THE ROTATION NUMBER OF PRIMITIVE VE SEQUENCES. Citation Osaka Journal of Mathematics. 52(3)

Transcription

1 Title THE ROTATION NUMBER OF PRIMITIVE VE SEQUENCES Author(s) Suyaa, Yusuke Citation Osaka Journal of Matheatics 52(3) Issue Date Tet Version publisher URL DOI Rights Osaka University

2 Suyaa, Y Osaka J Math 52 (205), THE ROTATION NUMBER OF PRIMITIVE VECTOR SEQUENCES YUSUKE SUYAMA (Received October 0, 203, revised April 4, 204) Abstract We give a forula on the rotation nuber of a sequence of priitive vectors, which is a generalization of the forula on the rotation nuber of a uniodular sequence in [2] Introduction Let Ú,, Ú d ¾ 2 be a sequence of priitive vectors such that i det(ú i, Ú i ) 0 for all i,, d, and let a i i i det(ú i, Ú i ), where Ú 0 Ú d and Ú Ú The rotation nuber Rot(Ú,, Ú d ) of the sequence Ú,, Ú d around the origin is defined by 2 i L i y d dy 2 y 2, where L i is the line segent fro Ú i to Ú i The sequence is called uniodular if i for all i,, d Recently A Higashitani and M Masuda [2] proved the following: Theore A ([2]) If Ú,, Ú d is a uniodular sequence, then Rot(Ú,, Ú d ) 2 i (3 i a i ) When i for all i and the rotation nuber is one, Theore A is well known and forulated as 3d È d i a i 2 It can be proved in an eleentary way, but interestingly it can also be proved using toric geoetry, to be ore precise, by applying Nöther s forula to coplete non-singular toric varieties of cople diension two, see [] When i for all i but the rotation nuber is not necessarily one, Theore A was proved in [4] using toric topology The proof is a generalization of the proof above using toric geoetry The original proof of Theore A by Higashitani and 200 Matheatics Subect Classification A55, 57R9

3 850 Y SUYAMA Masuda was a slight odification of the proof in [4] but then they found an eleentary proof Another eleentary proof of Theore A is given by RT Živalević [3] Theore A does not hold when the uniodularity condition is dropped In this paper, we give a forula on the rotation nuber of a (not necessarily uniodular) sequence of priitive vectors Ú,, Ú d with i 0 for all i, see Theore 6 In this case, a i s are rational nubers but are not always integers The proof is done by adding priitive vectors in an appropriate way to the given sequence so that the enlarged sequence is uniodular and then by applying Theore A to the enlarged uniodular sequence This cobinatorial process, that is, aking the given sequence to a uniodular sequence by adding priitive vectors corresponds to resolution of singularity by blow-up in geoetry, see [] The structure of the paper is as follows: In Section 2, we discuss Hirzebruch Jung continued fractions used in our proof of the ain theore In Section 3, we state the ain theore and give an eaple In Section 4, we give a proof of the ain theore 2 Continued fractions Let 2 and ( ) be a positive integer prie to, and let n n 2, n 2 n L(,) be the continued fraction epansion This continued fraction epansion is unique, and is called a Hirzebruch Jung continued fraction We define L(, 0) 0 Lea Let 2 and ( ) be a positive integer prie to, and let n n 2, n 2 n L(,) be the continued fraction epansion Let y ( ) be a unique positive integer such that y (od ) Then the following identity holds: 0 0 n n 2 0 n L(,) y y Proof We prove this by induction on L(, )

4 ROTATION NUMBER OF PRIMITIVE VECTOR SEQUENCES 85 If L(, ), then we ust have, n and y So the lea holds when L(, ) Suppose that L(, ) 2 and the lea holds for L(, ) We have n n 2 n 3 n L(,) Since L(, ) 2, we have 2 Since the right hand side in the identity above is greater than, we have 0 n Since and are relatively prie, and n are relatively prie Moreover (y ) is a positive integer less than and (n )(y ) (od ) Hence by the hypothesis of induction, we obtain 0 n 2 0 Therefore we have 0 n L(,) n 0 0 n proving the lea for L(, ) n L(,) (n )(y ) (n ) y y n y (n ) y y n y (n ) y y, y Proposition 2 The following identity holds: y n L(,) n L(,) n Proof Let f Ï M 2 () M 2 () be the antihooorphis defined by f a c b d a c b d

