1. bookVolumen 17 (2022): Edición 1 (December 2022)
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2309-5377
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30 Dec 2013
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On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems

Publicado en línea: 31 May 2022
Volumen & Edición: Volumen 17 (2022) - Edición 1 (December 2022)
Páginas: 29 - 54
Recibido: 08 Jul 2021
Aceptado: 23 Nov 2021
Detalles de la revista
License
Formato
Revista
eISSN
2309-5377
Primera edición
30 Dec 2013
Calendario de la edición
2 veces al año
Idiomas
Inglés
Abstract

The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζβ(z)of the β-shift, for β> 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in 𝔽p and their factorizations, via Kronecker’s Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in 𝔽p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.

Keywords

MSC 2010

[1] ARTIN, M.—MAZUR, B.: On periodic points, Ann. Math. 81 (1965), 82–99.10.2307/1970384 Search in Google Scholar

[2] BALADI, V.—KELLER, G.: Zeta functions and transfer operators for piecewise monotone transformations,Comm. Math.Phys. 127 (1990), 459–479.10.1007/BF02104498 Search in Google Scholar

[3] BREUILLARD, E.—VARJÚ, P. P.: Irreducibility of random polynomials of large degree (2019); https://arxiv.org/pdf/1810.13360.pdf Search in Google Scholar

[4] BRANDL, R.: Integer polynomials that are reducible modulo all primes,Amer. Math. Monthly 93 (1986), 286–288.10.1080/00029890.1986.11971807 Search in Google Scholar

[5] CHEBOTAREV, N. G.: Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionklass gehören, Math. Ann. 95 (1925), 191–228. Search in Google Scholar

[6] COHEN, H.—STRÖMBERG, F.: Modular Forms Vol. 179. Graduate Studies in Mathematics. A Classical Approach. Amer. Math. Soc., Providence, RI, 2017.10.1090/gsm/179 Search in Google Scholar

[7] COHEN, S. D.—MOVAHHEDI, A.—SALINIER, A.: Galois groups of trinomials, J. Algebra 222 (1999), 561–573.10.1006/jabr.1999.8033 Search in Google Scholar

[8] CRESPO, T.: Galois representations, embedding problems and modular forms, Collect-anea Math. 48 (1997), 63–83. Search in Google Scholar

[9] DOBROWOLSKI, E.—FILASETA, M.—VINCENT, A. F.: The non-cyclotomic part of f (x)xn + g(x) and roots of reciprocal polynomials off the unit circle, Int.J.Number Theory 9 (2013), 1865–1877.10.1142/S1793042113500620 Search in Google Scholar

[10] DUTYKH, D.—VERGER-GAUGRY, J.-L.: On the reducibility and the lenticular sets of zeroes of almost Newman lacunary polynomials, Arnold Math. J. 4 (2018), no. 3–4, 315–344. Search in Google Scholar

[11] DUTYKH, D. —VERGER-GAUGRY, J.-L.: Alphabets, rewriting trails and periodic representations in algebraic bases, Res. Number Theory 7 (2021), art. no. 64. Search in Google Scholar

[12] FILASETA, M.: On the factorization of polynomials with small Euclidean norm, In: Number Theory in Progress, Vol. 1 (Zakopane-Kościelisko, 1997), De Gruyter, Berlin, 1999, pp. 143–163.10.1515/9783110285581.143 Search in Google Scholar

[13] FILASETA, M.—FINCH, C.—NICOL, C.: On three questions concerning 0, 1-polynomials,J. Théorie Nombres Bordeaux 18 (2006), 357–370.10.5802/jtnb.549 Search in Google Scholar

[14] FILASETA, M.—FORD, K.—KONYAGIN, S.: On a irreducibility theorem of A. Schinzel associated with coverings of the integers, Illinois J. Math. 44 (2000), 633–643.10.1215/ijm/1256060421 Search in Google Scholar

[15] FILASETA, M.—MATTHEWS, M.: On the irreducibility of 0, 1-polynomials of the form f (x)xn + g(x), Colloq. Math. 99 (2004), 1–5.10.4064/cm99-1-1 Search in Google Scholar

[16] FINCH, C.—JONES, L.: On the Irreducibility of −1, 0, 1- Quadrinomials, Integers 6 (2006), art. no. 16. Search in Google Scholar

