On the restricted Waring problem over ${\displaystyle \_\left\{2^n\right\}\left[t\right]}$

1. Introduction. The Waring problem for polynomial cubes over a finite field F of characteristic 2 consists in finding the minimal integer m ≥ 0 such that every sum of cubes in F[t] is a sum of m cubes. It is known that for F distinct from ₂, ₄, ${\displaystyle \_\left\{16\right\}}$, each polynomial in F[t] is a sum of three cubes of polynomials (see [3]). If a polynomial P ∈ F[t] is a sum of n cubes of polynomials in F[t] such that each cube A³ appearing in the decomposition has degree < deg(P)+3, we say that P is a restricted sum of n cubes. The restricted Waring problem for polynomial cubes consists in finding the minimal integer m ≥ 0 such that each sum of cubes S in F[t] is a restricted sum of m cubes. The best known result for the above problem is that every polynomial in ${\displaystyle \_\left\{2^n\right\}\left[t\right]}$ of sufficiently high degree that is a sum of cubes, is a restricted sum of eleven cubes. This result was obtained by the circle method in [1]. Here we improve this result using elementary methods. Let F be a finite field of characteristic 2, distinct from ₂, ₄, ${\displaystyle \_\left\{16\right\}}$. In Theorem 7, we prove that every polynomial in F[t] is a restricted sum of at most nine cubes, and that every polynomial in ${\displaystyle \_\left\{16\right\}\left[t\right]}$ is a restricted sum of at most ten cubes. We also prove, in Theorem 9, that by adding to a given ${\displaystyle P{\in }_{2^n}\left[t\right]}$ some square B² with deg(B²) < deg(P) + 2, the resulting polynomial is a restricted sum of at most four cubes, for all n ≠ 2.