On the enumeration of irreducible polynomials over GF(q) with prescribed coefficients

Granger, Robert

doi:10.1016/j.ffa.2019.01.001

Cited by 13 publications

(15 citation statements)

References 47 publications

(102 reference statements)

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“…where I q (n; φ −1 ( c)) counts all polynomials formed in (1) and the second term corresponds to all polynomials formed in (2). Equation (8) immediately implies the following bounds…”

Section: Properties Of Self-reciprocal Irreducible Monic Polynomialsmentioning

confidence: 97%

Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field

Gao¹

2021

Preprint

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A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients. Asymptotic expression with explicit error bound is derived, which is used to show that such polynomials with degree 2n always exist provided that the number of prescribed leading coefficients is slightly less than n/4. Exact expressions are also obtained for fields with two or three elements and up to two prescribed leading coefficients.

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“…where I q (n; φ −1 ( c)) counts all polynomials formed in (1) and the second term corresponds to all polynomials formed in (2). Equation (8) immediately implies the following bounds…”

Section: Properties Of Self-reciprocal Irreducible Monic Polynomialsmentioning

confidence: 97%

Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field

Gao¹

2021

Preprint

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“…Granger [8] gave the first systematic approach to enumerating irreducible polynomials with prescribed leading coefficients. He converted the problem into the problem of counting points on Artin-Schreier curves.…”

Section: (B) F Has Degree 2d If and Only If There Is A Monic Polynomi...mentioning

confidence: 99%

“…A generating function approach is used in [7] and the main result in this paper is stated as Theorem 2 below. Following Granger's notation [8], we set…”

Section: (B) F Has Degree 2d If and Only If There Is A Monic Polynomi...mentioning

confidence: 99%

Improved error bounds for the number of irreducible polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field

Gao¹

2021

Preprint

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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 always exist provided that the number of prescribed leading coefficients is slightly less than d/2.

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“…Also, an explicit expression for the number of irreducible polynomials over F 2 r with the first three coefficients prescribed zero was given by Ahmadi et al [1]; the proofs involve counting the number of points on certain algebraic curves over finite fields which are supersingular. More recently, Granger [14] carried out a systematic study on the problem with several prescribed leading coefficients. Through a transformation of the problem of counting the number of elements of F q n with prescribed traces into the problem of counting the number of elements for which linear combinations of the trace functions evaluate to 1, he converted the problem into counting points in Artin-Schreier curves of smaller genus and then computed the corresponding zeta functions using Lauder-Wan algorithm [21].…”

Section: Introductionmentioning

confidence: 99%

“…We demonstrate our method by computing these numbers for several concrete examples. Our method is also computationally simpler than that of Granger [14] in the case of prescribed leading coefficients only and it produces simpler formulas in some cases.…”

Section: Introductionmentioning

confidence: 99%

Counting irreducible polynomials with prescribed coefficients over a finite field

Gao¹,

Kuttner²,

Wang³

2021

Preprint

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We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by equivalent classes of polynomials with prescribed coefficients. Simplified expressions are derived for some special cases. Our results extend some earlier results.

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On the enumeration of irreducible polynomials over GF(q) with prescribed coefficients

Cited by 13 publications

References 47 publications

Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field

Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field

Improved error bounds for the number of irreducible polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field

Counting irreducible polynomials with prescribed coefficients over a finite field

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