|
| 1 | +# Asymptotic essential-dimension ratio of the symmetric groups |
| 2 | + |
| 3 | +## Description of constant |
| 4 | + |
| 5 | +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 |
| 6 | +$$ |
| 7 | +x^n + a_1 x^{n-1} + \cdots + a_n |
| 8 | +$$ |
| 9 | +can be reduced to a $d$-parameter form by a Tschirnhaus transformation. |
| 10 | +<a href="#ER2024-def">[ER2024-def]</a> |
| 11 | + |
| 12 | +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. |
| 13 | +<a href="#BR1997-cor42-dk-edSn">[BR1997-cor42-dk-edSn]</a> |
| 14 | + |
| 15 | +We define |
| 16 | +$$ |
| 17 | +C_{79}\ :=\ \limsup_{n\to\infty}\frac{ed_{\mathbb C}(S_n)}{n}. |
| 18 | +$$ |
| 19 | + |
| 20 | +Thus this entry's asymptotic constant is specifically the characteristic-$0$ quantity over $\mathbb C$. |
| 21 | + |
| 22 | +The best established range is |
| 23 | +$$ |
| 24 | +\frac12\ \le\ C_{79}\ \le\ 1. |
| 25 | +$$ |
| 26 | +<a href="#ER2024-intro-bounds-open">[ER2024-intro-bounds-open]</a> <a href="#BR1997-sym-bounds">[BR1997-sym-bounds]</a> |
| 27 | + |
| 28 | +It is widely believed that $ed_{\mathbb C}(S_n)=n-3$ for every $n\ge 5$, which would imply $C_{79}=1$. |
| 29 | +<a href="#ER2024-intro-bounds-open">[ER2024-intro-bounds-open]</a> |
| 30 | + |
| 31 | +## Known upper bounds |
| 32 | + |
| 33 | +| Bound | Reference | Comments | |
| 34 | +| ----- | --------- | -------- | |
| 35 | +| $1$ | <a href="#BR1997">[BR1997]</a> | Follows from $ed_{\mathbb C}(S_n)\le n-3$ for all $n\ge 5$, hence $C_{79}\le \limsup_{n\to\infty}(n-3)/n = 1$. <a href="#BR1997-sym-bounds">[BR1997-sym-bounds]</a> | |
| 36 | + |
| 37 | +## Known lower bounds |
| 38 | + |
| 39 | +| Bound | Reference | Comments | |
| 40 | +| ----- | --------- | -------- | |
| 41 | +| $1/2$ | <a href="#BR1997">[BR1997]</a>, <a href="#ER2024">[ER2024]</a> | In characteristic $0$, Buhler–Reichstein proved $ed_k(S_n)\ge \lfloor n/2 \rfloor$, and Duncan improved this to $ed_k(S_n)\ge \lfloor (n+1)/2 \rfloor$ for $n\ge 7$; ER2024 notes the $\lfloor n/2 \rfloor$ argument also works for $\mathrm{char}(k)\ne 2$. In particular, over $\mathbb C$ either estimate yields $C_{79}\ge 1/2$. <a href="#ER2024-intro-bounds-open">[ER2024-intro-bounds-open]</a> <a href="#BR1997-sym-bounds">[BR1997-sym-bounds]</a> | |
| 42 | + |
| 43 | +## Additional comments and links |
| 44 | + |
| 45 | +- **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 $n\ge 8$. |
| 46 | + <a href="#BR1997-sym-bounds">[BR1997-sym-bounds]</a> <a href="#Duncan2010-exact7">[Duncan2010-exact7]</a> <a href="#ER2024-intro-bounds-open">[ER2024-intro-bounds-open]</a> |
| 47 | + |
| 48 | +- **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. |
| 49 | + <a href="#BR1997-cor42-dk-edSn">[BR1997-cor42-dk-edSn]</a> <a href="#Duncan2010-h13">[Duncan2010-h13]</a> |
| 50 | + |
| 51 | +- **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. |
| 52 | + <a href="#ER2024-intro-bounds-open">[ER2024-intro-bounds-open]</a> <a href="#BR1997-sym-bounds">[BR1997-sym-bounds]</a> <a href="#ER2024-poschar">[ER2024-poschar]</a> |
| 53 | + |
| 54 | +- **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 $n\ge 5$. |
| 55 | + <a href="#Sutherland2023-status">[Sutherland2023-status]</a> |
| 56 | + |
| 57 | +- **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. |
| 58 | + <a href="#ER2026-h13-poschar">[ER2026-h13-poschar]</a> |
| 59 | + |
| 60 | +## References |
| 61 | + |
| 62 | +- <a id="BR1997"></a>**[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](https://doi.org/10.1023/A:1000144403695). Author-hosted PDF: [edtotal.pdf](https://www.math.ubc.ca/~reichst/edtotal.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Joe+Buhler+Zinovy+Reichstein+On+the+essential+dimension+of+a+finite+group) |
| 63 | + - <a id="BR1997-cor42-dk-edSn"></a>**[BR1997-cor42-dk-edSn]** |
