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The strict Waring problem for polynomial rings. (English) Zbl 1220.11151
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.

11T55Arithmetic theory of polynomial rings over finite fields
11D85Representation problems of integers
11P05Waring’s problem and variants
Full Text: DOI
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