Linear Forms in Monic Integer Polynomials

Dubickas, Artūras

Canadian mathematical bulletin, Tome 56 (2013) no. 3, pp. 510-519

Résumé

We prove a necessary and sufficient condition on the list of nonzero integers ${{u}_{1}},...,{{u}_{k}}$ , $k\,\ge \,2$ , under which a monic polynomial $f\,\in \,\mathbb{Z}\left| x \right|$ is expressible by a linear form ${{u}_{1}}\,{{f}_{1}}\,+\,\cdot \cdot \cdot \,+\,{{u}_{k}}\,{{f}_{k}}$ in monic polynomials ${{f}_{1}},...,\,{{f}_{k}}\,\in \,\mathbb{Z}\left| x \right|$ . This condition is independent of $f$ . We also show that if this condition holds, then the monic polynomials ${{f}_{1}},...,\,{{f}_{k}}$ can be chosen to be irreducible in $\mathbb{Z}\left[ x \right]$ .

Détail

DOI : 10.4153/CMB-2011-179-0

Mots-clés : 11R09, 11C08, 11B83, irreducible polynomial, height, linear form in polynomials, Eisenstein's criterion.

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