JOURNAL OF ALGEBRA 199, 499527 1998 ARTICLE NO. JA977206 O HlbertKuz Fuctos of Soe Hypersurfaces L Chag* Departet of Matheatcs, Natoal Tawa Uersty, Tape, Tawa ad Yu-Chg Hug Departet of Matheatcs, Natoal Tawa Noral Uersty, Tape, Tawa Coucated by Crag Huee Receved October 26, 1996 1. INTRODUCTION Let p be a pre, Ž O,. a coplete local Noethera Ž p. -algebra, ad the p th Frobeus power of,.e., the deal of O geerated by p all a wth a. By e Ž O. we deote the legth of O. The fucto h : e Ž O. s called the HlbertKuz fucto of O. a By a result of Mosy 11, Theore 1.8, t s ow that e O cp where a s the Krull deso of O, c a postve real costat, ad Ž Ž a1. O p.. I the sae paper, Mosy 11, Theore 3.10 also showed that f a 1, the c s the ultplcty of O ad s a evetually perodc fucto of. I geeral, to detere the HlbertKuz fucto of a rg s a hard proble. However, there are already soe results for soe classes of specal rgs, for exaple, 1, 4, 5, 711. Whle s d 1 d 1 s ž 1 / 2 OŽ p. X,..., X X X, * E-al address: chagl@ath.tu.edu.tw. E-al address: hugyc@ath.tu.edu.tw. 499 0021-869398 $25.00 Copyrght 1998 by Acadec Press All rghts of reproducto ay for reserved.
500 CHIANG AND HUNG s2 Žj. j Kuz 10 showed that c s ratoal ad j0 p wth each Žj. evetually perodc. I 8, Ha ad Mosy showed that f d1 dt 1 t Ž 1 t. O p X,..., X X X wth d 1, the c s ratoal ad s perodc for suffcetly large whle p 2ort3. If p 2 ad t 3, they also proved that there are tegers l ad, 1, 0 l p Žt3. such that l for 0 8, Theore 5.7. I 4, Coca detered the HlbertKuz fuctos of OFX,..., X 1 s I where I s a ooal deal or a prcpal deal geerated by a hoogeeous boal for Žcf. 4, Theore 2.1, Theore 3.1.. I ths artcle, by ag use of the represetato rg developed by Ha ad Mosy, we ca geeralze the results 8, Theore 5.7; 4, Theore 2.1. Our a result s as follows: THEOREM 1.1. Let F be a feld of characterstc p 0; let t ž, j j1 / 1 d f X Ž 1. where t1 t2 ts 2 ts1 ts2 t 1. Let a Ž t. 1, S 1,...,s 4, t1 Ž t 2.For. 1 S each A S, de- fe Ž A. Ž t 2. A f A, ad 0. The the HlbertKuz fucto of the hypersurface fs Ž A. t a cp A, jž. jž., AS j0 j0 where c s a ratoal uber. For the ters Ž. ad Ž. A, j j, we ca fd poste tegers ad l A s such that A, jž. la p j A, jž. for 0 ad jž. fjž. jž., j where f p f. If p 2, we hae Ž. 0 f 0. j j
ON HILBERTKUNZ FUNCTIONS 501 Furtherore, the tegers l A hae the followg propertes: Ž. Ž A3. 1 For odd p, f A 3 the l p, ad f A A 3 the la 1. Ž. ŽA1. 2 For p 2, the l 2 f A, ad l 1. A By cobg Theore 1.1 ad E. Ž 21., we ca also obta the HlbertKuz fuctos of all hypersurfaces of the for t,j 1 j1 d f X, j 2 for d, j 1. Actually, f we tae t1 t2 ts 2 ts1 Ž t t 1, a t. 1, S 1,...,s4 s2 1 ad t 1 Ž t 2., we wll have the followg slar result: S COROLLARY 1. E. Ž. 2 s where c s ratoal ad The fuctos re 1.1. The HlbertKuz fucto for the hypersurface f defed Ž A. t a cp A, j j, 3 AS j0 j0 Ž A. Ž t 1. A. 1 s ad s hae the sae propertes as those Theo- A, j j 2. REPRESENTATION RING AND MULTIPLICATION FORMULAE There s a useful tool called the represetato rg preseted 8. Let F be a feld. A F-object s a ftely geerated FT-odule o whch T acts lpotetly. Let M ad N be two F-objects. The M N ad M N are also F-objects wth TŽ. T T ad TŽ. F Ž T. Ž T.. Let be the set of ordered pars Ž M, N. F of two F-objects. I we defe the followg euvalece relato: F Ž M 1, N1. Ž M 2, N2. M N s soorphc to M N as FT-odules. 1 2 2 1
