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].
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