Author

Evan M. O'Dorney

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Preprint Sep 2026

Galois groups of twisted reciprocal polynomials: a uniform asymptotic

We study the Galois group $G_f$ of a random polynomial $f$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x) = x^{2n}/b^n \cdot f(b/x)$. We use a Euclidean height adapted to this relation. Our main result is an asymptotic theorem of van der Waerden--Bhargava type: for fixed $b...

Evan M. O'Dorney · 0 citations

Preprint Sep 2026

Galois groups of integer polynomials with prescribed coefficients

Fix a degree $n$, a nonzero integer constant term, and some further coefficients of a monic integer polynomial, leaving $s$ coefficients to vary in $[-H,H]$. We first prove that, when only the leading and constant coefficients are fixed, the number of polynomials whose Galois group is not $S_n$ is of order $H^{n-2}$ fo...

Evan M. O'Dorney · 0 citations