# 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)
Quantum $(abc)$-theorems. (English) Zbl 0990.11015
We recall that, very loosely speaking, the $abc$ theorem for relatively prime polynomials $a$, $b$, and $c$ not all constant says that if $a+b=c$, then not too many of the zeros of $a$, $b$, and $c$ can have high multiplicity (one says a polynomial $f$ is constant’ if $f'=0$; note the subtlety in characteristic $p$). Here the author replaces zeros of high multiplicity’ by sets of zeros in progression’ -- that too in a fairly generalised sense: thus one may have a multiset of zeros in progression’, namely a divisor (in the sense used in algebraic geometry $\alpha+f\alpha+\cdots+f^{m-1}\alpha$, where $f$ is a map of the set of potential zeros into itself. Very familiar examples include running powers’ $(x)^h_n=x(x+h)\cdots(x+(n-1)h)$ and the $q$-version’ of $x^n$, namely $(1-x;q)_n=(1-(1-x))(1-(1-x)q)\cdots(1-(1-x)q^{n-1})$. The author provides less familiar generalisations.
##### MSC:
 11D88 $p$-adic and power series fields 11D99 Diophantine equations
Full Text:
##### References:
 [1] Beukers, F.: The Diophantine equation axp+Byq=Czr. Duke math. J. 91, 61-88 (1998) [2] Darmon, H.; Granville, A.: On the equations $zm=F(x,~y)$ and axp+Byq=Czr. Bull. London math. Soc. 27, 513-543 (1995) · Zbl 0838.11023 [3] Dizik, M.: Polynomial solutions of generalized Fermat equations. Master paper (1996) [4] Mason, R. C.: Diophantine equations over function fields. London math. Soc. lecture note series 96 (1984) [5] J. Oesterlé, Nouvelles approaches du ”théorème” de Fermat, (French) [New approaches to Fermat’s ”theorem”], Séminaire Bourbaki, Vol. 1987/88. Astérique, No. 1988), Exp. No. 694, 4, 165--186 (1989).