zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Vanishing polynomial sums. (English) Zbl 1073.12001
The main result of the paper gives an improved version of the generalization of the abc-theorem for function fields of zero characteristic. Precisely, if $y_{1}+\dots+y_{m}=y_{0}$, where $m\geq 2$, $y_{i} \in F[t]$ ($i=0,\dots,m$), $\text{char}(F)=0$, $\gcd(y_{1},\dots,y_{m})=1$, not all $y_{i}$ are constant, and no subsum of $y_{1}$, $\dots$, $y_{m}$ vanishes, then $\deg(y_{0}) < (m-1)\sum_{i=0}^{m}\nu(y_{i})$, and $\deg(y_{0}) \leq (\nu(y_{0}\dots y_{m})-1)m(m-1)/2$. This improves on known previous results by {\it W. D. Brownawell} and {\it D. W. Masser} [Math. Proc. Camb. Philos. Soc. 100, 427--434 (1986; Zbl 0612.10010)]. The result is applied to the generalized Fermat equation, $c_{0}x_{0}^{n_{0}}+\dots+c_{m}x_{m}^{n_{m}}=0$, in $m+1$ unknowns $x_{i}$ in the polynomials ring $F[t]$, with $\text{char}(F)=0$, $n\geq 2$, and $c_{i}\in F\setminus\{0\}$. The authors prove that such equation has no non--trivial solutions when $\sum 1/n_{i} \leq 1/(m-1)$, improving on a previous result by {\it D. J. Newman} and {\it M. Slater} [J. Number Theory 11, 477--487 (1979; Zbl 0407.10039)].

12E05Polynomials over general fields
11D41Higher degree diophantine equations
Full Text: DOI