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In the Mathoverflow question Examples of nice families of irreducible polynomials over Z, user trew mentions a family of irreducible polynomials over the integers of the following form: $$ p(x) = x^4 \prod_{i=1}^{n-4} (x - b_i) - (-1)^n (2x + 4), \quad b_i \in \mathbb{Z}, \text{ pairwise distinct}, b_i \neq -2. $$

He attributes it to Philipp Furtwängler but provides no reference, and up to now I have not been able to locate it.

Edit: as Sil rightly points out, these conditions are not sufficient to make $p$ irreducible. I do not know what other conditions (on the growth of the $b_i$?) are required (and if there is anything behind the claim.

Here is Furtwängler's publication list on zbMathOpen: https://www.zbmath.org/?q=ai%3Afurtwangler.philipp

I'd be very glad if anyone has an idea on how to dig this up!

It might be an interesting question to the whole community as irreducibility arguments are frequently sought and notoriously difficult once one moves beyond the standard techniques. This family can be interpreted as a small perturbation of a reducible polynomial with real roots and Furtwängler's proof might add an additional technique to the irreducibility toolbox: empirically many more polynomials of this type seem to be irreducible.

wandersam
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  • What do you mean by "pairwise" different ? All of them different ? – Jean Marie Apr 12 '21 at 09:39
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    Yes, i.e. no two of them equal to each other. The referenced mathoverflow question puts, equivalently, "strictly increasing". – wandersam Apr 12 '21 at 09:45
  • Surely you need to assume $b_i\neq-2$ for all $i$? – Jyrki Lahtonen Apr 12 '21 at 09:54
  • Thanks, very true. I do not know if there are further assumptions since all I know on this situation is from the linked post... – wandersam Apr 12 '21 at 09:58
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    Some kind of "perturbation preserving irreducibility" is discussed in the comments under this answer. – Jyrki Lahtonen Apr 12 '21 at 09:58
  • Thanks! I think the difference is between density / measure etc. arguments on the one hand and a "deterministic" proof for Furtwängler's family on the other. – wandersam Apr 12 '21 at 10:05
  • Reposted to mathoverflow: https://mathoverflow.net/questions/405502/furtw%c3%a4nglers-family-of-irreducible-polynomials – wandersam Oct 05 '21 at 11:53
  • There are many counterexamples, for example $n=6$ any of $(b_1,b_2)=(-1,1),(0,1),(2,7),(3,4)$ (e.g. $(x-1) \mid x^4(x-3)(x-4)-(-1)^6(2x+4)$ or $x^4(x-1)(x+1)-(-1)^6(2x+4)=(x^3+2)(x^3-x-2)$. For higher $n$ there are also examples... – Sil Oct 17 '21 at 12:50
  • You are right. That is why I'd wish to find the original claim.. – wandersam Oct 17 '21 at 13:07

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