The number of primitive unassociated third-order matrices of a given determinant

R. A. Dokhov ; U. M. Pachev

Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 129-137.

Résumé

When studying questions of the asymptotic distribution of integer points over domains on hyperboloids, as well as integer matrices of the second and third orders, it becomes necessary to use primitive unassociated matrices of the second and third orders of a given determinant. Counting the number of integer matrices of the same order and a given determinant requires the selection of pairwise unassociated matrices among them. Non-associated second-order matrices appear when considering preliminary ergodic theorems for flows of integer points on hyperboloids when applying the discrete ergodic method to the problem of representing integers by ternary quadratic forms. The number of unassociated second-order matrices is also used to express the number of binary quadratic forms, the arithmetic minimum of which is divisible. In addition, formulas for the number of primitive unassociated matrices of the second and third orders make it possible to determine the orders of the principal terms in asymptotic formulas for the number of integer matrices of large norm(determinant). In this paper, based on the canonical triangular form of the third-order integer matrices, a formula is obtained for the number of primitive unassociated third-order matrices represented by the canonical decomposition. A formula is also obtained for the number of primitive unassociated matrices of the third order of a given determinant, divisible by a given matrix. The main results related to the question of the number of non-associated integer matrices of a given determinant belong to Yu. V. Linnik, B. F. Skubenko, A.V. Malyshev and the authors of this work, the results of which can be further transferred to integer matrices of any order.

Détail

Keywords: integer matrix, divisibility of matrices, primitive matrix, non-associated right (left) matrices.

Bibliographie

[1] Linnik Yu. V., Ergodicheskie svoistva algebraicheskikh polei, Izdatelstvo Leningradskogo universiteta, L., 1967, 210 pp.

[2] Malyshev A. V., Pachev U. M., “Ob arifmetike matrits vtorogo poryadka”, Issledovaniya po teorii chisel. 6, Zap. nauchn. sem. LOMI, 93, Izd-vo «Nauka», Leningrad. otd., L., 1980, 41–86 | DOI

[3] Pachev U. M., “O raspredelenii tselykh tochek na nekotorykh dvupolostnykh giperboloidakh”, Issledovaniya po teorii chisel. 6, Zap. nauchn. sem. LOMI, 93, Izd-vo «Nauka», Leningrad. otd., L., 1980, 87–141 | DOI

[4] Pachev U. M., “O chisle privedennykh tselochislennykh neopredelennykh binarnykh kvadratichnykh form s usloviem delimosti pervykh koeffitsientov”, Chebyshevskii sb., 4:3 (2003), 92–105 | MR | Zbl

[5] Pachev U. M., “Predstavlenie tselykh chisel izotropnymi ternarnymi kvadratichnymi formami”, Izv. RAN. Ser. matem., 70:3 (2006), 167–184 | DOI | MR | Zbl

[6] Pachev U. M., “Ob asimptotike chisla privedennykh tselochislennykh binarnykh kvadratichnykh form s usloviem delimosti pervykh koeffitsientov”, Sib. matem. zhurn., 48:2 (2007), 376–388 | DOI | MR | Zbl

[7] Pachev U. M., “O chisle primitivnykh neassotsiirovannykh matrits vtorogo poryadka opredelitelya n, delyaschikhsya na zadannuyu matritsu”, Vladikavk. matem. zhurn., 17:2 (2015), 62–67 | MR | Zbl

[8] Pachev U. M., “Ob arifmetike koltsa tselykh matrits $n$-go poryadka”, Vladikavkazskii matematicheskii zhurnal, 10:1 (2008), 75–78 | MR | Zbl

[9] Pachev U. M., Dokhov R. A., “O primitivnykh neassotsiirovannykh matritsakh tretego poryadka zadannogo opredelitelya”, Materialy XVII Mezhdunarodnoi konferentsii “Algebra, teoriya chisel i diskretnaya geometriya: sovremennye problemy, prilozheniya i problemy istorii”, posvyaschennoi 100-letiyu so dlya rozhdeniya professora N.I. Feldmana i 90-letiyu so dnya rozhdeniya professorov A.I. Vinogradova, A.V. Malysheva i B.F. Skubenko, Izd-vo TGPU im. L.N. Tolstogo, Tula, 2019, 121–123

[10] Linnik Yu. V., Skubenko B. F., “K asimptotike tselochislennykh matrits tretego poryadka”, Dokl. AN SSSR, 146:5 (1962), 1007–1008 | Zbl

[11] Linnik Yu. V., Skubenko B. F., “Asimptoticheskoe raspredelenie tselochislennykh matrits tretego poryadka”, Vestn. LGU: Ser. Matem. Mekh. Astron., 1964, no. 13, 26–36

[12] Pachev U. M., Shakova T. A., “O edinitsakh kvaternionnogo poryadka neopredelennoi anizotropnoi ternarnoi kvadratichnoi formy”, Chebyshevskii sb., 20:4 (2019), 270–280 | DOI | MR | Zbl

[13] Newman M., Integral matrices, Academic Press, New York, 1972, 244 pp. | MR | Zbl

[14] Venkov B. A., “Ob integralnom invariante gruppy unimodulyarnykh lineinykh podstanovok”, Uchen. zap. Lening. un-ta, 144:23 (1952), 3–25

[15] Vinogradov I. M., Osnovy teorii chisel, Nauka, M., 1981, 180 pp. | MR

[16] Sidorov S. V., “O klassakh podobiya matrits vtorogo poryadka s nulevym sledom nad koltsom tselykh chisel”, Izv. vuzov. Matem., 2016, no. 4, 79–86 | DOI | Zbl

[17] Sidorov S. V., “O podobii matrits tretego poryadka nad koltsom tselykh chisel, imeyuschikh privodimyi kharakteristicheskii mnogochlen”, Vestn. NNGU.: Ser. matem. modelir. i optimalnogo upravleniya, 2009, no. 1, 119–127