Gallardo, Luis H.; Vaserstein, Leonid N. The strict Waring problem for polynomial rings. (English) Zbl 1220.11151 J. Number Theory 128, No. 12, 2963-2972 (2008). For any ring \(A\) and any integer \(k\geq 1\), let \(A_k\subset A\) be the set of all sums of \(k\)-th powers in \(A\). For any \(a\in A_k\), let \(w_k(a,A)\) be the least \(s\) such that \(a\) is the sum of \(s\) \(k\)-th powers. Let \(w_k(A)\) be the supremum of \(w_k(a,A)\) where \(a\) ranges over \(A_k\). Let \(k\geq 2\), \(F\) a field such that \(-1\in F_k\) and \(k\neq 0\) in \(F\). The authors prove: if \(\text{char}(F)=0 \), then \[ w_k(F)\leq w_k(F[t])\leq\frac{k^2(k-1)(w_k(-1,F)+1)}{4}; \] if \(\text{char}(F)\neq 0\), then \[ w_k(F)\leq \frac{k^2(k-1)}{2} \] and \[ w_k(F[t])\leq k+1+\frac{k^2(k-1)}{2}; \] if \(\text{char}(F)=p\neq 0 \) then \[ w_k(F)\leq p(p-1)^2k(\log_p(k)+3) \] and \[ w_k(F[t])\leq k+1+p(p-1)^2k(\log_p(k)+3); \] every polynomial in \(F[t]\) which is a strict sum of \(k\)-th powers is the strict sum of at most \(k^6\) \(k\)-th powers; every polynomial in \(F[t]_k\) of degree \(\geq k^5-1\) is the strict sum of at most \(\frac{k^3}{2}\) \(k\)-th powers. Assume that \({F^*}^k\cap F_k\) has a finite index \(K\) in \((F_k)^*\). Then \[ w_k(F)\leq K; \] if \(F\) is infinite, then \[ F_k=F,w_k(F)\leq 1+w_k(-1,F) \] and \[ F[t]_k=F[t], w_k(F[t])\leq\frac{k(K+1)}{2}. \] Assume that \(\text{card}(F_k)\geq k\). Then: \[ w_k(F[t])\leq w_k(F) (k-1)+1; \] every polynomial \(a\in F[t] \) of degree \(D\geq k^4-k^2-k+1\) is the strict sum of at most \(k(w_k(F)+\ln(k+1))+1\) \(k\)-th powers; every polynomial \(a\in F[t] \) of degree \(D\geq k^3-2k^2-k+1\) is the strict sum of at most \(k(w_k(F)+3\ln(k))+2\) \(k\)-th powers; every polynomial \(a\in F[t] \) which is the strict sum of \(k\)-th powers is the strict sum of \((k^3-2k^2-k+1)w_k(F)\) \(k\)-th powers. Reviewer: Florin Nicolae (Berlin) Cited in 2 ReviewsCited in 9 Documents MSC: 11T55 Arithmetic theory of polynomial rings over finite fields 11D85 Representation problems 11P05 Waring’s problem and variants Keywords:Waring problem; polynomial rings PDFBibTeX XMLCite \textit{L. H. Gallardo} and \textit{L. N. Vaserstein}, J. Number Theory 128, No. 12, 2963--2972 (2008; Zbl 1220.11151) Full Text: DOI References: [1] Bergelson, V.; Shapiro, D. B., Multiplicative subgroups of finite index in a ring, Proc. Amer. Math. 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