5 852 Y SUYAMA By Lea, we have 0 n L(,) 0 f proving the proposition y y n 0 f y y n 0, n L(,) REMARK 3 A siilar assertion holds for regular continued fractions Let n, n n 2 n l be a continued fraction epansion, and let y ( ) be a unique positive integer such that y ( ) l (od ) Then the following identity holds: y 0 0 n n 2 0 ( )l n l The proof is siilar to Lea The following identity can be deduced by taking transpose at the identity above: y n l n l n y 3 The ain theore Let Ú,, Ú d ¾ 2 be a sequence of vectors such that i det(ú i, Ú i ) 0 for all i,, d We define Ú 0 Ú d and Ú Ú We assue that each vector is priitive, ie its coponents are relatively prie Lea 4 For each i,, d, there eists a unique non-negative integer i i such that i and i are relatively prie and P i (Ú i, Ú i ) i 0 i

6 ROTATION NUMBER OF PRIMITIVE VECTOR SEQUENCES 853 is a uniodular atri Proof Let Ú i a and Ú i c b d We assue that i 0 Since Ú i is priitive, there eist p, q ¾ such that ap bq Then we have p b q a (Ú i, Ú i ) cp dq 0 i There eists a unique n ¾ satisfying i cp dq n i 0 So we put i (cp dq n i ) Then we have Hence n 0 p b q a P i a q b p (Ú i, Ú i ) i n 0 0 i is a uniodular atri When i 0, we can show that the assertion holds by a siilar arguent Since i i P i Ú i is priitive, i and i are relatively prie Note that det(p i ) i i Siilarly, there eists a unique non-negative integer y i i such that (3) Q i (Ú i, Ú i ) y i 0 i is a uniodular atri Lea 5 Let a i i i det(ú i, Ú i ) Then a i satisfies (32) i Ú i i Ú i a i Ú i 0 Proof It is easy to check that det(ú i, Ú i )Ú i det(ú i, Ú i )Ú i det(ú i, Ú i )Ú i 0 Dividing both sides by i i det(ú i, Ú i ) det(ú i, Ú i ), we obtain (32) Let 2 and ( ) be a positive integer prie to, and let n n 2, n 2 n L(,)

7 854 Y SUYAMA be the Hirzebruch Jung continued fraction epansion We set N(, ) n n L(,) and N(, 0) 0 The following is our ain theore: Theore 6 Let Ú,,Ú d be a sequence of priitive vectors and i det(ú i,ú i ), a i i i det(ú i, Ú i ) Let i and y i be the integers defined in Lea 4 and (3) Then Rot(Ú,, Ú d ) is given by (33) 2 i (3(L( i, i )) N( i, i )) i i a i i y i i In particular, the value of (33) is an integer For i such that i 2, let i i n (i) n (i) 2 n (i) L( i, i ), n (i) 2, be the Hirzebruch Jung continued fraction epansion EXAMPLE 7 Ú 0 Let d 5 and, Ú 2 3 Then we have the following:, Ú 3 2, Ú 4 2, Ú i i a i i y i L( i, i ) n (i) n (i) 2 N( i, i )

8 ROTATION NUMBER OF PRIMITIVE VECTOR SEQUENCES 855 So we have i Fig A sequence of priitive vectors (3(L( i, i )) N( i, i )) i i (3(2) 4)(3(2) 5) (3() 4)3(3() 3) 3, i i a i , i y i i Therefore the value (33) is (2)(332 52), while the rotation nuber of the sequence Ú,, Ú 5 in Fig is clearly one 4 Proof of Theore 6 In this section, we give a proof of Theore 6 We will use the notation in Section 3 freely We need the following lea

