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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.
11D88$p$-adic and power series fields
11D99Diophantine equations
Full Text: DOI
[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).