Every strict sum of cubes in <b>F</b>₄[<var>t</var>] is a strict sum of 6 cubes

Abstract

It is easy to see that an element P(t) ∈ F4[ t ] is a strict sum of cubes if and only if P(t) ∈ M(4) where

M(4) = { P(t) ∈ F4[ t ] | P(r) ∈ {0,1} for all r ∈ F4 and such that either 3 does not divide deg(P(t)), or 3 does divide deg(P(t)) and P(t) is monic }.

We say that P(t) is a “strict” sum of cubes A1(t)3 +···+ Ag(t)3 if deg(Ai3) < deg(P) + 3 for each i, and we define g(3,F4[ t ]) as the least g such that every element of M(4) is a strict sum of g cubes. The main result is that

g(3,F4[ t ]) ≤ 6.

This improves an earlier result of the author that g(3,F4[ t ]) ≤ 9.