riv with sll oifiction of f t ( Consir th function g g ( ( ( ( 0 ( ( ta A A A 0 ( ( ta( ( ( A A A ( ( If ( 0 is insrt in th nurtor of th bov qution w gt ta( ( 0 g ( A A ( ( ( g ( 0 0 ( A ( ( A ( A 0 ta( ( ( ( A ( ft( ( Rcll tht L(P t R s(f t( 0 using this rltion w obtin R s(g ( 0 R s( ( ( 0 Rs(f 0 t( ( (L(P( t ( (L(P ( t 0 ( f t( ( 0 ( ( ( (L(P ( t ( 0 g ( Th following l is n to riv th forul of th cofficints of th hrhrt polynoil L ( [] Suppos tht L(Pt R s(g c t c ( 0! t c0 thn for S ( c t whr S ( nots th Stirling nubr (nubr 0

3 ournl of Al-Nhrin Univrsity Vol( April 008 pp05-9 Scinc of prtition of n -st into -blocs of th scon in of n n c 0 = Thor ( [] : Lt P b lttic -polytop givn by xprssion ( with th hrhrt polynoil L(P t c t c t c0 thn for S ( c t (R s(g (! R s(g ( whr { C \ {}: A } Corollry ( []: For > 0 c is th cofficint of (R s(g( R s(g(! t in Thor ( []: Lt b lttic -polytop with vrtics ( 00 0 ( 0 0(00 whr r pirwis rltivly pri intgrs Th first nontrivil hrhrt cofficints c is givn by c (C S(A S(A (! whr S ( b nots th Din su n C ( A A A A ( A Coputing c of Th hrhrt Polynoil using VBP: As sn bfor th ling cofficint of th hrhrt polynoil rprsnts th volu of th polytop th scon cofficint rprsnts hlf of th surfc r of th polytop n th constnt tr is on whil th othr cofficints r unnown In this sction w fin th non trivil cofficints c for th polytop with whr P is rprsnt by list of vrtics ( 00 0 ( ( (00 0 such tht r pirwis rltivly pri positiv intgrs Thor (5: Lt P not th polytop in ( with vrtics ( 000( 00(00 whr r pirwis rltivly pri positiv intgrs Thn c is givn by c D (! S(A S(A C Whr S(b is th Din su of n b D (B B B B! (A C (A (A ( (A (A!! ( (0! (tb (tb!! (A (A!! A A â th fctor!! ( â ns is oitt B n B â Proof: By corollry ( if w fin g ( g ( ( ( ( ta s A A A ( ( WhrA A â n â ns tht th fctor is oitt thn th pols of th function g ( r t = 0 n th roots of unity W fin th rsius of th function g ( t ths pols Sinc r pirwis rltivly pri thrfor g ( hs sipl pols t -th roots of unity Lt 07

4 Shth Ass Al-Nr n sinc A A A thrfor g ( ( Now t 0 ( ( t( n 0 Thrfor A chng of vribls ( ( xp log is whr suitbl brnch of logrith such tht thus xp log( Rs(g( A ( ( Whrn ( Rs ( tb B ( B B B â Sinc Rs(f ( Rs( f ( 0 thn ( Rs ( tb ( Rs B ( Lt = tb thn ( Rs( B ( B B ( tb ( B tb ( B 0 0 ( = Rs( 0 B B ( ( By writing th Mclurin sris for xponntil function on cn gt ( ( (!! Rs (B (B B B!! 0 Aftr sipl coputtions th bov rsiu cn b writtn s ( ( (!! R s ( B ( B (B (B (B (B!!!! 0 Lt I!!! A A A!!! B B B!!! B B B!!! Thn B B B!!! ( Rs 0 B ( B ( ( R s (I 0 ( B ( B ( ( B ( B For th function (I w hv pol of orr two t ro n Lt ( I ( ( B ( B Aftr sipl coputtions on w gt t By th forul for fining th B rsius if w consir ( f ( thn Rs(f ( (0 0 whr! ( ( I I Lt K I K I 08

