On the factorization of polynomials with small Euclidean norm. (English) Zbl 0928.11015

Filaseta, Michael

Győry, Kálmán (ed.) et al., Number theory in progress. Proceedings of the international conference organized by the Stefan Banach International Mathematical Center in honor of the 60th birthday of Andrzej Schinzel, Zakopane, Poland, June 30–July 9, 1997. Volume 1: Diophantine problems and polynomials. Berlin: de Gruyter. 143-163 (1999).

This paper deals with univariate polynomials over the integers. The non-reciprocal part of \(f\) is the quotient of \(f\) by the product of its reciprocal irreducible factors. The paper contains the two following results:
Theorem 1: Let \(a<b<c<d<n\) be positive integers. Then the non-reciprocal part of the polynomial \(f=x^n+x^d+x^c+x^b+x^a+1\) is reducible if, and only if, \(f(x)\) is a variation (in a natural sense) of \(g(x)=x^{5s+3t}+x^{4s+2t}+x^{2s+2t}+x^t+x^s+1\), which factorizes as \[ g(x)= (x^{3s+2t}-x^{s+t}+x^t+1)(x^{2s+t}+x^s+1), \] where \(s\) and \(t\) denote arbitrary positive integers.
Theorem 2: Let \(f(x)\) be an irreducible non-reciprocal polynomial with coefficients \(0\) or \(1\) such that \(f(0)\not=0\). Then for each positive integer \(l\), the polynomial \(f(x^l)\) is irreducible.
The methods of the proofs are algebraic and related to those used previously by Ljunggren (1960) and by Schinzel (1969).
For the entire collection see [Zbl 0911.00025].

Reviewer: Maurice Mignotte (Strasbourg)

MSC:

11C08	Polynomials in number theory
12E05	Polynomials in general fields (irreducibility, etc.)
12D05	Polynomials in real and complex fields: factorization

Keywords:

Factorization of polynomials; cyclotomic factors; fewnomials