502 CHIANG AND HUNG The euvalece class Ž M, N., deoted by M N, s called the foral dfferece of M ad N. Let F be the set of the foral dffereces of F-objects. I we defe two bary operatos: F Ž M1 N1. Ž M2 N2. Ž M1 M2. Ž N1 N 2., Ž M N. Ž M N. Ž M M. Ž N N. 1 1 2 2 1 F 2 1 F 2 Ž M N. Ž N M.. 1 F 2 1 F 2 The Ž,,. F s a coutatve rg, whch s called the represetato rg Žcf.. 8. We call the ultplcato defed here the HaMosy ultplcato. We wll defe aother ultplcato at the ed of ths secto. Note that we ca decopose ay F-object to a drect su of FTT Ž. s. Let FTŽ T. 0. The Ž. 4 F, s a free -odule wth bass 1. We have a ap fro the set of F-objects to F, : M M 0 wth Ž M N. Ž M. Ž N. ad Ž M N. Ž M. Ž N.. Throughout ths artcle, F wll always deote a feld of characterstc p 0, ad a power of p. Here we defe a -hooorphs D F: as follows: For ay F-objects M ad N,f M FTT Ž. a F 1 ad N FTT Ž. b, the ad 1 It s easy to see that D Ž MN. Ž a b., F 1 d Ž M N. Ž a b.. F 1 D Ž, j.. F j For coveece, as 8, we tae aother bass for the free -odule : ½ 1 f 0, Ž 1. Ž. f 1. 1 We fd soe forulae 8, 3 to decopose to the su of. Oe ca also refer to 2. Here we lst soe of the.
ON HILBERTKUNZ FUNCTIONS 503 PROPOSITION 2.1 8, Lea 3.3, Theore 3.4. p. We hae ad, for 0 s, we hae Let 0 s,1 s s, Ž 4. 1s Ž s1. s, Ž 5. 1s Ž s1. s, Ž 6. s s, Ž 7. 1s 1s. Ž 8. Ž. Ž. By E. 4 E. 8, the followg forulae are edate. For s, 1, s Ž 1., Ž 9. s Ž 1. s s, Ž 10. 1 Ž 1. 1s s. Ž 11. Note here that E. 10 ad E. 11 are also vald whe s. Let 0 f 0, H ½ 1 f 0. We have PROPOSITION 2.2 3, Proposto 3.3. If 1, j p, the jžj1. Žj3. b Ž j p. HŽ jp. p, Ž 12. where b j 1, 2 p 1 j. PROPOSITION 2.3 3, Proposto 3.4. If 1, j p ad j 1 p the j1 p for all 1, p, p 2,.... By E. 12 ad Proposto 2.3, we edately have j Ž Žj1. Žj3. b p. Ž j p. HŽ jp.. Ž 13.
504 CHIANG AND HUNG PROPOSITION 2.4 3, Proposto 3.5. hae For 0 p 1, 1 j p, we Ž 1. j HŽ j. j Žj1. c HŽ jp. p, Ž 14. where c j,2p1j. PROPOSITION 2.5 3, Proposto 3.6. For 0, j p 1, we hae c c, Ž 15. j Žj2. Žj2.1 0 1 where c, j, p 1, p 1 j. Let 0 b p 1, ad t b 1. PROPOSITION 2.6 3, Proposto 4.1. Let p be a odd pre, ad 1, p, p 2,.... We hae u u u u 0 0 2p1 21 2p 2 4p1 41 u u Ž p1.p1 Ž p1.1 Ž p1.p Ž p1. f b s ee, ad f b u u u u p1 1 p 3p1 31 3p 3 u u 2 Ž p2.p Ž p2. p 1 p1 s odd. Here the u s are o-egate tegers. There are soe propertes about the coeffcets u. PROPOSITION 2.7 3, Proposto 4.2. for all. u p t2 Whle t 2 we hae PROPOSITION 2.8 3, Corollary to Proposto 4.2. If t 3 ad 2 2 0, p 1, p p, p 1,we 4 hae u p t3.