5 ournl of Al-Nhrin Univrsity Vol( April 008 pp05-9 Scinc K I thrfor ( ( K K K t = 0 w coput (0 ftr sipl coputtions w gt ( ( 0 (B B B!! thrfor (B B B t! R s(f ( 0 B (! Lt D (B B B B! Thrfor D R s(g ( t A ( ( ll th -th root of unity r up to gt Dt Rs(g( A ( ( Lt b priitiv thrfor thn Dt Dt A ( ( ( ( A A th roots of unity A ( ( ( ( A A A A A A A A A A ( A A A A A A Now sinc thn by using th forul for fining th roots in th coplx pln n r i = 0 W obtin = ( ( A A i i A A A ( i A ( i ( i ( i i cos( But cot( i sin( i A hnc ( i ( i A ( i ( i n = cot i A ( i A ( i A cot A A 09

6 Shth Ass Al-Nr thrfor A cot cot A cot cot ( A A A A ( i A cot cot A cot cot Th iginry trs isppr n thn th bov qution cn b writtn s A cot cot S(A whr S(A is th Din su of A n Hnc R s(g( Dt ( S(A Siilr xprssions r obtin for th rsius t th othr roots of unity Now w fin th rsiu t g ( t = w hv Rs(g ( Rs( g( 0 thn Rs(g ( ta ( R s A ( A ( A ( ( 0 By writing th Mclurin sris for xponntil function w gt ( ( (!! R s (A (A A A!!! 0 whr ta thn th bov rsiu bcos ( (A (A ( (!! R s (A (A (A (A!!!! 0!! th function for which w wnt to fin th rsiu hs pol of orr four t ro Lt ( (!! ( (A (A (A (A!!!! ( n f ( AA By th forul for fining th rsiu w gt ( (0 Rs(f ( 0! Lt n I h!!!!!! A A A!!! A A A!!! A A A!!! Thn ( I h n ( I h I lt I h I h!! h 0

7 ournl of Al-Nhrin Univrsity Vol( April 008 pp05-9 Scinc K I h K I h K I h n K I h Hnc K K K n K K K Now thrfor ( K ( K ta (ta I!! I (ta! ta (ta (ta!!! ( ta (ta!! Diffrntiting I to gt I n I thn put = 0 in th obtin xprssion to gt For I(0 I (0 I (0 ta (! ( ( ( (ta! ta! (ta! ta I (0 ( ( 5(! ta (ta (! A A A!!! A A!! A A!! A! A! ( ( ta!(ta! Diffrntit to gt n thn put = 0 in th obtin xprssions to gt A (0 A (0 (0 n (0 0!! In siilr wy w gt th othr iffrntition of n h thn (ta R s (A (A Lt ( ( A t (A (A! (A C (A (A ( 0 ( (0 (0! So by corollry ( w gt for c which is th cofficint of t of (Rs(g( (! So c Rs(g ( D S(A (! S(A C Gnrl forul for th iffrntition of I h In this sction w gt gnrl for for th iffrntition of th trs I n h tht pprs throughout th procss of fining th cofficints of th hrhrt polynoil w bgin by consiring I I h whr I ns tht only I in th xprssion I h is iffrntit tis Lt thn I I I I I (I I I ( I I (I I I (I I (5 5 ( I I (I I I (I I (I I I I I I

8 Shth Ass Al-Nr I I I ( (7 (8 5 (5 I (I I I I ( I (I I I I I I (I I (I ( I I I 7 ( 5 I (5I I 0 I 0I 5 5 (5 5 I I (0I I ( (0I 0I (5I 0I ( (I (5 5I I 0 I 0 I 0 I 0 I 0 I 5 I 5 I 5 I 0 I 8 7 (7 5 I (I I 5 I 0 I 5 ( 5 I I (5I I 5 (5 ( (0I 5I (5I 0I ( (5 (I 5I (I I ( I 0 I 5 90 I 0 I 5 I 0 I 0 I 0 I 0 I 5 I ( I 5 I I 90 I 5 I 0 I 5 I 0 I 5( I In orr to iffrntit w n to fin gnrl forul for ths iffrntitions so w wor on ths lnts n fin gnrl forul To illustrt this consir for xpl w w (!! w w w w! Th w By ssuing th iplicit iffrntition for both sis of th bov qution w gt w ( ( w n th scon rivtiv of th bov qution is w ( ( 0 whn w iffrntit w -tis w gt shp li binoil forul ( b = ( b b b! = i i b i i Thrfor whr w ( ( w ( 0 is th -th rivtiv of sinc w is constnt thrfor w ns w ris to th powr For xpl w lt h thn h h w w w w ( w w w w B w w ( w w w B h ( w w w n so on Thrfor [] I whr [] I I [] [] ( [] [] [] [] w n siilrly for highst rivtiv By rrnging th togthr w obtin I [] [] [] ( [] [] ( [] ( ( []

