Constants as sums of polynomial cubes in characteristic 2 (Q2869042)

In a finite field \(\mathbb{F}_q\) any element can be represented as the sum of two squares or three cubes (except for \(q=4\)) of elements of \(\mathbb{F}_q\). The authors consider the similar problem of representing the elements of \(\mathbb{F}_q\) as the sums of squares or cubes of polynomials of one variable over \(\mathbb{F}_q\) when at least one of the polynomials is nonconstant.NEWLINENEWLINEIt is shown that for any \(q\) all elements of the finite field \(\mathbb{F}_q\) can be written as the sum of squares of at least three elements from \(\mathbb{F}_q[t]\). Also partial results are given for the representation as the sum of cubes of polynomials for the characteristic 2 case.