Sums and strict sums of biquadrates in \(\mathbb F_q[t]\), \(q \in \{3,9\}\)
Let \({\mathbb F}_q[t]\) be the polynomial ring over \({\mathbb F}_q\), the finite field consisting of \(q\) elements. The main subject of this paper is Waring's problem for biquadrates in \({\mathbb F}_q[t]\), that is, additive representation for elements of \({\mathbb F}_q[t]\) by using fourth powers of polynomials in \({\mathbb F}_q[t]\), for \(q=3\) and 9. Some special terms are required in order to state the main theorems. First, a sum of the form \(e_1A_1^4+e_2A_2^4+\cdots+e_sA_s^4\) with \(e_i=\pm1\) is called a \textit{mixed} sum of \(A_i^4\). Next, when a polynomial \(P\) is written as a sum (or, a mixed sum) of \(A_i^4\) with \(A_i\in{\mathbb F}_q[t]\), the degree of at least one of the \(A_i\) is trivially greater than or equal to \(\lceil \deg(P)/4\rceil\), the least integer \(\geq\deg(P)/4\). Therefore the (mixed) sum is called \textit{strict}, if one has \(\deg(A_i)\leq\lceil\deg(P)/4\rceil\) for all \(i\). Now let \(A_4(q)\) be the set of all the polynomials that can be written as a sum of biquadrates in \({\mathbb F}_q[t]\), regardless of the number of summands. Similarly, let \(SA_4(q)\), \(MA_4(q)\) and \(MSA_4(q)\), respectively, be the sets of polynomials that can be written as a strict sum, a mixed sum and a mixed strict sum of biquadrates in \({\mathbb F}_q[t]\), regardless of the number of summands. These sets eventually coincide with the entire ring \({\mathbb F}_q[t]\) unless \(q=3\) or 9, in which case these sets are concretely described in this paper. Then, the main theorems of this paper assert that for \(q\in\{3,9\}\), (i) every \(P\in A_4(q)\) can be written as a sum of at most 8 biquadrates, (ii) every \(P\in SA_4(q)\) can be written as a strict sum of at most 14 biquadrates, (iii) every \(P\in MA_4(q)\) can be written as a mixed sum of at most 6 biquadrates, (iv) every \(P\in MSA_4(q)\) can be written as a mixed strict sum of at most 10 biquadrates. The proofs are elementary, and take advantage of several special identities, together with extensive check, executed by a computer, for polynomials of degree at most 8. The paper also discusses the case of cubes shortly, and shows that \(P\in{\mathbb F}_4[t]\) can be written as a sum of cubes of polynimials in \({\mathbb F}_4[t]\), if, and only if, \(P(r)\in{\mathbb F}_2\) for all \(r\in{\mathbb F}_4\).
scientific article; zbMATH DE number 5837630
scientific article