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on Asymptotic essential-dimension ratio of the symmetric groups ($C_{79}$)

The statement

For each integer $n \ge 1$ , let $S_n$ be the symmetric group on $n$ letters. Over a base field $k$ , the essential dimension $ed_k(S_n)$ is the smallest integer $d$ such that the general degree- $n$ polynomial

x^n + a_1 x^{n-1} + \cdots + a_n

can be reduced to a $d$ -parameter form by a Tschirnhaus transformation. [ER2024-def]

Buhler and Reichstein proved that this polynomial-theoretic parameter count is exactly $ed_k(S_n)$ , so the asymptotic problem for $ed_{\mathbb C}(S_n)$ is equivalent to the asymptotic Tschirnhaus-simplification problem for the general polynomial. [BR1997-cor42-dk-edSn]

We define

C_{79}\ :=\ \limsup_{n\to\infty}\frac{ed_{\mathbb C}(S_n)}{n}.

Thus this entry's asymptotic constant is specifically the characteristic- $0$ quantity over $\mathbb C$ .

The best established range is

\frac12\ \le\ C_{79}\ \le\ 1.

[ER2024-intro-bounds-open] [BR1997-sym-bounds]

It is widely believed that $ed_{\mathbb C}(S_n)=n-3$ for every $n\ge 5$ , which would imply $C_{79}=1$ . [ER2024-intro-bounds-open]

What gets accepted

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Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.