On the irreducibility of $0,1$-polynomials of the form ${f(x) x^{n} + g(x)}$

Michael Filaseta ¹   ; Manton Matthews, Jr. ¹

¹ Mathematics Department University of South Carolina Columbia, SC 29208, U.S.A.

Colloquium Mathematicum, Tome 99 (2004) no. 1, pp. 1-5

Résumé

If $f(x)$ and $g(x)$ are relatively prime polynomials in $\mathbb Z[x]$ satisfying certain conditions arising from a theorem of Capelli and if $n$ is an integer $> N$ for some sufficiently large $N$, then the non-reciprocal part of $f(x) x^{n} + g(x)$ is either identically $\pm1$ or is irreducible over the rationals. This result follows from work of Schinzel in 1965. We show here that under the conditions that $f(x)$ and $g(x)$ are relatively prime $0,1$-polynomials (so each coefficient is either $0$ or $1$) and $f(0) = g(0) = 1$, one can take $N = \deg g + 2\max\{ \deg f, \deg g \}$.

Détail

DOI : 10.4064/cm99-1-1

Keywords: relatively prime polynomials mathbb satisfying certain conditions arising theorem capelli integer sufficiently large non reciprocal part either identically irreducible rationals result follows work schinzel here under conditions relatively prime polynomials each coefficient either deg max deg deg