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Irreducible Polynomials Over a Finite Field with Restricted Coefficients

  • Sam Porritt (a1)

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.
[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.
[3] Ha, J., Irreducible polynomials with several prescribed coefficients . Finite Field Appl. 40(2016), 1025.
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[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.
[7] Pollack, P., Irreducible polynomials with several prescribed coefficients . Finite Fields Appl. 22(2013), 7078.
[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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