9 ournl of Al-Nhrin Univrsity Vol( April 008 pp05-9 Scinc (5 ( (7 (8 [5] [] ( ( [] ( [] [] ( ( ( [] [5] (5 ( [] [] ( ( ( [] ( [] [] [] ( [] [] ( [] [] [] [7] [] ( [5] [] (5 0 0 [] 5 [] [5] (5 [] (0 5 (0 ( [] ( [] (5 0 ( 5 (5 [] [] [] [] [] [] 0 5 [] [] [] 5( 0 [8] [7] (7 [] [5] ( 5 0 [] 5 [] [] [] (5 ( [5] (5 [] (0 5 (5 ( ( [] ( (5 [] ( ( ( [5] [] 0 90 [] [] [] 5 0 [] [] [] 0 0 [] ( [] [] 5( [] [] 90 ( 5 ( [] [] [] [] [] 0( ( [] [] [] 05 5 ( [] ( [] (5 [] 7 [5] [] 5 ( 5 [] [] [] 05 ( 05( [] [] ( 0 05 [] ( [] [] 70( 0 [] [] [] ( 5 0 [] [] ( ( 05 0 By siilr procur w gt th rivtivs of n h tht r us in th finition of ( in th prcing sctions Whn w rrng th obtin rsults w gt tringl li Poly tringl [8 p0] whr th contnts of th tringl r th cofficints of in th xprssion ( ( (5 ( (7 (8 (9 ( ( Also th first trs of th cofficints of in th xprssion of r ( (5 (9 [9] 5 [5] (7 [7] (7 [] ( 7 5 [5] (5 ( [] ( [] 7 ( [] (5 [] [] [] [8] (8 5 [] 7 0 [5] 5 [] [] 0 [7] [] [] 7 [] 5 (7 [] [5] [] 0 0 5

10 Shth Ass Al-Nr ( (5 ( (7 (8 ( ( (5 5 ( 7 ( Th scon trs of th cofficints of in th xprssion of r ( (5 ( (7 (8 (9 ( ( ( ( Th cofficints of in th xprssion of r rrng s follows 0 0 ( (5 ( (7 (8 ( (5 (7 ( ( (8 (5 (7 ( ( (5 ( ( (5 5 Th igonl of th bov rsults is th scon colun of th prcing Poly tringl n th first colun for th bov rsults is obtin s follows: By ultiplying th igonl by 5 w gt th lin unr th igonl which r ((= ((5=0 (0(=0 (5(7=05 Th gnrl forul of th iffrntition is givn by W whr < 8 n W cn b obtin fro th givn tbls s follow whn = thn whn = thn W fro th tbls W cn b foun s follows W

11 ournl of Al-Nhrin Univrsity Vol( April 008 pp05-9 Scinc Strt ntr S=su (A (i/ (i i= to Fill A (i i= to C = (+S+ (/A+S// A=prouct of (i i= to A (i = A/ (i i= to B=prouct of (i i= to B (i = B/ (i i= to S=su (B (i D= (-05S/B M=su {cot (πia (i/ (icot (πi/ (i} i= - C - =/ ((-!( C -S (n An I n ( n ( = (n I t =0 (n h n h t =0 ( ( n I / I i i ( n (n n I / i * i Clcult cofficints of xprssion of [n] = (pp n in M th uppr tringl of ro S [i]a[i] =(im/ i= to (ii= i= p S = su (S [i]a[i] A ii+ =A/((i(i+ i= to - S=su (A ii+ i= to - (i = (i- + i = p 5

12 Shth Ass Al-Nr Nxt i= p = p If < > i (i=(i--+(i- Clcult cofficints of xprssion of (n = (pp ( ' n in th M th uppr tringl of ro (ii =(i+ i= p Nxt i (i+ i = (i+ (i+ i= p Clculbl cofficints of xprssion of (n [n] = (P P = (pp in th (i = (i i- i= p Clculbl cofficints of xprssion of (n [n] = (P P in th Clcult cofficints of xprssion of (n =5 (pp ( in th ( = M th uppr tringl of 5 ro M th uppr tringl of ro 5 (ii =(i+ i= p (i =+i i= p 5 (i+ = (+i 5 (ii i= p i= p 5 (i =5 (i i- i= p = p (i= (i- Clcult cofficints of xprssion of [n] = (pp ( in th

