Every sum of cubes in F2 [t] is a strict sum of 6 cubes

It is easy to see that an element P (t) ∈ F<inf>2</inf> [t] is a sum of cubes if and only ifP (t) ∈ M (2) : = {P (t) : P (t) ≡ 0  or  1 (mod t² + t + 1)} . We say that P (t) is a "strict" sum of cubes A<inf>1</inf> (t)³ + ⋯ + A<inf>g</inf> (t)³ if we have deg (A<inf>i</inf>³) ≤ deg (P) + 2 for each i, and we define g (3, F<inf>2</inf> [t]) as the least g such that every element of M (2) is a strict sum of g cubes. Our main result is then that5 ≤ g (3, F<inf>2</inf> [t]) ≤ 6 . This improves on a recent result 4 ≤ g (3, F<inf>2</inf> [t]) ≤ 9 of the first named author. © 2007 Elsevier Inc. All rights reserved.