TY - JOUR
T1 - A lower bound for periods of matrices
AU - Corvaja, Pietro
AU - Rudnick, Zéev
AU - Zannier, Umberto
N1 - Funding Information:
This work was supportedb y gran~f rom the ConsiglioN azionaled elle Ricerche,R oma, and from the FAPESP, Sao Paulo, Brasil. We are in-debtedt o, and wish to thank Dr. E. Nonato, InstitutoO ceanograficoS,i o Paulo, and Dr. A. BerberianI,n stitutod e Tecnotogiad e.~dimentos, Se~aod eP escadosS,a ntosB, razil,f ort hec ollection of the Brazilianm olluscs.
PY - 2004/12
Y1 - 2004/12
N2 - For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values of N for which the order of A modulo N is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of A modulo N goes to infinity faster than any constant multiple of log N.
AB - For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values of N for which the order of A modulo N is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of A modulo N goes to infinity faster than any constant multiple of log N.
UR - http://www.scopus.com/inward/record.url?scp=11244271621&partnerID=8YFLogxK
U2 - 10.1007/s00220-004-1184-6
DO - 10.1007/s00220-004-1184-6
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AN - SCOPUS:11244271621
VL - 252
SP - 535
EP - 541
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
SN - 0010-3616
IS - 1-3
ER -