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Low-Degree Factors of Random Polynomials

Author

Listed:
  • Sean O’Rourke

    (University of Colorado at Boulder)

  • Philip Matchett Wood

    (University of Wisconsin-Madison)

Abstract

We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers, which is equivalent to having a low-degree algebraic root. It is known in certain cases that random polynomials with integer coefficients are very likely to be irreducible, and our project can be viewed as part of a general program of testing whether this is a universal behavior exhibited by many random polynomial models. Our main result shows that pointwise delocalization of the roots of a random polynomial can be used to imply that the polynomial is unlikely to have a low-degree factor over the integers. We apply our main result to a number of models of random polynomials, including characteristic polynomials of random matrices, where strong delocalization results are known.

Suggested Citation

  • Sean O’Rourke & Philip Matchett Wood, 2019. "Low-Degree Factors of Random Polynomials," Journal of Theoretical Probability, Springer, vol. 32(2), pages 1076-1104, June.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:2:d:10.1007_s10959-018-0839-8
    DOI: 10.1007/s10959-018-0839-8
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    References listed on IDEAS

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    1. Robert M. Bond & Christopher J. Fariss & Jason J. Jones & Adam D. I. Kramer & Cameron Marlow & Jaime E. Settle & James H. Fowler, 2012. "A 61-million-person experiment in social influence and political mobilization," Nature, Nature, vol. 489(7415), pages 295-298, September.
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