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Improved Error Bounds for the Number of Irreducible Polynomials and Self-Reciprocal Irreducible Monic Polynomials with Prescribed Coefficients over a Finite Field

Author Zhicheng Gao



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Zhicheng Gao
  • School of Mathematics and Statistics, Carleton University, Canada

Acknowledgements

I would like to thank the referees for their helpful comments which improve the presentation of the paper.

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Zhicheng Gao. Improved Error Bounds for the Number of Irreducible Polynomials and Self-Reciprocal Irreducible Monic Polynomials with Prescribed Coefficients over a Finite Field. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 9:1-9:13, Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik (2022)
https://doi.org/10.4230/LIPIcs.AofA.2022.9

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 number of irreducible polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field. The improved bounds imply that self-reciprocal irreducible monic polynomials with degree 2d and prescribed 𝓁 leading coefficients always exist provided that 𝓁 is slightly less than d/2.

Subject Classification

ACM Subject Classification
  • Mathematics of computing β†’ Generating functions
  • Mathematics of computing β†’ Enumeration
Keywords
  • finite fields
  • irreducible polynomials
  • prescribed coefficients
  • generating functions
  • Weil bounds
  • self-reciprocal

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References

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