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

Luis Gallardo, Olivier Rahavandrainy, Leonid N. Vaserstein

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)648-658
Number of pages11
JournalFinite Fields and their Applications
Volume13
Issue number3
DOIs
StatePublished - Jul 1 2007

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Galois field
Polynomials
Polynomial
Quadratic Polynomial
Ternary
Odd
Analogue
Distinct

All Science Journal Classification (ASJC) codes

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

Cite this

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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.",
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Representations of polynomials over finite fields of characteristic two as A2 + A + B C + D3. / Gallardo, Luis; Rahavandrainy, Olivier; Vaserstein, Leonid N.

In: Finite Fields and their Applications, Vol. 13, No. 3, 01.07.2007, p. 648-658.

Research output: Contribution to journalArticle

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