Author

W. Sawin

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Preprint Oct 2026

Counting matrices with given characteristic polynomial

Let $f$ be a monic polynomial of degree $n$ with integer coefficients. We use Fourier analysis to prove a new upper bound on the number of $n\times n$ matrices with characteristic polynomial $f$ and integer entries in $[-T,T]$, as $T\to \infty$. The implied constant in our bound only depends on $n$ and is otherwise uni...

Tim Browning, Fabian Gundlach, W. Sawin · 0 citations

Preprint Aug 2026

The strong form of Van der Waerden's conjecture via twisted Chowla

Determining the properties of a random polynomial has fuelled significant investigation over the past century. One driving force of this research is a 1936 paper of Van der Waerden. Fix $n \ge 3$ and let $E_n(B)$ be the number of monic, irreducible, non-$S_n$ polynomials $f = X^n + a_1 X^{n-1} + \cdots + a_n$ with $|a_...

Theresa C. Anderson, Sam Chow, R. Dietmann et al. · 1 citation