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arXiv:1810.13360math.NT

Irreducibility of random polynomials of large degree

Emmanuel Breuillard, Péter P. Varjú

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Abstract

We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are irreducible and their Galois groups contain the alternating group with high probability as the degree goes to infinity. This settles a conjecture of Odlyzko and Poonen conditionally on RH for Dedekind zeta functions.

arXiv comments
50 pages, this is the accepted version for publication in Acta Math., minor changes and corrections based on referees' reports
Journal reference
Acta Math. 223(2): 195-249 (December 2019)
DOI
doi:10.4310/ACTA.2019.v223.n2.a1
Subjects
math.NTNumber Theory math.PRProbability

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Discussed in comments1 thread

  1. Mathematicians Seal Loophole to Breaking RSA Encryption 2 comments

    • This is the actual paper. https://arxiv.org/abs/1810.13360 It's worth noticing that Riemann Hypothesis is assumed in the proof, as I layman, I think it means the proof merely gives us more confidence about the difficulty of integer factorization. But integer factorization itself…

      segfaultbuserrView on HNParent context

    • https://arxiv.org/abs/1810.13360

      segfaultbuserrView on HNParent context

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