Upper Bounds on Resolvent Degree and Its Growth Rate

Abstract

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].

40 pages. Comments welcome!

References in corpus (2)

• Felix Klein's "About the Solution of the General Equations of Fifth and Sixth Degree (Excerpt from a letter to Mr. K. Hensel)"
• Anders Wiman's "On the Application of Tschirnhaus Transformations to the Reduction of Algebraic Equations''