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

For any ring A and any integer k1, let A k A be the set of all sums of k-th powers in A. For any aA 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 k2, F a field such that -1F k and k0 in F. The authors prove: if char(F)=0, then

w k (F)w k (F[t])k 2 (k-1)(w k (-1,F)+1) 4;

if char(F)0, then

w k (F)k 2 (k-1) 2


w k (F[t])k+1+k 2 (k-1) 2;

if char(F)=p0 then

w k (F)p(p-1) 2 k(log p (k)+3)


w k (F[t])k+1+p(p-1) 2 k(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 k 5 -1 is the strict sum of at most k 3 2 k-th powers. Assume that F * k F k has a finite index K in (F k ) * . Then

w k (F)K;

if F is infinite, then

F k =F,w k (F)1+w k (-1,F)


F[t] k =F[t],w k (F[t])k(K+1) 2·

Assume that card(F k )k. Then:

w k (F[t])w k (F)(k-1)+1;

every polynomial aF[t] of degree Dk 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 aF[t] of degree Dk 3 -2k 2 -k+1 is the strict sum of at most k(w k (F)+3ln(k))+2 k-th powers; every polynomial aF[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