| 64 | + **loc:** Author-hosted PDF p.11, §4, file `edtotal.pdf` |
| 65 | + **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 $ed_k(S_n)$.” |
| 66 | + - <a id="BR1997-sym-bounds"></a>**[BR1997-sym-bounds]** |
| 67 | + **loc:** Author-hosted PDF p.17, §6.3 “Symmetric Groups”, file `edtotal.pdf` |
| 68 | + **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) $ed_k(S_4) = ed_k(S_5) = 2$, $ed_k(S_6) = 3$.” |
| 69 | + |
| 70 | +- <a id="Duncan2010"></a>**[Duncan2010]** Duncan, Alexander. *Essential dimensions of $A_7$ and $S_7$.* Mathematical Research Letters **17** (2010), no. 2, 263–266. arXiv PDF: [0908.3220](https://arxiv.org/pdf/0908.3220.pdf). DOI: [10.4310/MRL.2010.v17.n2.a5](https://doi.org/10.4310/MRL.2010.v17.n2.a5). [Google Scholar](https://scholar.google.com/scholar?q=Alexander+Duncan+Essential+dimensions+of+A7+and+S7) |
| 71 | + - <a id="Duncan2010-exact7"></a>**[Duncan2010-exact7]** |
| 72 | + **loc:** arXiv v1 PDF p.1, Introduction, file `0908.3220.pdf` |
| 73 | + **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. $ed_k(A_7) = ed_k(S_7) = 4$.” |
| 74 | + - <a id="Duncan2010-h13"></a>**[Duncan2010-h13]** |
| 75 | + **loc:** arXiv v1 PDF p.2, Introduction, file `0908.3220.pdf` |
| 76 | + **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.” |
| 77 | + |
| 78 | +- <a id="ER2024"></a>**[ER2024]** Edens, Oakley; Reichstein, Zinovy. *Essential dimension of symmetric groups in prime characteristic.* Comptes Rendus. Mathématique **362** (2024), 639–647. arXiv PDF: [2308.10096](https://arxiv.org/pdf/2308.10096.pdf). DOI: [10.5802/crmath.577](https://doi.org/10.5802/crmath.577). [Google Scholar](https://scholar.google.com/scholar?q=Oakley+Edens+Zinovy+Reichstein+Essential+dimension+of+symmetric+groups+in+prime+characteristic) |
| 79 | + - <a id="ER2024-def"></a>**[ER2024-def]** |
| 80 | + **loc:** Numdam PDF p.2, Abstract, file `10.5802/crmath.577.pdf` |
| 81 | + **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.” |
| 82 | + - <a id="ER2024-intro-bounds-open"></a>**[ER2024-intro-bounds-open]** |
| 83 | + **loc:** Numdam PDF p.3, §1 “Introduction”, file `10.5802/crmath.577.pdf` |
| 84 | + **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 $0$.” |
| 85 | + - <a id="ER2024-poschar"></a>**[ER2024-poschar]** |
| 86 | + **loc:** Numdam PDF p.2, Abstract, file `10.5802/crmath.577.pdf` |
| 87 | + **quote:** “In this paper we show that for every prime $p$ there are infinitely many positive integers $n$ such that $ed_{\mathbb{F}\_p}(S_n) \le n - 4$.” |
| 88 | + |
| 89 | +- <a id="Sutherland2023"></a>**[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](https://arxiv.org/pdf/2312.04430.pdf). DOI: [10.48550/arXiv.2312.04430](https://doi.org/10.48550/arXiv.2312.04430). [Google Scholar](https://scholar.google.com/scholar?q=Alexander+J+Sutherland+A+Summary+of+Known+Bounds+on+the+Essential+Dimension+and+Resolvent+Degree+of+Finite+Groups) |
| 90 | + - <a id="Sutherland2023-status"></a>**[Sutherland2023-status]** |
| 91 | + **loc:** arXiv v1 PDF p.3, §3 “Lower Bounds on ed(G)”, Case 2, file `2312.04430.pdf` |
| 92 | + **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 $n \ge 5$.” |
| 93 | + |
| 94 | +- <a id="ER2026"></a>**[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](https://arxiv.org/pdf/2406.15954.pdf). DOI: [10.4171/DM/984](https://doi.org/10.4171/DM/984). [Google Scholar](https://scholar.google.com/scholar?q=Oakley+Edens+Zinovy+Reichstein+Hilbert%E2%80%99s+13th+problem+in+prime+characteristic) |
| 95 | + - <a id="ER2026-h13-poschar"></a>**[ER2026-h13-poschar]** |
| 96 | + **loc:** Doc. Math. PDF p.1, Abstract, file `dm984.pdf` |
| 97 | + **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.” |
| 98 | + |
| 99 | +## Contribution notes |
| 100 | + |
| 101 | +Prepared with assistance from ChatGPT 5.2 Pro. |
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