### Abstract

Let F_{q} be a finite field of characteristic 2, with q elements. If q ≥ 8 then every polynomial P ∈ F_{q} [t] has a strict representation P = A^{2} + A + B C, i.e.:max (deg (A^{2}), deg (B^{2}), deg (C^{2})) < deg (P) + 2 . When q ≤ 4 we display the finite list of polynomials that are not of the above form. More generally, the representation of P by a ternary quadratic polynomial Q (A, B, C) is studied. Furthermore, we show that every polynomial P ∈ F_{q} [t] has a strict representation P = A^{2} + A + B C + D^{3}, i.e.:{(max (deg (A^{2}), deg (B^{2}), deg (C^{2})) < deg (P) + 2,; deg (D^{3}) < deg (P) + 3 .). This is an analogue of a result of Serre: for q odd, every polynomial in F_{q} [t] is a strict sum of 3 squares, where either q ≠ 3, or q = 3 and P is distinct from some finite number of polynomials.

Original language | English (US) |
---|---|

Pages (from-to) | 648-658 |

Number of pages | 11 |

Journal | Finite Fields and their Applications |

Volume | 13 |

Issue number | 3 |

DOIs | |

State | Published - Jul 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Algebra and Number Theory
- Engineering(all)
- Applied Mathematics

### Cite this

^{2}+ A + B C + D

^{3}.

*Finite Fields and their Applications*,

*13*(3), 648-658. https://doi.org/10.1016/j.ffa.2005.11.007

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^{2}+ A + B C + D

^{3}',

*Finite Fields and their Applications*, vol. 13, no. 3, pp. 648-658. https://doi.org/10.1016/j.ffa.2005.11.007

**Representations of polynomials over finite fields of characteristic two as A ^{2} + A + B C + D^{3}.** / Gallardo, Luis; Rahavandrainy, Olivier; Vaserstein, Leonid N.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Representations of polynomials over finite fields of characteristic two as A2 + A + B C + D3

AU - Gallardo, Luis

AU - Rahavandrainy, Olivier

AU - Vaserstein, Leonid N.

PY - 2007/7/1

Y1 - 2007/7/1

N2 - Let Fq be a finite field of characteristic 2, with q elements. If q ≥ 8 then every polynomial P ∈ Fq [t] has a strict representation P = A2 + A + B C, i.e.:max (deg (A2), deg (B2), deg (C2)) < deg (P) + 2 . When q ≤ 4 we display the finite list of polynomials that are not of the above form. More generally, the representation of P by a ternary quadratic polynomial Q (A, B, C) is studied. Furthermore, we show that every polynomial P ∈ Fq [t] has a strict representation P = A2 + A + B C + D3, i.e.:{(max (deg (A2), deg (B2), deg (C2)) < deg (P) + 2,; deg (D3) < deg (P) + 3 .). This is an analogue of a result of Serre: for q odd, every polynomial in Fq [t] is a strict sum of 3 squares, where either q ≠ 3, or q = 3 and P is distinct from some finite number of polynomials.

AB - Let Fq be a finite field of characteristic 2, with q elements. If q ≥ 8 then every polynomial P ∈ Fq [t] has a strict representation P = A2 + A + B C, i.e.:max (deg (A2), deg (B2), deg (C2)) < deg (P) + 2 . When q ≤ 4 we display the finite list of polynomials that are not of the above form. More generally, the representation of P by a ternary quadratic polynomial Q (A, B, C) is studied. Furthermore, we show that every polynomial P ∈ Fq [t] has a strict representation P = A2 + A + B C + D3, i.e.:{(max (deg (A2), deg (B2), deg (C2)) < deg (P) + 2,; deg (D3) < deg (P) + 3 .). This is an analogue of a result of Serre: for q odd, every polynomial in Fq [t] is a strict sum of 3 squares, where either q ≠ 3, or q = 3 and P is distinct from some finite number of polynomials.

UR - http://www.scopus.com/inward/record.url?scp=33947656483&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33947656483&partnerID=8YFLogxK

U2 - 10.1016/j.ffa.2005.11.007

DO - 10.1016/j.ffa.2005.11.007

M3 - Article

VL - 13

SP - 648

EP - 658

JO - Finite Fields and Their Applications

JF - Finite Fields and Their Applications

SN - 1071-5797

IS - 3

ER -

^{2}+ A + B C + D

^{3}. Finite Fields and their Applications. 2007 Jul 1;13(3):648-658. https://doi.org/10.1016/j.ffa.2005.11.007