ON HILBERTKUNZ FUNCTIONS 505 Whle p 2, we have 00 0, 0 ad 0, thus we have the followg sple forulae: PROPOSITION 2.9 3, Proposto 4.3. Let p 2. The 0 f b s ee, ad f b s odd. For a pre power ad l 1, defe 1 l l l l j1 D 2j j1 j1, Ž p1. GŽ. Ž 1. Ž p1.ž p1.1ž p3. 0 ; 0 Ž p1. G 1 f s odd ad Ž p1. Ž p2. Ž p3. 0 G 0 Ž. Ž 0. f s eve. If 1, we defe E to be the addtve subgroup geerated by the, 0or1Ž od 2. ad O to be the addtve subgroup geerated by the, or 1 Ž od 2. Žcf. 8, Defto 3.12.. PROPOSITION 2.10 8, Theore 3.15. If r s ee, r E E ad r O O. If r s odd, r E O ad r O E. PROPOSITION 2.11 8, Corollary 3.16. E E E, E O O ad OOE. Let j A u u u u u 0 0 p1 1 p 2p1 21 2p 2 where u ad defe u 2 u 2, p p p p 1 p1 u 2 u 2 p p p 1 WŽ A.. u u p1 0
506 CHIANG AND HUNG For a eleet K 1 u 0 F T Ž K. u. where u, we defe 1 PROPOSITION 2.12. Let, j. The 1 T 1 H j j. j Ž. Proof. Let u. By E. 6, we have j 1 Thus 1 1 Ž j. j 1 1 1 u Ž.. 1 1 u 1 u u Ž, j. u j 1 1 1 because u D Ž, j.. The Ž 1. 1 T Ž. 1 F j j Ž, j. ad the proposto follows. Q.E.D. By Proposto 2.12, t s easy to verfy the followg lea: LEMMA 2.1. l DF u D Ž. u ž ž / / 1 1 l l Ž., l l T u D Ž. u. 1 1 ž ž / / By ths lea, we have ž ž / / ž ž / / 2 2 F 1 1 1 D u D Ž. T u D Ž. 2 u. Ž 16. If p s odd, we have soe portat relatos betwee G 0 Ž. ad G. Let 0 Ž. Ž. G Ž. E O,
ON HILBERTKUNZ FUNCTIONS 507 where E Ž. E PROPOSITION 2.13. Proof. ad O Ž. O. We have Ž. Ž. 1 GŽ. E O. Ž 17. It s easy for 1. For 1, we have ž / 0 Ž1. Ž1. Ž1. Ž1. G Ž. Ž E O. Ž E. Ž O.. 0 Hece ž / ž 21/ Ž. Ž1. 2 Ž1. 2 E Ž E. Ž O., 2 Ž. Ž1. 21 Ž1. 21 O Ž E. Ž O.. O the other had, we have ž / Ž1. Ž1. Ž1. Ž1. 1 1 0 GŽ. E O Ž E. O. Applyg 2, we the have 1 0 ž / Ž1. 2 Ž1. 2 GŽ. Ž E. Ž O. 2 For b p, we have 1 ž / Ž1. 21 Ž1. 21 Ž E. Ž O.. 21 Q.E.D. b 0 0 b 1 G 1 G 1. It follows that ad p1 0 2 0 0 0 G Ž 1. Ž 1. G Ž 1. pg Ž 1. b 0 1 0 b Ž 1. G Ž 1. p G Ž 1..
508 CHIANG AND HUNG Thus, f we wrte b 0 b 0 0 p1 1 p 1 G u u u, we wll have up už1.p1 p 1, ad the W Ž Ž 1. b G 0 Ž.. s of the for b By otg that u p u 1 2 2 p p p p 1 p u0 u0. ad that b 0 Ž 1. b G Ž. 001 21 Ž p1. 0 p1 b 1 0 b 1 0 1 Ž p1. p1 1 G 0 0, we have u u 0 Ž p1. p ad the ž / p u u 1 b u0 p u 0 0 b 1 0 0 W 1 G. 18 O the other had, by coparg wth b b Ž. Ž. Ž 1. b GŽ. Ž 1. b Ž E 1O. b 0 b Ž. Ž. b b we have ether 1 G 1 E O, or ž / 0 2u p 2u p 1 b 0 0 b 1 0 W 1 G ž / p 2u 0 1 b 0 p 2u 0 b 1 0 W 1 G.
