On the inverse stability of $z^{n}+c$

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Abstract

Let ϕ(z) = z^d + c be a polynomial over a field K. We study the inverse stability of ϕ(z) over K. In this paper, we establish some sufficient conditions for the inverse stability of ϕ(z) over the field of rational numbers and a function field. Furthermore, we also provide necessary and sufficient conditions for the inverse stability of ϕ(z) over a finite field.

Keywords

stability / polynomial irreducibility / iterations / arithmetic dynamics

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References

[1] R.W.K. Odoni, On the prime divisors of the sequence w_n+1 = 1 + w₁ ... w_n , J. London Math. Soc., 32 (1985), 1-11. https://doi.org/10.1112/jlms/s2-32.1.1

[2] M. Ayad, D.L. McQuillan, Irreducibility of the iterates of a quadratic polynomial over a field, Acta Arith., 93 (2000), 87-97. https://doi.org/10.4064/aa-93-1-87-97

[3] D. Gómez, A.P. Nicolás, An estimate on the number of stable quadratic polynomials, Finite Fields Appl., 16 (2010), 401-405. https://doi.org/10.1016/j.ffa.2010.06.005

[4] R. Jones, N. Boston, Settled polynomials over finite fields, Proc. Am. Math. Soc., 140 (2012), 1849-1863. https://doi.org/10.1090/S0002-9939-2011-11054-2

[5] P. Illig, R. Jones, E. Orvis, Y. Segawa, N. Spinale, Newly reducible polynomial iterates, Int. J. Number Theory, 17 (2021), 1405-1427. https://doi.org/10.1142/S1793042121500433

[6] L. Danielson, B. Fein, On the irreducibility of the iterates of xⁿ − b , Proc. Am. Math. Soc., 130 (2002), 1589-1596. https://doi.org/10.1090/S0002-9939-01-06258-X

[7] A. Fernandes, D. Panario, L. Reis, Stable binomials over finite fields, Finite Fields Appl., 101 (2025), 102520. https://doi.org/10.1016/j.ffa.2024.102520

[8] L. Shanta, S. Ritumoni, S. Himanshu, Stability of certain higher degree polynomials, Int. J. Number Theory, 20 (2024), 229-240. https://doi.org/10.1142/S1793042124500118

[9] L. Tong, W. Qiang, On the stable polynomials of degrees 2, 3, 4, Finite Fields Appl., 99 (2024), 102474. https://doi.org/10.1016/j.ffa.2024.102474

[10] N. Ali, Stabilité des polynômes, Acta Arith., 119 (2005), 53-63. https://doi.org/10.4064/aa119-1-4

[11] L. Mrai, A. Ostafe, Dynamical irreducibility of polynomials modulo primes, Math. Z., 298 (2021), 1187-1199. https://doi.org/10.1007/s00209-020-02630-5

[12] R. Lucas, On the factorization of iterated polynomials, Rev. Mat. Iberoam., 36 (2020), 1957-1978. https://doi.org/10.4171/rmi/1187

[13] R. Jones, L. Alon, Eventually stable rational functions, Int. J. Number Theory., 13 (2017), 2299-2318. https://doi.org/10.1142/S1793042117501263

[14] K. Cheng, A new direction on constructing irreducible polynomials over finite fields, Finite Fields Appl., 95 (2024), 102368. https://doi.org/10.1016/j.ffa.2024.102368

[15] H. Kriger, Primitive prime divisors in the critical orbit of z^d + c , Int. Math. Res. Not., 23 (2013), 5498-5525. https://doi.org/10.1093/imrn/rns213

[16] P. Ingram, Lower bounds on the canonical height associated to the morphism z^d + c , Monatsh. Math., 157 (2007), 69-89. https://doi.org/10.1007/s00605-008-0018-6

[17] P. Ingram, Canonical heights and preperiodic points for certain weighted homogeneous families of polynomials, Int. Math. Res. Not., 15 (2019), 4859-4879. https://doi.org/10.1093/imrn/rnx291

[18] S. Huczynska, G.L. Mullen, D. Panario, D. Thomson, Existence and properties of k-normal elements over finite fields, Finite Fields Appl., 24 (2013), 170-183. https://doi.org/10.1016/j.ffa.2013.07.004

[19] G. Karpilovsky, Topics in Field Theory , North-Holland Mathematics Studies, North-Holland Publishing Company, Amsterdam, 1989.

[20] S. Lang, Algebra, Graduate Texts in Mathematics , Springer, Verlag, 2002.

[21] J.H. Silverman, The Arithmetic of Elliptic Curves , 2^nd edition, Graduate Texts in Mathematics, Springer, Dordrecht, 2009.

[22] R. Lidl, H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications , Cambridge University Press, Cambridge, 1996.

[23] G. Dattoli, C. Cesarano, On a new family of Hermite polynomials associated to parabolic cylinder functions, Appl. Math. Comput., 141 (2003), 143-149. https://doi.org/10.1016/S0096-3003(02)00328-4

[24] G. Dattoli, P.E. Ricci, C. Cesarano, The Lagrange polynomials, the associated generalizations, and the umbral calculus, Integr. Transforms Spec. Funct., 14 (2003), 181-186. https://doi.org/10.1080/1065246031000098186

[25] M.M. Bhatti, M. Marin, R. Ellahi, I.M. Fudulu, Insight into the dynamics of EMHD hybrid nanofluid (ZnO/CuO-SA) flow through a pipe for geothermal energy applications, J. Therm. Anal. Calorim., 148 (2023), 14261-14273. https://doi.org/10.1007/s10973-023-12565-8

[26] A.K. Yadav, E. Carrera, M. Marin, M.I.A. Othman, Reflection of hygrothermal waves in a Nonlocal Theory of coupled thermo-elasticity, Mech. Adv. Mater. Struct., 31 (2022), 1083-1096. https://doi.org/10.1080/15376494.2022.2130484

[27] M. Marin, I. Abbas, R. Kumar, Relaxed Saint-Venant principle for thermoelastic micropolar diffusion, Struct. Eng. Mech., 51 (2014), 651-662. https://doi.org/10.12989/sem.2014.51.4.651