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Linear Forms in Monic Integer Polynomials

  Published:2011-09-15
 Printed: Sep 2013
  • Artūras Dubickas,
    Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
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Abstract

We prove a necessary and sufficient condition on the list of nonzero integers $u_1,\dots,u_k$, $k \geq 2$, under which a monic polynomial $f \in \mathbb{Z}[x]$ is expressible by a linear form $u_1f_1+\dots+u_kf_k$ in monic polynomials $f_1,\dots,f_k \in \mathbb{Z}[x]$. This condition is independent of $f$. We also show that if this condition holds, then the monic polynomials $f_1,\dots,f_k$ can be chosen to be irreducible in $\mathbb{Z}[x]$.
Keywords: irreducible polynomial, height, linear form in polynomials, Eisenstein's criterion irreducible polynomial, height, linear form in polynomials, Eisenstein's criterion
MSC Classifications: 11R09, 11C08, 11B83 show english descriptions Polynomials (irreducibility, etc.)
Polynomials [See also 13F20]
Special sequences and polynomials
11R09 - Polynomials (irreducibility, etc.)
11C08 - Polynomials [See also 13F20]
11B83 - Special sequences and polynomials
 

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