ON HILBERTKUNZ FUNCTIONS 509 The we get the followg propostos: PROPOSITION 2.14. If p s odd, the egealues of t b 0 b ž 1 / W 1 G 1 Ž t are p ad the poste egealue of W Ž 1. b G. 1 b. Moreoer, p t2 3 f t 2 3. If p 2, 1, ad 1, we hae Ž t Ž 1. b. G 0 Ž. 2 1 Ž.. 1 b 0 If p s odd, the proposto ca be proved by Proposto 2.8, ad by the 2 fact that G. The case of p 2 s easy. Ž p1.2 Aother Multplcato F. Now we troduce aother ultplcato represetato rgs. If M ad N are two F-objects, o M F N we defe the T-acto to be T T. The M F N s a F-object. We deote the correspodg ultplcato F to be ad call ths ultplcato the crcle product. It s easy to see that Ž,,. F fors a coutatve rg wthout detty. Actually, f j, we have the followg ultplcato forula Ž,,.: j 2Ž 1 2 1. Ž j 1.. F Fx. Let 3. PROOF OF THE THEOREM ad p p M F X,Y Ž X,Y. d, dj M d p FXŽ X. be FT-odules such that the T-acto o M Ž resp. M. d, d d s ultplca- j d d j Ž d. to by X Y resp. X.If p dar where 0 r d ad d1 d 2, the by the theory of odules ad soe sple calculatos, we have For M d 1, d2, we ay let a d a a M d r Ž 1.. Ž 19. a 1 2 M u. d 1, d2 1
510 CHIANG AND HUNG The we have for j a, ad 2 d F Md, d T j M 1 2 d 1, d2 u 2u 3u ju ju ju 1 2 3 j j1 a21 2 2 1 p p jd a21 2 u p. 1 Fro ths, we ca evaluate u s. The results are j 2 2 2 j 1 1 1 u 2 p jd p j 1 d p j 1 d 2dd 1 2 for 1 j a 1; 2 ad Thus we have 2 2 a Ž 2. Ž 2. 2 1 1 u 2 p a d p a 1 d p a d Ž d r. d Ž d r., 2 1 2 2 1 2 2 2 a Ž 21 2 1 2 1. 2 u p a d p a d r. M dd 2 2 2 d 1, d2 1 2 1 2 a21 a2 ž / r2 2 d1 a 2 d1 2dr 1 2 a2 Ž 1. a2 p 1 Ž M d2.. Ž 20. d d By E. 19 ad E. 20, we get 2 2 ž / d1 d 1 Ž Md, d. Ž Md, d. p 1 Ž M d., 1 2 2 2 2 d d 2 2
ON HILBERTKUNZ FUNCTIONS 511.e., ž / d1 d 1 Ž Md. Ž Md. Ž Md. Ž Md. p 1 Ž M d., Ž 21. 1 2 2 2 2 d d 2 2 f d1 d 2. Fro ow o, we wll always tae the ultplcato F to be the HaMosy ultplcato f we do ot specfy t. Cosder the FTodule p Ž, j, j. N F X j1,...,t X j1,...,t such that the T-acto o N s ultplcato by t defto of the crcle product, we have j1, j N M M M. d,1 d,2 d, t Tae d d d d. By E. 21, we have,1,2,t X d, j. The by the t Ž t1. 2 Ž t2.,0 d,1 d,2 d N x M p x M p x M p Žt 1. x M,, t1 d where the x, j s are ratoal fuctos of d,1, d,2,...,d,t ad Md eas the th power of Ž M. d wth respect to the crcle product. Now Ž we have to evaluate D Ž N..,.e., the D value of F 1 F t Ž t1. 2 Ž t2. x,0 Ž M d. p x,1 Ž M d. p x,2 Ž M d. 1 Žt1. p x, t 1 M d. 22 We cosder frstly the case of d d d. Let d d.,1,2,t, j For a fxed, tae p dar where 0 r d. Oe ay fd t a t t a a Ž N. r Ž 1. Ž r d. r a1 t ½ j1 2 r jd r j 1 d t t aj r Ž j1. d Ž 23. 5
512 CHIANG AND HUNG by a slar process as above. Wrte 1d to p-adc for: 1 1 2 b Ž 0. b Ž 1. p b Ž 2. p. d For fxed, let p adr wth 0 r d. The abž 0. p bž 1. p 1 bž 2. p 2 bž.. For ay, defe cž,. b p bž 1. p 1 bž., t x t t K x r 1 r d r x x x1 t t ½2rjd r Ž j 1. d j1 t 5 xj r j1 d. For 1 l t, p, defe ž / b l t tl l 1 Žb. L, d p od d 1 d Žb Ž.. 1, l where the sybol a od b deotes the reader of a dvded by b. Let e be a postve teger, 1 l e 1, ad ay power of p. Defe ž/ p1 l e l Ž p1. GeŽ. Ž 1. Ž p1.. l 1 Ž. By E. 5 through E. 11, oe ca easly chec that ad that e1 e 0 e l e l el l1 D Ž p. G Ž. D Ž. G Ž. D Ž. Ž 24. b Ž. b Ž. Ž. t1 l l tl l0 K b c,1 1 K c,1 L Ž,. D Ž.. Ž 25.