13 ournl of Al-Nhrin Univrsity Vol( April 008 pp05-9 Scinc 5 M th uppr tringl of ro (ii =(i+ i= p (i+ i = (i+ (ii i= p Clcult cofficints of xprssion of [n] =7 M th uppr tringl of 7 ro 7 (i=(i+ i= p 7 (i+ i = (i+5 7 (ii i= p 7 (i = 7 (i i- i= p ( in th W ( = ( [ (i ( n- + (i ( n -] i= p W ( = ( [ ( n - (i ] i= p W (5 = ( [ ( n - (i + (i ( n-] i= p W ( = ( [ ( n - (i ] i= p W (7 = [ ( n - (i ] i= P W (8 = ( [ ( n -5 (i + (i ( n-] i= p W (9 = ( ( [ ( n - (i ] i= p W=w + w (i i= (n W (0 = ( [ ( n - 5 (i ] i= p W ( = ( [(n- ( n - + (i ( n-] i= p W ( = ( [ ( n - (i ] i= p W ( = [ ( n - (i ] i= p W ( = 5 [ ( n -5 (i ] i= P 5 7

14 Shth Ass Al-Nr 7 W ( = (5 [ ( n - (i 5 + (i 5 ( n-] i= p W ( = ( ( [ ( n -5 (i ] i= p W (5 = ( [ ( n -5 (i ] i= p W ( = ( [ ( n -5 (i ] i= p W ( = ( [ ( n -7 (i + (i ( n-7] W ( = ( [ ( n - (i ] i= p W ( = (7 [ ( n -7] + R [n] = ( n I * ( i / ( * ( ( n n i= ( n ( n [( n * ( n ( n ] W W (7 = ( [ ( n - 7 (i ] i= p n ( n su( ( i n i= W (8 = ( ( [ ( n -5] Y= prouct of (A (i i= W (9 = ( ( [ ( n -5] i= P W (0 = ( ( [ ( n - 5 (i ] i= p C= A (-n (n / (Yn! C (- =-/ (-! [D (/-/ (/A-su-C] C n 7 Fig ( : Progr Flowchrt for fining th cofficint of hrhrt polynoil 8

15 ournl of Al-Nhrin Univrsity Vol( April 008 pp05-9 Scinc Rfrncs [] A Brvino n Porshi An lgorithic thory of lttic points in polyhr nw prspctivs in gotric cobintorics MSRI publictions 8 ( [] M Bc A D Lor M Dvlin Pifl n R P Stnly Cofficints n roots of hrhrt polynoils confrnc on intgr points in polyhr (-7 uly in Snowbir (00 - [] M Brion n M Vrgn Lttic points in sipl polytops Ar Mth Soc 0 ( ( [] R Di S Robins Th hrhrt polynoil of lttic polytop Annls of Mth 5 ( [5] T M Apostol Moulr functions n Dirichlt sris in nubr thory Springr Vrlg Inc 97 [] M Bc Counting lttic points by ns of th rsiu thor Rnun ( ( [7] R P Stnly nurtiv cobintorics Wsworth & Broos / Col Avnc Boos & softwr Cliforni 98 [8] G Poly n R Trn n DR Woos Nots on introuctory cobintorics Birhüsr Bostn Inc 98 الخالصة حساب حجم متعدد األضالع وكذلك حساب عدد النقاط التي احداثياتها أعداد صحيحة في المجال R هو موضوع مهم جدا في فروع الرياضيات المختمفة مثل نظرية االعداد نظرية التمثيل متعدد حدود ايرهارت في التوافيقية التجفير و النظام االحصائي تم حساب متعدد حدود ايرهارت باستخدام بعض الطرق ا حدى هذه الطرق طورت واستنتجنا مبرهنة لحساب معامالت متعددة الحدود ايرهارت كذلك كتب برنامج بمغة فيجوال بيسك لحساب المشتقات لمدالة التي أستخدمت لحساب معامالت متعددة الحدود ايرهارت 9