Quantitative Hilbert Irreducibility and Almost Prime Values of Polynomial Discriminants.

The article focuses on two polynomial counting problems in arithmetic statistics, employing a combination of Fourier analytic and arithmetic methods. First, it establishes new quantitative bounds related to Hilbert's Irreducibility Theorem (HIT) for degree \( n \) polynomials whose Galois groups lie within the alternating group \( A_n \), improving previous results for both monic and non-monic polynomials. Second, it provides lower bounds on the number of degree \( n \) monic polynomials and corresponding number fields with discriminants having few prime factors ("almost prime" discriminants), extending known results to all degrees \( n \geq 3 \). The approach introduces a modified Selberg sieve leveraging Fourier transforms of arithmetic functions over finite fields, enabling refined control over local conditions and error terms in counting polynomials with specified Galois groups or discriminant properties.