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 $N = \deg g + 2\max\{ \deg f, \deg g \}$.
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
Affiliations des auteurs :
Michael Filaseta
1
;
Manton Matthews, Jr.
1
1
Mathematics Department University of South Carolina Columbia, SC 29208, U.S.A.
Michael Filaseta; Manton Matthews, Jr. On the irreducibility of
$0,1$-polynomials of the form ${f(x) x^{n} + g(x)}$. Colloquium Mathematicum, Tome 99 (2004) no. 1, pp. 1-5. doi: 10.4064/cm99-1-1
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title = {On the irreducibility of
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journal = {Colloquium Mathematicum},
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year = {2004},
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