Vukovic, Andrej2023-08-282023-08-282023-08-282023-08-23http://hdl.handle.net/10012/19767We prove several square-divisibility results about the discriminant of homogeneous polynomials of arbitrary degree and number of variables, when certain coefficients vanish, and give characterizations for when the discriminant is divisible by $p^2$ for $p$ prime. We also prove several formulas about a certain polynomial $\Delta_d'$, first introduced in (Bhargava, Shankar, Wang, 2022), which behaves like an average over the partial derivatives of $\Delta_d$, the discriminant of degree $d$ polynomials. In particular, we prove that $\Delta_d'$ is irreducible when $d\geq 5$.enmathematicsalgebraic geometrynumber theoryDoctoral Thesis