ON HILBERTKUNZ FUNCTIONS 513 Let I 0*,0,1,2,...,t 1, 4 II1 I2 I. For cove- ece, we defe 0* for all 0*. We say that Ž 1,...,. Ž,...,. 1 f for all. We deote f ad. Such a order I ca be exteded to a lear order the followg way. Let S Ž j. : j4 1,2,..., 4. Ž. 1 If S Ž 0*. S Ž 0*., ad S Ž 0*. S Ž 0*. the. If S Ž 0*. S Ž 0*. ad S Ž 0*. S Ž 0*. the accordg to the lexcographc order of S Ž 0*. ad S Ž 0*. Žcf.. 6 ; e.g., Ž 0*, 2, 1, 0*, 3. Ž 1, 0, 0*, 4, 2. ad Ž 3, 1, 0*, 2, 0*. Ž 1, 2, 1, 0*, 0*.. Ž 2. If S Ž j. S Ž j. for all j ad S Ž. S Ž. the. If S Ž j. S Ž j. for all j ad S S ad S Ž. S Ž. the accordg to the lexcographc order of S Ž. ad S Ž. ; e.g., Ž 2, 0*, 0, 1, 0. Ž 1, 0*, 0, 1, 1. ad Ž 0, 0*, 1, 2, 0. Ž 0, 0*, 2, 1, 0.. It s easy to see that f. Let ad for 0 l t 1. Let v Ž,. K c Ž,.,0* l t l, l v, p D p v Ž,. v Ž,., 1, 0 f Ž 0*,...,0*., ½ f otherwse 0* for all Ž,...,. 1 I. For exaple, f 5, t1 t2 t5 4 ad Ž 1, 0*, 0, 3, 2. I, the Ž. t 1 1 t v, pd p K2 c2, D 3 p 3 t 3 2 4 t 5 2 p D p p D p. Now to evaluate the HlbertKuz fucto of the hypersurface defed by Ž. 1, we have to evaluate Fž / F 1 D K c Ž,0. D v Ž,0.
514 CHIANG AND HUNG for Ž 0*, 0*,..., 0*.. We wll gve a recursve algorth to evaluate t. I geeral, we fx soe Ž,...,. 1 I ad soe o-egatve teger. Let b b Ž., c c Ž,1.. By E. Ž 24. ad E. Ž 25., we have vž,., 1 v Ž,. K b c p D t p 0* 0* t1 b l l tl b l0 Ž 1. K Ž c. LŽ,. D Ž. 0* Ž. 0 t 0* p G D t1 l l t l Gt D l1 t1 b l l tl b l0 Ž 1. K Ž c. LŽ,. D Ž. 0* 0 t p G D 0* t1 l l tl p Gt Ž. D Ž.. l1 There wll be Ž t 1. Ž t. 0* 0* ters whe we ultply out the above euato. Let b, b 0* 0*, 0* A, 1 L, f ad A, t p G 0 p G Ž., 0* 0*, Ž,. 0 otherwse. The v Ž,. A Ž,. v Ž, 1.., I
ON HILBERTKUNZ FUNCTIONS 515 Let Ž,. A Ž,. v Ž, 1.,, where the su taes over all wth soe t 1. By otg that t t t K c r dc p od d, l D t l tl l t ad by E. 13 ad E. 14, we have Ž,. 1 t A,, p p od d 0* p a, j j j1 p, Ž. where a 1 t 1 ad, j s are fuctos of b ad p od d, thus are evetually perodc fuctos of. Let The vž,. p Ž,. p, j j. j1 t a t l tl l b l2 0* b Ž,. L Ž,. D Ž. Ž 1. K Ž c. t1 tl tl l 0 t t 0* l2 p G Ž. D Ž. G Ž. D Ž.. Multplyg out the product of the above euato, we have a v, A,, v, 1,, J where J I: t 1, for all 4. Arrage v Ž,. to a colu vector vž,. v Ž,. J the lear order defed above such a way that v Ž,. coes after v Ž,. f. It s easy to see that the correspodg atrx AŽ,. t
516 CHIANG AND HUNG A Ž,. s a upper tragular atrx the atrx rg M Ž.,, J J F. We ca also create the colu vector Ž,. Ž,. J by the sae way. The we get the recursve forula vž,. AŽ,. vž, 1. p až. Ž,.. Note that A Ž,., E O. Let Ã, WŽ A,Ž,.., TpŽ v Ž,.. Ž v,., D v Ž,. F Ž. Ž. Ž. Tp,. D, The by E. Ž. 7 ad E. Ž. 8, we have až v Ž,. A v Ž, 1. p. Ž., where v Ž,. ad are 2J 1 atrces ad A s a Ž2J. Ž2J. atrx over. Moreover, A ad are evetually perodc as fuctos of. We ay fd tegers ad such that A ad are perodc for wth a coo ultple of ther perods. For large, there exst Ž. ad Ž. such that 1 Ž. Ž. where 0 Ž.. Because AŽ,. s a upper tragular atrx, the set of egevalues of A s the uo of the sets of egevalues of A, for J. Cosder A Ž,. p G Ž. Ž 1.., F 0 b 0* 0* For odd p, by Proposto 2.14, f Ž 0*,..., 0*., the egevalues of A, are p Ž SŽ 0*.,. Ž 26. 1 0* Ž 1. ad p, where ŽS Ž 0*.,. s the postve egevalue of Let b ž b Ž. / 0* 0* b W Ž 1. GŽ.. Ž 27. 1 1 Q. 2 1 1
