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Linear Forms in Monic Integer Polynomials

Published online by Cambridge University Press:  20 November 2018

Artūras Dubickas
Artūras Dubickas*
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania e-mail: [email protected] Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, Vilnius LT-08663, Lithuania
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Abstract.

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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 $\mathbb{Z}\left[ x \right]$.

Keywords

11R0911C0811B83irreducible polynomialheightlinear form in polynomialsEisenstein's criterion.
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 3 , 01 September 2013 , pp. 510 - 519
Copyright
Copyright © Canadian Mathematical Society 2013

References

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