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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   $\textrm {Gal}(f) \subseteq A_n$  . 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.

Quantitative Hilbert Irreducibility and Almost Prime Values of Polynomial Discriminants