Low-Degree Factors of Random Polynomials

Sean O’Rourke () and Philip Matchett Wood ()
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Sean O’Rourke: University of Colorado at Boulder
Philip Matchett Wood: University of Wisconsin-Madison

Journal of Theoretical Probability, 2019, vol. 32, issue 2, 1076-1104

Abstract: 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.

Keywords: Random polynomials; Irreducible; Random matrices; Delocalization; 11C08; 15B52 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-018-0839-8

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