9 856 Y SUYAMA Lea 8 0 n (i) For each i,, d, the following identity holds: 0 n (i) 2 0 n (i) L( i, i ) i y i i i Proof If i, then i y i L( i, i ) 0 and the left hand side above is understood to be the identity atri Assue i 2 By Lea 4, i and i are relatively prie Since Q i P i y i i 0 i y i 0 i (Ú i, Ú i ) (Ú i, Ú i ) i 0 i i i y i y i i i i y i i i is congruent to odulo i Therefore the lea fol- is a uniodular atri, i y i lows fro Lea Proof of Theore 6 (4) Û (i) P i 0 n (i) P i 0 n (i) For 0,, L( i, i ), we define 0 n (i) 0 n (i) 0 0 (0 L( i, i )), ( L( i, i )) Note that both epressions at the right hand side of (4) are equal if L( i, i ) By the definition of Û (i), it follows that (42), Û (i) det(p i ) det for any 0,, L( i, i ) So the sequence 0, 0 i i ¾ { } (43), Ú i Û (i) 0, Û(i),, Û(i) L( i, i ) Ú i, is uniodular

10 ROTATION NUMBER OF PRIMITIVE VECTOR SEQUENCES 857 Fig 2 Adding Û (i) to the given vector sequence (44) Hence by Theore A, the rotation nuber of Ú,, Ú d is given by 4 i 2 L( i, i ) 0 i, Û (i) 2 L( i, i ), Û(i) i (i ) det Û L( i, i ), Ú i, Û(i) (i) det(û, Û (i) As for the first suand in (44), it follows fro (42) that L( i, i ) 0, Û (i) (L( i, i )) i i ), Û(i L( i, i ) det Úi, Û (i) As for the second suand in (44), we first observe that it follows fro Lea 5 that P i P i i 0 i i i 0 i i i 0 i i (Ú i, Ú i ) (Ú i, Ú i ) i 0 i (Ú i, Ú i ) (Ú i, i ( i Ú i a i Ú i )) i i a i i 0 i i 0 i i i i i i a i i i i i i ( i a i i ) i i 0

11 858 Y SUYAMA So it follows fro (43), (4), (42), and Lea 8 that (i ) det Û L( i, i ), Ú i (i ) det Û i i i i i i i i i det ), Û(i L( i, i ) L( i, i ) det det Úi, Û (i) ), Û(i L( i, i ) ), Û(i L( i, i ) P i 0 det(p i ) det det Û (i) 0, P i n P 0 i P i, 0, Û(i) (i ) i y i i y i i i i i i y i a i i i a i i i y i i i i ( i a i i ) y i 0 (i ) n L( i, i ) As for the last suand in (44), it follows fro (4) and (42) that, Û(i), Û(i) det (i) Û, Û (i) 0 det(p i ) det n (i) det(p i ) det i i det n (i) 0 0 n (i) 0 n (i), n (i), n (i) 0 n (i) 0 i i 0 4 Therefore (44) reduces to i 2 (L( i, i )) i i 2 i i a i i i y i i 2 (3(L( i, i )) N( i, i )) i i a i i y i i i, L( i, i ) n (i) i i proving the theore

12 ROTATION NUMBER OF PRIMITIVE VECTOR SEQUENCES 859 REMARK 9 It soeties happens that a sequence of priitive vectors Ú,, Ú d is not uniodular but is uniodular with respect to the sublattice of 2 generated by vectors Ú,, Ú d Such a sequence is called an l-refleive loop and studied in [5] Theore A can be applied to an l-refleive loop with respect to the sublattice generated by the vectors in the l-refleive loop, but it is unclear whether the resulting forula can be obtained fro Theore 6 ACKNOWLEDGEMENT The author wishes to thank Professor Mikiya Masuda for his valuable advice and continuing support References [] W Fulton: Introduction to Toric Varieties, Annals of Matheatics Studies 3, Princeton Univ Press, Princeton, NJ, 993 [2] A Higashitani and M Masuda: Lattice ulti-polygons, arxiv: v3 [3] RT Živalević: Rotation nuber of a uniodular cycle: an eleentary approach, arxiv: v4, to appear in Discrete Math [4] M Masuda: Unitary toric anifolds, ulti-fans and equivariant inde, Tohoku Math J (2) 5 (999), [5] AM Kasprzyk and B Nill: Refleive polytopes of higher inde and the nuber 2, Electron J Cobin 9 (202) Departent of Matheatics Graduate School of Science Osaka City University Sugioto, Suiyoshi-ku Osaka Japan e-ail: 3saU0r3@eediaosaka-cuacp