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Composite factors of binomials and linear systems in roots of unity

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Abstract

In this paper we completely classify binomials in one variable which have a nontrivial factor which is composite, i.e., of the shape g(h(x)) for polynomials g, h both of degree > 1. In particular, we prove that, if a binomial has such a composite factor, then deg g ≤ 2 (under natural necessary conditions). This is best-possible and improves on a previous bound deg g ≤ 24.

This result provides evidence toward a conjecture predicting a similar bound when binomials are replaced by polynomials with any given number of terms.

As an auxiliary result, which could have other applications, we completely classify the solutions in roots of unity of certain systems of linear equations.

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References

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127, Pisa, Italia

    Roberto Dvornicich

  2. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italia

    Umberto Zannier

Authors
  1. Roberto Dvornicich
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  2. Umberto Zannier
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Correspondence to Roberto Dvornicich.

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Dvornicich, R., Zannier, U. Composite factors of binomials and linear systems in roots of unity. Isr. J. Math. 229, 381–391 (2019). https://doi.org/10.1007/s11856-018-1806-x

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  • DOI: https://doi.org/10.1007/s11856-018-1806-x

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