Counting polynomials with a prescribed Galois group

Algebra and Number Theory Seminar
Event time: 
Tuesday, March 1, 2022 - 4:30pm
Vlad Matei
Speaker affiliation: 
Tel Aviv University
Event description: 

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 $n \ge 3$ and $n\neq 7,8,10$.
In joint work with Lior Bary-Soroker, and Or Ben-Porath we prove a lower bound in 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. This in turn induces a linear map $A^{n/2}$ to $A^n/ A_n$. For the upper bound for $C_2$ wreath $S_{n/2}$ we prove a monic version of Widmer's result four counting polynomials with imprimitive Galois group.

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