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On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems


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[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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