  For each $n$, let $\text{RD}(n)$ denote the minimum $d$ for which there
exists a formula for the general polynomial of degree $n$ in algebraic
functions of at most $d$ variables. In 1945, Segre called for a better
understanding of the large $n$ behavior of $\text{RD}(n)$. In this paper, we
provide improved thresholds for upper bounds on $\text{RD}(n)$. Our techniques
build upon classical algebraic geometry to provide new upper bounds for small
$n$ and, in doing so, fix gaps in the proofs of A. Wiman and G.N. Chebotarev in
[Wim1927] and [Che1954].
