Abstract
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.
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 2007 |
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Algebra and Number Theory
- General Engineering
- Applied Mathematics