ON HILBERTKUNZ FUNCTIONS 517 We have 1 2, 2 Q A Q 1 0* Ž1. p 0 0 for all Ž 0*,..., 0*.. For Ž 0*,..., 0*., the A Ž., s are ether 0 dagoal or of the for 0. 00 10 For p 2, by Proposto 2.14, we have A or f, 01 00 1 0* Ž1. 10 0*,..., 0*, ad A 2 f Ž 0*,..., 0*.,. Thus, all the absolute values of the egevalues of 01 Ž A. wll be less tha a a p ad the all the egevalues of B wll ever be p. The followg atrces ca be well-defed: ad The CA Ž 0. A Ž., a až bp Ž 0. p 1. AŽ 0. Ž 1. A Ž 0. A Ž 1. Ž., BA 1 A, až1. až2. v 1 p A 1 2 Oe ca easly chec that A 1 A 2 A 1, a a 1 up p IB v, zp až 1. Cu p a b, Ž. Ž v, Ž.. p a Ž. u. v,0 C B Ž. p a z. Ž. Ž. 1 a a p I B p v, v,. Now suppose that p s odd ad that t 2 for soe. We wll show that the -copoet of Ž., deoted by Ž., s a scalar 1 ultple of. Let 1 p v Ž,. u. j, j j 1
518 CHIANG AND HUNG The ad by E. 16, we have O the other had, p p t 2 2 ž / 1 v, u p D p, p t 1 pž. FŽ. 1 T v, D v, 2 p u. ž j / F j, FŽ j,. u d v Ž,. d v Ž,., j 1 j j tj tj FŽ j,. FŽ jž j.. Ž j. j d v, d K c, p p od d f 0*; j j t t j j j d v Ž,. d Ž p. D Ž p. Ž p. f 0*. Hece j F j, j F Ž 1 t. t j1 j j p F j T v, D v, 2 p p od d, where p. The 0* j p a v, Ž. v, Ž. T Ž.1 p Ž vž, Ž... p a D v, Ž. F Ž. Ž Ž.. Ž Ž.. T Ž.1 p v,. D v, Because p od d p od d for 0, we have p a vž, Ž.. v Ž, Ž.. a p T Ž.1 v, Ž. p F Ž. 1 T Ž.1 p Ž vž, Ž.... 1
ON HILBERTKUNZ FUNCTIONS 519 Let w p a v Ž, Ž.. v Ž, Ž... The the -copoet of w s ad a w p B,, B A Ž 1. A Ž 2.,,Ž 1. Ž 1,2. A Ž 3. A Ž., 2,3 1, where the su s tae over all Ž. J such that Ž 1. Ž 2. Ž 1.. Because t 2 for soe, we have t 2 for all. Thus all AŽ,. have the factor G 0 Ž. Ž., Žj., ad the by E. Ž 18., we have u u WŽ AŽ,. Ž.,Ž j.. u u 0 p1 p1 0 for soe o-egatve tegers u s. The B, s also of the for as above. We ow prove by ducto o that Ž. s also a scalar 1 ultple of. If s the axal eleet J, the 1 a a w p Ž. B Ž. p Ž. B Ž..,, 1 1 1 1 1 1 1 1, 1t Žt 1. 2 1 1 1 1 Suppose that s ot a scalar ultple of. The u for soe u 0. Because u s a egevector of B wth respect to the egevalue p, the -copoet of w s ot zero. Ths s cotrary to the fact that w s a scalar ultple of. For a geeral wth soe t 2, we have a wp B,. 1 1 By ductve hypothess, the vector s a scalar ultple of for all,sos B Ž.. Suppose that Ž., s ot a scalar ultple 1 of. The by a slar dscusso as above, oe ca easly see that w s 1
520 CHIANG AND HUNG ot a scalar ultple of 1 1 1 1, ether. We the get a cotradcto ad thus s a scalar ultple of. I ths case, the egevalue 1 0* Ž 1. p wll ot volve ay etry of v Ž,0.. Let Q dagž I, Q, Q,...,Q. 2 2 2 2. We ca, by doublg the teger f ecessary, ae the atrx Q1 BQ upper tragular. We wll gve soe dscussos o the dagoalzablty of B. Let A : 0* 4 S 1,...,s 4. Cosder frstly for the case that p s odd. If Ž 0*,..., 0*., let By 26, we have Ž.. 1 p Ž SŽ 0*.,.. 1 Let l ŽS Ž 0*.,.. The, we have A 1 1 2, 2 1 0* Ž1. p 0 Q B Q. 0 If Ž 0*,..., 0*., t s easy to see that oly oe of the egevalues of B s volved the secod copoet of vž, 0.,, ad we tae ths egevalue to be l. We are gog to prove that la lb f A B ad that l la f oreover l 1. Wthout loss of geeralty, oe ay assue that A 1,..., j4 ad that B 1,..., j1 4. By Ž 27., we copare the followg two eleets the represetato rg: wth L G A j j1 b Ž j1. j2 L G. B b Wrte u00 u1 21u2 2 j GŽ. b or Ž 28. j2 u u u. 0 1 1 2 31
