Asymptotic essential-dimension ratio of the symmetric groups

Description of constant

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]

Known upper bounds

Known lower bounds

Additional comments and links

• Exact values through degree $7$. Over $\mathbb C$, one has $ed_{\mathbb C}(S_4)=ed_{\mathbb C}(S_5)=2$, $ed_{\mathbb C}(S_6)=3$, and $ed_{\mathbb C}(S_7)=4$. The exact value is open for every $n\ge 8$. [BR1997-sym-bounds] [Duncan2010-exact7] [ER2024-intro-bounds-open]

• Connection with Hilbert’s $13$th problem. The sequence $ed_{\mathbb C}(S_n)$ measures the complexity of simplifying the general degree-$n$ polynomial by Tschirnhaus transformations; already the case $n=7$ is tied to algebraic variants of Hilbert’s $13$th problem. [BR1997-cor42-dk-edSn] [Duncan2010-h13]

• Characteristic bookkeeping. The bounds split by characteristic as follows: $ed_k(S_n)\le n-3$ $(n\ge 5)$ is valid over arbitrary fields; $ed_k(S_n)\ge \lfloor n/2 \rfloor$ is proved in characteristic $0$ and extends to $\mathrm{char}(k)\ne 2$; the stronger $ed_k(S_n)\ge \lfloor (n+1)/2 \rfloor$ is currently quoted in characteristic $0$; and in prime characteristic ($\mathrm{char}(k)=p>0$), for every prime $p$ there are infinitely many $n$ with $ed_{\mathbb{F}_{p}}(S_n)\le n-4$. Here $\mathrm{char}(k)\ne 2$ means both characteristic $0$ and odd prime characteristics. [ER2024-intro-bounds-open] [BR1997-sym-bounds] [ER2024-poschar]

• Recent synthesis of the $\mathbb C$ status. A 2023 summary of known bounds on essential dimension and resolvent degree over $\mathbb C$ reiterates that for symmetric groups, the exact value of $ed(S_n)$ is open for every $n\ge 8$, and that the expected value is $n-3$ for $n\ge 5$. [Sutherland2023-status]

• Related invariant in positive characteristic. A 2026 follow-up by Edens and Reichstein shows that the sextic, 13th-problem, and octic resolvent-degree conjectural equalities can fail over fields of positive characteristic, highlighting parallel characteristic-sensitivity for complexity invariants attached to general polynomials. [ER2026-h13-poschar]

