Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants

Recently AndersonGafniLemke OliverLowry-DudaZhang and myself uploaded ”Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants” to the arXiv. Let {H} be a large integer,

\displaystyle V_n(H) = \{x^n + a_{n_1} x^{n-1} + \cdots + a_0 : a_{n-1} , \ldots , a_0 \in \mathbb{Z} \cap [-H,H]\},

and

\displaystyle E_n(H) = \#\{f \in V_n(H) : {\rm Gal}(f) \neq S_n\}.

A classical principle going back to at least Hilbert is that a typical polynomial should have Galois group {S_n}, and this would suggest {\#E_n(H)} is significantly smaller than {\#V_n(H)}. Van der Waerden conjectured that {E_n(H) = o(\#V_n(H) /H)}. Van der Waerden’s conjecture was almost proved quite recently by Chow and Dietmann (for {n \neq 7,8,10}), except they are not able to estimate 

\displaystyle E_n(H , A_n) = \#\{f \in V_n(H) : {\rm Gal}(f) = A_n\}.

Gallagher showed, using the large sieve, that 

\displaystyle E_n(H) = O\left(\frac{\#V_n(H) \log H}{H^{1/2}} \right).\ \ \ \ \ (1)

We show the following.

{\ } Theorem 1: Let {H} be a positive integer. then 

\displaystyle \#E_n(H , A_n) = O\left(\frac{\#V_n(H) H^{O(1/n)}}{H^{2/3}} \right). \ \ \ \spadesuit{\ }

We note that Bhargava recently announced a proof of {E_n(H , A_n) = O(\#V_n(H) / H)}, 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 {f} is {A_n}, then {\mu_p(f) = (-1)^{n+1}}, where {\mu_p} is the Mobius function for {\mathbb{F}_p[x]}. Thus we can employ a sieve to estimate {\#E_n(H , A_n) }. 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 

\displaystyle \widehat{\mu_p}(\xi_{n-1} , \ldots , \xi_0) =,\ \ \ \ \

\displaystyle = \frac{1}{p^{n}}\sum_{a_0 , \ldots , a_{n-1} \in \mathbb{F}_p} \mu(x^n + a_{n-1} x^{n-1} + \ldots + a_0) \psi_p(\xi_{n-1} a_{n-1} + \cdots + a_0 \xi_0),\ \ \ \ \ (2)

where {\psi_p : \mathbb{F}_p \rightarrow \mathbb{F}_p} is an additive character. This is morally the same problem as estimating  \displaystyle \sum_{n \leq x} \mu(n) e(nx), where {\mu} is the classical Mobius function. Baker and Harman showed that GRH implies that for any {\epsilon > 0}

\displaystyle |\sum_{n \leq x} \mu(n) e(nx)| \ll x^{3/4 + \epsilon}.

As the Riemann hypothesis is known in the function field setting, such a bound should also hold for (2). Indeed Porritt showed this is the case and this allows us to establish Theorem 1.

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