### 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) |
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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 |

### All Science Journal Classification (ASJC) codes

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

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## 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