Recently Anderson, Gafni, Lemke Oliver, Lowry-Duda, Zhang and myself uploaded ”Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants” to the arXiv. Let be a large integer,
A classical principle going back to at least Hilbert is that a typical polynomial should have Galois group , and this would suggest is significantly smaller than . Van der Waerden conjectured that . Van der Waerden’s conjecture was almost proved quite recently by Chow and Dietmann (for ), except they are not able to estimate
We show the following.
Theorem 1: Let be a positive integer. then
We note that Bhargava recently announced a proof of , which is stronger than Theorem 1. We remark that our techniques also apply to counting polynomials with almost prime discriminants (see Theorem 1.2 and the related 1.3).
We briefly mention some of the ideas used to prove Theorem 1. Classical algebraic number theory reveals that if the Galois group of a polynomial is , then , where is the Mobius function for . Thus we can employ a sieve to estimate . However, instead of using the large sieve, we use the small sieve (more precisely, the Selberg sieve). Using this naively gives a bound similar to that of Gallagher in (1), as expected. The key advantage of using the small sieve is we can seek improvements if we have good distribution results. After employing Poisson summation our task in then to understand local factors, which turns out to be basically