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Irreducible Polynomials Over a Finite Field with Restricted Coefficients

  • Sam Porritt (a1)
Abstract

We prove a function field analogue of Maynard’s celebrated result about primes with restricted digits. That is, for certain ranges of parameters $n$ and $q$ , we prove an asymptotic formula for the number of irreducible polynomials of degree $n$ over a finite field $\mathbb{F}_{q}$ whose coefficients are restricted to lie in a given subset of $\mathbb{F}_{q}$ .

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This work was supported by the Engineering and Physical Sciences Research Council EP/L015234/1 via the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London.

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[1] Dartyge, C., Mauduit, C., and Sárközy, A., Polynomial values and generators with missing digits in finite fields . Funct. Approx. Comment. Math. 52(2015), 6574. https://doi.org/10.7169/facm/2015.52.1.5.
[2] Dietmann, R., Elsholtz, C., and Shparlinski, I., Prescribing the binary digits of squarefree numbers and quadratic residues . Trans. Amer. Math. Soc. 369(2017), 83698388. https://doi.org/10.1090/tran/6903.
[3] Ha, J., Irreducible polynomials with several prescribed coefficients . Finite Field Appl. 40(2016), 1025. https://doi.org/10.1016/j.ffa.2016.02.006.
[4] Hayes, D. R., The expression of a polynomial as a sum of three irreducibles . Acta Arith. 11(1966), 461488. https://doi.org/10.4064/aa-11-4-461-488.
[5] Maynard, J., Primes with restricted digits. 2016. arxiv:1604.01041.
[6] Oppenheim, A. and Shusterman, M., Squarefree polynomials with prescribed coefficients . J. Number Theory 187(2018), 189197. https://doi.org/10.1016/j.jnt.2017.10.025.
[7] Pollack, P., Irreducible polynomials with several prescribed coefficients . Finite Fields Appl. 22(2013), 7078. https://doi.org/10.1016/j.ffa.2013.03.001.
[8] Tuxanidy, A. and Wang, Q., Irreducible polynomials with prescribed sums of coefficients. 2016. arxiv:1605.00351.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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