zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

MSC:
11T55Arithmetic theory of polynomial rings over finite fields
11D85Representation problems of integers
11P05Waring’s problem and variants
WorldCat.org
Full Text: DOI
References:
[1] Bergelson, V.; Shapiro, D. B.: Multiplicative subgroups of finite index in a ring, Proc. amer. Math. soc. 116, No. 4, 885-896 (1992) · Zbl 0784.12002 · doi:10.2307/2159464
[2] Car, M.: Le problème de Waring pour l’anneau des polynômes sur un corps fini, C. R. Acad. sci. Paris sér. A -- B 273, A141-A144 (1971) · Zbl 0221.10060
[3] M. Car, Le problème de Waring pour l’anneau des polynômes sur un corps fini, Séminaire de Théorie des Nombres, 1972 -- 1973, Univ. Bordeaux I, Talence, Exp. No. 6, 13 pp. Lab. Théorie des Nombres, Centre Nat. Recherche Sci., Talence, 1973 · Zbl 0282.10027
[4] Car, M.; Gallardo, L.: Sums of cubes of polynomials, Acta arith. 112, No. 1, 41-50 (2004) · Zbl 1062.11078 · doi:10.4064/aa112-1-4 · http://journals.impan.gov.pl/aa/Inf/112-1-3.html
[5] Cohen, E.: Sums of an even number of squares in GF[pn,x], Duke math. J. 14, 251-267 (1947) · Zbl 0030.10401 · doi:10.1215/S0012-7094-47-01418-X
[6] Effinger, G. W.; Hayes, D. R.: Additive number theory of polynomials over a finite field, Oxford math. Monogr. (1991) · Zbl 0759.11032
[7] Gallardo, L.: Sums of biquadrates and cubes in fq[t], Rocky mountain J. Math. 33, No. 3, 865-873 (2003) · Zbl 1077.11070 · doi:10.1216/rmjm/1181069932
[8] Gallardo, L.: Waring’s problem for cubes and squares over a finite field of even characteristic, Bull. belg. Math. soc. Simon stevin 12, No. 3, 349-362 (2005) · Zbl 1111.11059
[9] Kubota, R. M.: Waring’s problem for fq[x], Dissertationes math. (Rozprawy mat.) 117, 60 (1974) · Zbl 0298.12008
[10] Paley, R. E. A.C.: Theorems on polynomials in a Galois field, Q. J. Math. 4, 52-63 (1933) · Zbl 0006.24703
[11] Pólya, G.; Szegö, G.: Problems and theorems in analysis, I. Series, integral calculus, theory of functions, (1998) · Zbl 1053.00002
[12] Tornheim, L.: Sums of n-th powers in fields of prime characteristic, Duke math. J. 4, 359-362 (1938) · Zbl 0019.00302 · doi:10.1215/S0012-7094-38-00427-2
[13] Vaserstein, L. N.: Waring’s problem for algebras over fields, J. number theory 26, 286-298 (1987) · Zbl 0624.10049 · doi:10.1016/0022-314X(87)90085-0
[14] Vaserstein, L. N.: Sums of cubes in polynomial rings, Math. comp. 56, 349-357 (1991) · Zbl 0711.11013 · doi:10.2307/2008546
[15] Vaserstein, L. N.: Ramsey’s theorem and the Waring’s problem for algebras over fields, , 435-442 (1992) · Zbl 0817.12002
[16] Webb, W. A.: Waring’s problem in GF[q,x], Acta arith. 22, 207-220 (1973) · Zbl 0258.12014
[17] Webb, W. A.: Numerical results for Waring’s problem in GF[q,x], Math. comp. 27, 193-196 (1973) · Zbl 0255.10049
[18] Weil, A.: Numbers of solutions of equations in finite fields, Bull. amer. Math. soc. 55, 497-508 (1949) · Zbl 0032.39402
[19] Wooley, T. D.: Large improvements in Waring’s problem, Ann. of math. (2) 135, No. 1, 131-164 (1992) · Zbl 0754.11026 · doi:10.2307/2946566