Beijing Institute of Mathematical Sciences and Applications

Hansheng Diao , Yueke Hu , Emmanuel Lecouturier , Cezar Lupu

Vlad Matei

Tuesday, October 18, 2022 3:30 PM - 4:30 PM

1131

Zoom 293 812 9202 (BIMSA)

An old problem, dating back to Van der Waerden, asks about counting irreducible polynomials degree $n$ polynomials with coefficients in the box $[-H,H]$ and prescribed Galois group. Van der Waerden was the first to show that $H^n+O(H^{n-\delta})$ have Galois group $S_n$ and he conjectured that the error term can be improved to $o(H^{n-1})$. Recently, Bhargava almost proved van der Waerden conjecture showing that there are $O(H^{n-1+\varepsilon})$ non $S_n$ extensions, while Chow and Dietmann showed that there are $O(H^{n-1.017})$ non $S_n$, non $A_n$ extensions for $n\geq 3$ and $n\neq 7,8,10$. In joint work with Lior Bary-Soroker, and Or Ben-Porath we use a result of Hilbert to prove a lower bound for the case of $G=A_n$, and upper and lower bounds for $C_2$ wreath $S_{n/2}$ . The proof for $A_n$ can be viewed, on the geometric side, as constructing a morphism $\varphi$ from $A^{n/2}$ into the variety $z^2=\Delta(f)$ where each $varphi_i$ is a quadratic form. For the upper bound for $C_2$ wreath $S_{n/2}$ we improve on the monic version of Widmer's result on counting polynomials with an imprimitive Galois group. We also pose some open problems/conjectures.