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

Volume 99 / 2004

Michael Filaseta, Manton Matthews, Jr. Colloquium Mathematicum 99 (2004), 1-5 MSC: 11R09, 12D05, 12E05. DOI: 10.4064/cm99-1-1

Abstract

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 $N = \deg g + 2\max\{ \deg f, \deg g \}$.

Authors

  • Michael FilasetaMathematics Department
    University of South Carolina
    Columbia, SC 29208, U.S.A.
    e-mail
  • Manton Matthews, Jr.Mathematics Department
    University of South Carolina
    Columbia, SC 29208, U.S.A.

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