[17] FLATTO, L.—LAGARIAS, J. C.—POONEN, B.: The zeta function of the beta transformation, Ergodic Theory Dynam. Systems. 14 (1994), 237–266.10.1017/S0143385700007860 Search in Google Scholar

[18] FROBENIUS, F. G.: Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe, Sitz. Akad. Wiss. Berlin (1896), 689–703. Search in Google Scholar

[19] GUPTA, S.: Irreducible polynomials in ℤ[x] that are reducible modulo all primes,Open Journal of Discrete Mathematics 9 (2019), 52–61.10.4236/ojdm.2019.92006 Search in Google Scholar

[20] R. GURALNICK, R.—SCHACHER, M. M.—SONN, J.: Irreducible polynomials wich are locally reducible everywhere, Proc. Amer. Math. Soc. 133 (2005), 3171–3177.10.1090/S0002-9939-05-07855-X Search in Google Scholar

[21] HARRINGTON, J.—VINCENT, A.—WHITE, D.: The factorization of f (x)xn + g(x) with f (x) monic and of degree ≤ 2, J. Théor. Nombres Bordeaux 25 (2013), 565–578.10.5802/jtnb.849 Search in Google Scholar

[22] ITÔ, S.—TAKAHASHI, Y.: Markov subshifts and realization of β-expansions,J. Math. Soc. Japan 26 (1974), 33–55.10.2969/jmsj/02610033 Search in Google Scholar

[23] KRONECKER, L.: Über die Irreductibilität von Gleichungen, Sitz. Akad. Wiss. Berlin (1880), 689–703 (Berl. Monatsber. 1880, 155–162). Search in Google Scholar

[24] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences, John Wiley & Sons, New York, 1974. Search in Google Scholar

[25] LMFDB-COLLABORATION: The L-functions and Modular Forms Database, [Online; accessed March 15, 2022] http://www.lmfdb.org/ Search in Google Scholar

[26] LAGARIAS, J. C.: Number theory zeta functions and dynamical zeta functions,Contemp. Math. 237 (1999), 45–86.10.1090/conm/237/1710789 Search in Google Scholar

[27] LENSTRA, JR., H. W.—STEVENHAGEN, P.: Artin reciprocity and Mersenne primes, Nieuw Arch. Wiskd. 1 (2000), no. 5, 44–54. Search in Google Scholar

[28] LOTHAIRE, M.: Algebraic Combinatorics on Words,In: Encylopedia of Mathematics and its Applications Vol. 90, Cambridge University Press, Cambridge 2002. Search in Google Scholar

[29] ONO, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series,In: CBMS Regional Conference Series in Mathematics Vol. 102 (Published for the Conference Board of the Mathematical Sciences Washington, DC), Amer. Math. Soc, Providence, RI, 2004.10.1090/cbms/102 Search in Google Scholar

[30] PARRY, W.: On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.10.1007/BF02020954 Search in Google Scholar

[31] POLLICOTT, M.: Dynamical zeta functions, In: (Anatole Katok, ed. et al.) Smooth ergodic theory and its applications. (Seattle, WA, 1999), Proc. Sympos. Pure Math. Vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 409–427.10.1090/pspum/069/1858541 Search in Google Scholar

[32] PRASOLOV, V. V.: Polynomials. In: Algorithms and Computation in Mathematics Vol. 11, Springer-Verlag, Berlin, 2004. Search in Google Scholar

[33] ROSEN, M.: Polynomials modulo p and the theory of Galois sets. In: (Michel Lavrauw, (ed.) et al.), Theory and Applications of Finite Fields. (The 10th International Conference on Finite Fields and Their Applications, July 11-15, 2011, Ghent, Belgium.) In: Contemp. Math. Vol. 579, Amer. Math. Soc., Providence, RI, 2012. pp. 163–178. Search in Google Scholar

[34] RUBINSTEIN, M.—SARNAK, P.: Chebyshev’s Bias, Experiment. Math. 3 (1994), 173–197.10.1080/10586458.1994.10504289 Search in Google Scholar

[35] SAWIN, W.—SHUSTERMAN, M.—STOLL, M.: Irreducibility of polynomials with a large gap,Acta Arith. 192 (2020), 111–139. Search in Google Scholar

