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New Estimates and Existence Results About Irreducible Polynomials and Self-Reciprocal Irreducible Polynomials with Prescribed Coefficients Over a Finite Field

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Abstract

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we obtain improved error bounds for the numbers of irreducible monic polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field \({\mathbb F}_{q}\). The new lower bounds are used to derive some existence results about irreducible monic polynomials of degree d and self-reciprocal irreducible monic polynomials of degree 2d with roughly d/2 coefficients prescribed at positions including the middle range \(d/2-\log _q d\le j\le d/2+\log _q d\).

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References

  1. Carlitz, L.: Some theorems on irreducible reciprocal polynomials over a finite field. J. Reine Angew. Math. 227, 212–220 (1967)

    MathSciNet  Google Scholar 

  2. Cohen, S.D.: On irreducible polynomials of certain types in finite fields. Proc. Camb. Philos. Soc. 66, 335–344 (1969)

    Article  MathSciNet  Google Scholar 

  3. Cohen, S.D.: Explicit theorems on generator polynomials. Finite Fields Appl. 11, 337–357 (2005)

    Article  MathSciNet  Google Scholar 

  4. Effinger, G.W., Hayes, D.R.: Additive Number Theory of Polynomials over a Finite Field, Oxford Mathematical Monographs. Oxford University Press, Oxford (1991)

    Book  Google Scholar 

  5. Z.C. Gao, Improved error bounds for the number of irreducible polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field, Proceedings of The 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, Philadelphia (2022)

  6. Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite fEnumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite Gao, Z.C.: Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field. Finite Fields Appl. 83, 102083 (2022)

    Article  MathSciNet  Google Scholar 

  7. Gao, Z., Kuttner, S., Wang, Q.: Counting irreducible polynomials with prescribed coefficients over a finite field. Finite Fields Appl. 80, 102023 (2022)

    Article  MathSciNet  Google Scholar 

  8. Garefalakis, T.: Irreducible polynomials with consecutive zero coefficients. Finite Fields Appl. 14, 201–208 (2008)

    Article  MathSciNet  Google Scholar 

  9. Garefalakis, T., Kapetanakis, G.: On the Hansen-Mullen conjecture for self-reciprocal irreducible polynomials. Finite Fields Appl. 69, 832–841 (2012)

    Article  MathSciNet  Google Scholar 

  10. Granger, R.: On the enumeration of irreducible polynomials over \(GF(q)\) with prescribed coefficients. Finite Fields Appl. 57, 156–229 (2019)

    Article  MathSciNet  Google Scholar 

  11. Ha, J.: Irreducible polynomials with several prescribed coefficients. Finite Fields Appl. 40, 10–25 (2016)

    Article  MathSciNet  Google Scholar 

  12. Hayes, D.R.: The distribution of irreducibles in \(GF[q, x]\). Trans. Am. Math. Soc. 117, 101–127 (1965)

    MathSciNet  Google Scholar 

  13. Hsu, C.N.: The distribution of irreducible polynomials in \(\mathbb{F} _q[t]\). J. Number Theory 61(1), 85–96 (1996)

    Article  MathSciNet  Google Scholar 

  14. Meyn, H.: On the construction of irreducible self-reciprocal polynomials over finite fields. AAECC 1, 43–53 (1990)

    Article  MathSciNet  Google Scholar 

  15. Panario, D.: Open problems for polynomials over finite fields and applications. In: Chapter 5 of Proceedings on the Open Problems in Mathematics and Computer Science Conference, Springer, pp. 111–126 (2015)

  16. Panario, D., Tzanakis, G.: A generalization of the Hansen–Mullen conjecture on irreducible polynomials over finite fields. Finite Fields Appl. 18, 303–315 (2012)

    Article  MathSciNet  Google Scholar 

  17. Steinberg, B.: Representation Theory of Finite Groups. An Introductory Approach, Springer, New York (2012)

    Book  Google Scholar 

  18. Wan, D.: Generators and irreducible polynomials over finite fields. Math. Comp. 219, 1195–1212 (1997)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank Simon Kuttner for his contribution to Proposition 6(a) and Proposition 8. The research is partly supported by Carleton University Development Grant (189035).

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  1. School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S5B6, Canada

    Zhicheng Gao

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  1. Zhicheng Gao
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Gao, Z. New Estimates and Existence Results About Irreducible Polynomials and Self-Reciprocal Irreducible Polynomials with Prescribed Coefficients Over a Finite Field. La Matematica 2, 789–815 (2023). https://doi.org/10.1007/s44007-023-00062-1

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  • DOI: https://doi.org/10.1007/s44007-023-00062-1

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