On the resolvent degree of PSU(3,q)

Abstract

Resolvent degree ($\operatorname{RD}$) is an invariant of finite groups in terms of the complexity of their algebraic actions. We address the problem of bounding $\operatorname{RD}(G)$ for all finite simple groups using the methods established by G\'{o}mez-Gonz\'{a}les-Sutherland-Wolfson in terms of $\operatorname{RD}^{\leq d}_{\mathbb{C}}$-versality and special points. We give upper bounds on $\operatorname{RD}(\operatorname{PSU}(3,q))$ and $\operatorname{RD}(\operatorname{PSU}(2, q))$ in terms of classical invariant theory. In the $\operatorname{PSU}(3,q)$ case, stability of low-degree invariants permit an asymptotic bound on $\operatorname{RD}$ growing in $q$.