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arXiv · 2107.02914

Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants

Theresa C. Anderson·Ayla Gafni·Robert J. Lemke Oliver·David Lowry-Duda·George Shakan·Ruixiang Zhang
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Abstract

We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree $n$ polynomials $f$ with $\mathrm{Gal}(f) \subseteq A_n$. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree $n$ monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree $n$ number fields with almost prime discriminants.

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Theresa C. Anderson, Ayla Gafni, Robert J. Lemke Oliver, David Lowry-Duda, George Shakan, Ruixiang Zhang. 2021-07-06. Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants. https://doi.org/10.1093/imrn%2Frnab296

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