ON HILBERTKUNZ FUNCTIONS 521 The L u u u A bj1 0 0 1 21 2 2 u u bj1 0 p1bj1 p1 or L u u u A bj1 0 1 1 2 31 u u, bj1 1 p1bj1 Ž p1. ad LB u0 0u1 21u2 2 GŽ. or p1 p1 u u 0 0 21 p1 0 LB u0 1u1 u2 31 GŽ. p1 p1 u u. 0 1 Ž p1. 0 I ether case, u 0 for all. Thus u u, ad we have la l B.If for soe 1,..., 4, : 0, 1,..., p 1, u 04 2, the the ozero etres W Ž L. ust be larger tha those W Ž L. B A, ad we wll have l l. Note that f j 0,.e., A, the u A B 0 for 0,..., p1, thus, we have la l B.If la1, there ust be at least two s such that u 0 for soe 1, 2,..., 4, the la l B. Thus, the oly possble Jorda blocs of B that are ot dagoal are those whch correspod to the egevalues of powers of p. If oe of these l s a power of p, the t s clear that B A ca be dagoalzed, ad the j the fj Theore 1.1 s p whe. For the case of p 2, the dagoalzablty of B s sple. The oly thg we have to cosder s the dagoalzablty of the bloc A A,, 0 A,
522 CHIANG AND HUNG for 0*,..., 0*, 0* ad 0*,..., 0*, 0. We have A A L j b j Ž,.,,, 0 A 0, 0 Ž 1. where j 1 b od 2. If b 0, we have L Ž,. 0. I ths case, 1 0 0 0 0 0 0 0 A A,, 0 0 0 0 0 1 0 0 or. 0 A, 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 If b 1, we have L,. I ths case, j j 1 0 0 0 0 0 0 A A,, 0 0 0 0 1 0 0 or. 0 A, 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 It s easy to see that the products of the above four atrces are dagoalzable. Thus the correspodg bloc of B s dagoalzable, ad the B s dagoalzable. Theore 1.1 ow ca be easly proved f we trasfor the atrx B to ts Jorda for. For the proof of Corollary 1, we ultply out E. Ž 22. ad obta a lear cobato Ž ratoal coeffcets. of the ters where w s p Ž M d., 1 w s t, Ž 29. j1 ad 1 s Ž s j j t j. The value DF j1 Md. j s just the value of the s HlbertKuz fucto for the hypersurface X d 1 j1, j. By Theo- re 1.1 the doate ter for ths value s cp s 1 for soe ratoal w uber c ad by E. 29, the D s F-value of p 1 Md s doated by cp Ž 1 t 1.. By our algorth for the evaluato of l A, the set 4 s l correspodg to X d A A 1 j1, j ust be cotaed that to t X d. Aga, by Theore 1.1 ad E. Ž 29., oe ay easly 1 j1, j 1
ON HILBERTKUNZ FUNCTIONS 523 Ž. chec that the secod su E. 3 s tae fro j 0to j Ž s 2. Ž t s. Ž t 1. AŽ A. A 1 1 ad that the thrd su s tae fro j 0to 1 Ž s2. Ž ts. 1 Ž t 2. t. s2 1 t2 Ths proves Corollary 1. We ow gve soe exaples below. Soe of the data s obtaed by a progra wrtte Maple V. EXAMPLE 1. Let F Ž. 5. By the algorth we have lsted above, we ca obta the HlbertKuz fucto of the hypersurface Ž XY. 4 T 4 U 4 W 4. By the coputer progra wrtte Maple V, we ay fd that 1 0 0 0 0 3 16 0 Ã, 0 0 4 3 0 0 3 4 0 1852, 248 1000 for all 1. Let B A Ž. 1. 1 1839 Ž v 1,1.. 249 1001 Fro these, oe ca fd that the HlbertKuz fucto of the hypersurface s 285166 145 752 4 5 3 7. 96099 311 309 EXAMPLE 2. Let F Ž. 5. We fd by our algorth that the HlbertKuz fucto of the hypersurface Ž XYZ. 4 T 4 U 4 W 4 s 378803300 5837 53520 4042 5 5 3 35. 125571523 1561 13277 1751
524 CHIANG AND HUNG By the progra Maple V, we ay fd that à 1 0 0 0 0 0 0 3 64 0 48 0 0 0 4 3 18 12 0 0 3 4 24 30 0 0 0 0 20 15 0 0 0 0 15 20 Ž. for all 1. Let B A 1. Arrage the egevectors of B to P ad oe has 1 0 0 0 0 0 0 1 32 16 144 129 P 0 0 1 1 3 39 0 0 1 1 3 63, 0 0 0 0 2 34 0 0 0 0 2 34 0 1 378803300 11047859203 125571523 125571523 28339 3950075 160886 160886 z 321995, Ž. 6327885, 160886 160886 121 6580 309 309 497 6580 309 309 1 17511 1561 3345 1 P Ž. 26554 0 0 3290 5253
