## 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 Oesterlé. 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

Zbl 0703.30004
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