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 language English (US) 648-658 11 Finite Fields and their Applications 13 3 https://doi.org/10.1016/j.ffa.2005.11.007 Published - Jul 1 2007

Galois field
Polynomials
Polynomial
Ternary
Odd
Analogue
Distinct

All Science Journal Classification (ASJC) codes

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

Cite this

title = "Representations of polynomials over finite fields of characteristic two as A2 + A + B C + D3",
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.",
author = "Luis Gallardo and Olivier Rahavandrainy and Vaserstein, {Leonid N.}",
year = "2007",
month = "7",
day = "1",
doi = "10.1016/j.ffa.2005.11.007",
language = "English (US)",
volume = "13",
pages = "648--658",
journal = "Finite Fields and Their Applications",
issn = "1071-5797",
publisher = "Academic Press Inc.",
number = "3",

}

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

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.

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U2 - 10.1016/j.ffa.2005.11.007

DO - 10.1016/j.ffa.2005.11.007

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