ON HILBERTKUNZ FUNCTIONS 525 ad C I. The oe ay easly fd that the secod copoet of s v,0 C B Ž. p a z 378803300 5837 53520 4042 5 5 3 35. 125571523 1561 13277 1751 Cobg the two exaples above wth the exaple preseted 8 for the HlbertKuz fucto of the hypersurface X 4 T 4 U 4 W 4 ŽŽ 16861. 5 3 Ž 10761. 3., ad by E. Ž 21., oe ay fd that the HlbertKuz fucto for the hypersurface s X d 1 Y d 2 Z 4 T 4 U 4 W 4 dd 378803300 5837 53520 4042 1 2 5 5 3 35 16 125571523 1561 13277 1751 ž / ž / ž /ž / d1 d2 d2 d1 285166 145 752 5 1 1 5 15 35 4 4 4 4 96099 311 309 d1 d2 168 107 5 1 1 5 75, 4 4 61 61 where d 1, d2 4. EXAMPLE 3. Let F Ž. 7. We obta by our algorth that the HlbertKuz fucto of the hypersurface Ž XYZT. 6 U 6 W 6 S 6 s 2050747718345259 104440051 267597 6 3 7 3 7 429658164277013 294115 74480 312186671451 164565 7 7. 878707312 2801 Aga by the progra Maple V, we obta the atrces A ad as à 0 0 0 0 0 0 0 0 0 3 1296 0 864 0 216 0 0 0 4 3 24 16 72 48 0 0 3 4 56 72 408 504 0 0 0 0 28 21 126 84 0 0 0 0 21 28 294 378 0 0 0 0 0 0 196 147 0 0 0 0 0 0 147 196,
526 CHIANG AND HUNG ad 0 561666 2810 325834 8526 273378 29302 205898 for 1. The Jorda for of B A Ž. 1 s 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 D 0 0 0 7 1 0 0 0 0 0 0 0 7 0 0 0. 0 0 0 0 0 49 1 0 0 0 0 0 0 0 49 0 0 0 0 0 0 0 0 343 Ths s a exaple such that the atrx B caot be dagoalzed. We tha Dr. Mosy who gves us aother exaple such that the dagoalzablty also fals. Whe p 19 ad the hypersurface s XYZT U 19 V 19 W 19, fro the coputer data, we have that the Jorda for of B s D 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 19 1 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 2 19 1 0 0 0 0 0 0 0 2 19 0 0 0 0 0 0 0 0 19 3, ad the HlbertKuz fucto for ths hypersurface s 1901704366851459696550 65245737771 6 3 19 19 70272691919506647 371360700 2051538694880851 1404902 19 19. 51092177546700 2063415
ON HILBERTKUNZ FUNCTIONS 527 ACKNOWLEDGMENT We tha Dr. M. C. Kag, who gave us soe suggestos for ths artcle. REFERENCES 1. S. T. Chag, The Asyptotc Behavour of HlbertKuz Fuctos ad Ther Geeralzatos, Doctoral Thess, Uversty of Mchga, 1992. 2. L. Chag, HlbertKuz Fuctos, Doctoral Thess, Natoal Tawa Noral Uversty, 1996. I Chese 3. L. Chag ad Y. C. Hug, O HlbertKuz fucto ad represetato rg, Bull. Ist. Math. Acad. Sca Ž 1998., press. 4. A. Coca, HlbertKuz fucto of ooal deals ad boal hypersurfaces, Mauscrpta Math. 90 Ž 1996., 287300. 5. M. Cotessa, O the HlbertKuz fucto ad Koszul hoology, J. Algebra 175 Ž 1995., 757766. 6. D. Cox, J. Lttle, ad D. O Shea, Ideals, Varetes, ad Algorths, Sprger-Verlag, New YorBerlHedelburg, 1992. 7. C. Ha, The HlbertKuz Fucto of a Dagoal Hypersurface, Doctoral Thess, Brades Uversty, 1992. 8. C. Ha ad P. Mosy, Soe surprsg HlbertKuz fuctos, Math. Z. 214 Ž 1993., 119135. 9. E. Kuz, Characterzatos of regular local rgs of characterstc p, Aer. J. Math. 41 Ž 1969., 772784. 10. E. Kuz, O Noethera rgs of characterstc p, Aer. J. Math. 98 Ž 1976., 9991013. 11. P. Mosy, The HlbertKuz fucto, Math. A. 263 Ž 1993., 4349.