[36] SCHINZEL, A.: Reducibility of polynomials and covering systems of congruences,Acta Arith. 13 (1967/1968), 91–101.10.4064/aa-13-1-91-101 Search in Google Scholar

[37] SCHINZEL, A.: Reducibility of lacunary polynomials. I, Acta Arith. 16 (1969/1970), 123–159.10.4064/aa-16-2-123-160 Search in Google Scholar

[38] SCHINZEL, A.: On the number of irreducible factors of a polynomial, Colloq. Math. Soc. Janos Bolyai 13 (1976), 305–314. Search in Google Scholar

[39] SCHINZEL, A.: Reducibility of lacunary polynomials III, Acta Arith. 34 (1978), 227–266.10.4064/aa-34-3-227-266 Search in Google Scholar

[40] SCHINZEL, A.: On the number of irreducible factors of a polynomial II, Ann. Polon. Math. 42 (1983), 309–320.10.4064/ap-42-1-309-320 Search in Google Scholar

[41] SCHINZEL, A.: Polynomials with special regard to reducibility. (With an appendix by Umberto Zannier). In: Encyclopedia od Mathematics and its Applications Vol. 77, Cambridge University Press, Cambridge, 2000.10.1017/CBO9780511542916 Search in Google Scholar

[42] SELMER, E. S.: On the irreducibility of certain tinomials, Math. Scand. 4 (1956), 287–302.10.7146/math.scand.a-10478 Search in Google Scholar

[43] SERRE, J.-P.: On a theorem of Jordan, Bull. Amer. Math. Soc. (N.S.) 40 (2003), 429–440.10.1090/S0273-0979-03-00992-3 Search in Google Scholar

[44] SERRE, J.-P.: Number of points modulo p when p tends to infinity, Oppenheim Lecture, Institute for Mathematical Science, Jointly org. with Department of Mathematics, NUS, 2018; https://www.youtube.com/watch?v=CoGMWDCmfUQ Search in Google Scholar

[45] SERRE, J.-P.: Counting solutions mod p and letting p tend to infinity, Minerva Lectures 2012, Princeton University, Princeton, 2012; https://www.math.princeton.edu/events/inaugural-minerva-lectures-iii-counting--solutions-mod-p-and-letting-p-tend-infinity-2012-10 Search in Google Scholar

[46] SMYTH, C.: The Mahler measure of algebraic numbers: a survey.In: Number theory and polynomials,In: London Math. Soc. Lecture Note Ser. Vol. 352, Cambridge Univ. Press, Cambridge, 2008, pp. 322–349. Search in Google Scholar

[47] STEVENHAGEN, P.—LENSTRA, JR., H. W.: Chebotarev and his density theorem, Math. Intelligencer 18 (1996), 26–37.10.1007/BF03027290 Search in Google Scholar

[48] STRAUCH, O.: Distribution of Sequences: A Theory. VEDA, Publishing House of the Slovak Academy of Sciences; Bratislava; Academia, Centre of Administration and Operations of the CAS Prague, 2019. Search in Google Scholar

[49] TAKAHASHI, Y.: Isomorphisms of β-automorphisms to Markov automorphisms, Osaka J. Math. 10 (1973), 175–184. Search in Google Scholar

[50] VERGER-GAUGRY, J.-L.: On gaps in Rényi β-expansions of unity for β> 1 an algebraic number, Ann. Inst. Fourier (Grenoble), 56 (2006), 2565–2579.10.5802/aif.2250 Search in Google Scholar

[51] VERGER-GAUGRY, J.-L.: On the conjecture of Lehmer, limit Mahler measure of trinomials and asymptotic expansions, Unif. Distrib. Theory 11 (2016), 79–139.10.1515/udt-2016-0006 Search in Google Scholar

[52] VERGER-GAUGRY, J.-L.: A panorama on the minoration of the Mahler measure: from the problem of Lehmer to its reformulations in topology and geometry, (2020), HAL archives-ouvertes; https://hal.archives-ouvertes.fr/hal-03148129/document Search in Google Scholar

[53] VERGER-GAUGRY, J.-L.: A proof of the conjecture of Lehmer; http://arxiv.org/abs/1911.10590(29Oct2021) Search in Google Scholar

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