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On some polynomials allegedly related to the \(abc\) conjecture. (English) Zbl 0903.11025

For positive integers \(a\), \(b\), and \(c\), with \(a=b+c\), let the \(abc\) polynomial \(f_{abc}(x)\) be defined by \(f_{abc}(x)=(bx^a-ax^b+c)/(x-1)^2\). This paper shows, for certain infinite families of triples \((a,b,c)\), that \(f_{abc}(x)\) is irreducible. For example, it is shown that the set of coprime triples \((a,b,c)\) with \(f_{abc}\) irreducible has density \(1\). The special case \(b=1\) of this family of polynomials was studied earlier by J. L. Nicolas and A. Schinzel [Lect. Notes Math. 1415, 167-179 (1990; Zbl 0703.30004)].
The motivation for looking at these polynomials comes from the “abc conjecture” of Masser and OesterlĂ©. The analogue of this conjecture when \(\mathbb Z\) is replaced by \(\mathbb C[t]\) is well-known; the polynomials \(f_{abc}\) arise from a rather unusual attempt to carry this proof over to the situation over \(\mathbb Z\). The author admits, however, “I do not know if the irreducibility [of \(f_{abc}(x)\)] is related in any way to the \(abc\) conjecture”.
Reviewer: P.Vojta (Berkeley)

MSC:

11R09 Polynomials (irreducibility, etc.)
14G99 Arithmetic problems in algebraic geometry; Diophantine geometry
11C08 Polynomials in number theory

Citations:

Zbl 0703.30004
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