The Optimal Coefficients-Based Criterion for Primitive Quadratic Polynomials over Finite Fields

Abstract

Let 𝔽_q be a finite field and consider quadratic polynomials f(X)=X²+bX+c∈𝔽_q[X] with primitive constant term c. We construct an optimal coefficients-based determining polynomial for primitive quadratic polynomials over every finite field. More precisely, for every primitive c∈𝔽_q, its specialization is the unique monic square-free polynomial whose roots are exactly the coefficients b for which X²+bX+c is primitive. The construction is based on Lucas polynomials and Lucas atoms over finite fields. In odd characteristic, the optimal determining polynomial is the (q+1)-st Lucas atom. In characteristic $2$, this Lucas atom has multiplicity $2$ in the variable B, and the optimal polynomial is obtained by removing this multiplicity through the inverse Frobenius over A=𝔽_q[C]/(Φ_q−1(C)). We prove that the determining polynomial is unique for each fixed primitive constant term and, globally, unique as an element of A[B]. We also give equivalent criteria involving only recursively computable Lucas polynomials and, as an application, a first-zero coefficient description of the binomial order and order of an irreducible quadratic polynomial.

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BibTeX RIS

BibTeX

@misc{zhu2026optimal,
  title = {{The Optimal Coefficients-Based Criterion for Primitive Quadratic Polynomials over Finite Fields}},
  author = {Li Zhu and Hongfeng Wu},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.01327},
  eprint = {2608.01327},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.01327},
}

RIS

TY  - UNPB
TI  - The Optimal Coefficients-Based Criterion for Primitive Quadratic Polynomials over Finite Fields
AU  - Li Zhu
AU  - Hongfeng Wu
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.01327
UR  - https://arxiv.org/abs/2608.01327
AB  - Let 𝔽_q be a finite field and consider quadratic polynomials f(X)=X^2+bX+c∈𝔽_q[X] with primitive constant term c. We construct an optimal coefficients-based determining polynomial for primitive quadratic polynomials over every finite field. More precisely, for every primitive c∈𝔽_q, its specialization is the unique monic square-free polynomial whose roots are exactly the coefficients b for which X^2+bX+c is primitive. The construction is based on Lucas polynomials and Lucas atoms over finite fields. In odd characteristic, the optimal determining polynomial is the (q+1)-st Lucas atom. In characteristic $2$, this Lucas atom has multiplicity $2$ in the variable B, and the optimal polynomial is obtained by removing this multiplicity through the inverse Frobenius over A=𝔽_q[C]/(Φ_q−1(C)). We prove that the determining polynomial is unique for each fixed primitive constant term and, globally, unique as an element of A[B]. We also give equivalent criteria involving only recursively computable Lucas polynomials and, as an application, a first-zero coefficient description of the binomial order and order of an irreducible quadratic polynomial.
ER  -

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