References

• [BR1997] Buhler, Joe; Reichstein, Zinovy. On the essential dimension of a finite group. Compositio Mathematica 106 (1997), no. 2, 159–179. DOI: 10.1023/A:1000144403695. Author-hosted PDF: edtotal.pdf. Google Scholar
  • [BR1997-cor42-dk-edSn] loc: Author-hosted PDF p.11, §4, file edtotal.pdf quote: “COROLLARY 4.2. Let $G$ be a transitive subgroup of $S_n$, let $x_1, \ldots, x_n$ be independent variables over $k$ and let $F_G$ be the fixed field for the natural (permutation) action of $G$ on $E = k(x_1, \ldots, x_n)$. Then $ed_k(F_G(x_1)/F_G) = ed_k(G)$. In particular, the number $dk(n)$, defined in the introduction, is equal to $ed_k(S_n)$.”
  • [BR1997-sym-bounds] loc: Author-hosted PDF p.17, §6.3 “Symmetric Groups”, file edtotal.pdf quote: “THEOREM 6.5. Let $k$ be an arbitrary field of characteristic $0$. Then (a) $ed_k(S_{n+2}) \ge ed_k(S_n) + 1$ for any $n \ge 1$. (b) $ed_k(S_n) \ge [n/2]$ for any $n \ge 1$. (c) $ed_k(S_n) \le n - 3$ for any $n \ge 5$. (d) $ed_k(S_4) = ed_k(S_5) = 2$, $ed_k(S_6) = 3$.”
• [Duncan2010] Duncan, Alexander. Essential dimensions of $A_7$ and $S_7$. Mathematical Research Letters 17 (2010), no. 2, 263–266. arXiv PDF: 0908.3220. DOI: 10.4310/MRL.2010.v17.n2.a5. Google Scholar
  • [Duncan2010-exact7] loc: arXiv v1 PDF p.1, Introduction, file 0908.3220.pdf quote: “The purpose of this note is to show that the essential dimension of the alternating group $A_7$ and the symmetric group $S_7$ can be computed using the recent work of Prokhorov [12] on the classification of rationally connected threefolds with faithful actions of non-abelian simple groups. Our main result is the following: Theorem 1. $ed_k(A_7) = ed_k(S_7) = 4$.”
  • [Duncan2010-h13] loc: arXiv v1 PDF p.2, Introduction, file 0908.3220.pdf quote: “The values of $ed_k(S_n)$ are of special interest because they relate to classical questions of simplifying degree $n$ polynomials via Tschirnhaus transformations. In particular, the degree $7$ case features prominently in algebraic variants of Hilbert’s 13th problem.”
• [ER2024] Edens, Oakley; Reichstein, Zinovy. Essential dimension of symmetric groups in prime characteristic. Comptes Rendus. Mathématique 362 (2024), 639–647. arXiv PDF: 2308.10096. DOI: 10.5802/crmath.577. Google Scholar
  • [ER2024-def] loc: Numdam PDF p.2, Abstract, file 10.5802/crmath.577.pdf quote: “The essential dimension $ed_k(S_n)$ of the symmetric group $S_n$ is the minimal integer $d$ such that the general polynomial $x^n + a_1x^{n-1} + \cdots + a_n$ can be reduced to a $d$-parameter form by a Tschirnhaus transformation.”
  • [ER2024-intro-bounds-open] loc: Numdam PDF p.3, §1 “Introduction”, file 10.5802/crmath.577.pdf quote: “Finding $ed_k(S_n)$ is a long-standing open problem, which goes back to F. Klein [8]; cf. also N. Chebotarev [14]. Essential dimension of a finite group was formally defined by J. Buhler and the second author in [5], where the inequalities $ed_k(S_n) \ge \lfloor n/2 \rfloor$ and $ed_k(S_n) \le n - 3$ $(n \ge 5)$ were proved. The field $k$ was assumed to be of characteristic $0$ in [5], but the proof of the first inequality in (1) given there goes through for any field $k$ of characteristic different from $2$. The second inequality is valid over an arbitrary field $k$. A. Duncan [6] subsequently showed that in characteristic $0$, $ed_k(S_n) \ge \lfloor (n + 1)/2 \rfloor$ for any $n \ge 7$. The exact value of $ed_k(S_n)$ is open for each $n \ge 8$ and any field $k$, though it is widely believed that $ed_k(S_n)$ should be $n - 3$ for every $n \ge 5$, at least in characteristic $0$.”
  • [ER2024-poschar] loc: Numdam PDF p.2, Abstract, file 10.5802/crmath.577.pdf quote: “In this paper we show that for every prime $p$ there are infinitely many positive integers $n$ such that $ed_{\mathbb{F}_p}(S_n) \le n - 4$.”
• [Sutherland2023] Sutherland, Alexander J. A Summary of Known Bounds on the Essential Dimension and Resolvent Degree of Finite Groups. Preprint (2023), arXiv:2312.04430. arXiv PDF: 2312.04430. DOI: 10.48550/arXiv.2312.04430. Google Scholar
  • [Sutherland2023-status] loc: arXiv v1 PDF p.3, §3 “Lower Bounds on ed(G)”, Case 2, file 2312.04430.pdf quote: “the exact value of $ed(S_n)$ is open for every $n \ge 8$, though it is widely believed that $ed(S_n)$ should be $n - 3$ for every $n \ge 5$.”
• [ER2026] Edens, Oakley; Reichstein, Zinovy B. Hilbert’s 13th problem in prime characteristic. Documenta Mathematica 31 (2026), no. 2, 385–399. arXiv PDF: 2406.15954. DOI: 10.4171/DM/984. Google Scholar
  • [ER2026-h13-poschar] loc: Doc. Math. PDF p.1, Abstract, file dm984.pdf quote: “In this paper, we show that all three of Hilbert’s conjectures can fail if we replace $C$ with a base field of positive characteristic.”

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.