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01.On Even Integers Not of the Form 2k + σ(m) (pp. 263-268)
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Every Polymer Over a Finite Field of Even Cardinal q>4 is a Strict Sum of Four Cubes and One Expression A2 + A (pp. 213-224) $45.00
Authors:  Luis H. Gallardo
Abstract:
Let q be a power of 2. Let M(q) = {P ∈ Fq[t]: P is a strict sum of cubes}. We know that M(q) = F2[t] if and only if q > 4. Every polynomial P ∈ M(q) is a strict sum

P = A2+A+B3+C3+D3+E3.

The polynomials A,B,C,D,E are effectively obtained from the coefficients of P. The proof uses the new result that every polynomial Q ∈ Fq[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as:

Q = F2+F +tG2.

This improves a result of Gallardo, Rahavandrainy and Vaserstein that requires three polynomials F,G,H for the strict representation Q = F2+F+GH. Observe that the latter representation may be considered as an analogue in characteristic 2 of the strict representation of a polynomial Q by three squares in odd characteristic. 


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Every Polymer Over a Finite Field of Even Cardinal q>4 is a Strict Sum of Four Cubes and One Expression A<sup>2</sup> + A (pp. 213-224)