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Let  p  be a prime and let  K  and      A i {A_i}       denote polynomials whose coefficients are elements of the finite field with  p  elements. Problems concerning the expression of an arbitrary polynomial  K  as sums of a small number of squares or cubes of polynomials      A i {A_i}       are discussed. In the problems treated the degrees of the      A i {A_i}       are restricted to be as small as possible. In particular, it is shown that at least five cubes are necessary and that three squares seem to suffice in all but one special case.

Numerical results for Waring’s problem in 𝐺𝐹[𝑞,𝑥]