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Constants as sums of polynomial cubes in characteristic 2. (English) Zbl 1308.11102
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. It 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.
MSC:
11T55Arithmetic theory of polynomial rings over finite fields
11P05Waring’s problem and variants
11T06Polynomials over finite fields or rings