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MAG122pp. 4–15

From Hilbert’s 13th problem to essential dimension and back

  • Zinovy Reichstein

    University of British Columbia, Vancouver, Canada

1 Introduction

The problem of solving polynomial equations in one variable, i.e., equations of the form

f(x)=0,wheref(x)=xn+a1xn1++an,f(x)=0,\quad\textrm{where}\quad f(x)=x^{n}+a_{1}x^{n-1}+\cdots+a_{n},

goes back to ancient times. Here by “solving” I mean finding a procedure or a formula which produces a solution xx for a given set of coefficients a1,,ana_{1},\ldots,a_{n}. The terms “procedure” and “formula” are ambiguous; to get a well-posed problem, we need to specify what kinds of operations we are allowed to perform to obtain xx from a1,,ana_{1},\ldots,a_{n}. In the simplest setting, we are only allowed to perform the four arithmetic operations: addition, subtraction, multiplication and division. In other words, we are asking if the polynomial (1) has a root xx which is expressible as a rational function of a1,,ana_{1},\ldots,a_{n}. For a general polynomial of degree n2n\geqslant 2, the answer is clearly “no”; this was already known to the ancient Greeks. The focus then shifted to the problem of “solving polynomials in radicals”, where one is allowed to use the four arithmetic operations and radicals of any degree. Here the mmth radical (or root) of tt is a solution to


Mathematicians attempted to solve polynomial equations this way for centuries, but only succeeded for n=1n=1, 22, 33 and 44. It was shown by Ruffini, Abel and Galois in the early 19th century that a general polynomial of degree n5n\geqslant 5 cannot be solved in radicals. This was a ground-breaking discovery. However, the story does not end there.

Suppose we allow one additional operation, namely solving


That is, we start with a1,,ana_{1},\ldots,a_{n}, and at each step, we are allowed to enlarge this collection by adding one new number, which is the sum, difference, product or quotient of two numbers in our collection, or a solution to (2) or (3) for any tt in our collection. In 1786, Bring [16 A. Chen, Y.-H. He and J. McKay, Erland Samuel Bring’s “Transformation of Algebraic Equations”, arXiv:1711.09253 (2017) ] showed that every polynomial equation of degree 55 can be solved using these operations.

Note that the coefficients of (2) and (3) only depend on one parameter tt. Thus roots of these equations can be thought of as ”algebraic functions” of one variable. By contrast, the coefficients of the general polynomial equation (1) depend on nn independent parameters a1,,ana_{1},\ldots,a_{n}. With this in mind, we define the resolvent degree rd(f)\operatorname{rd}(f) of a polynomial f(x)f(x) in (1) as the smallest positive integer rr such that every root of f(x)f(x) can be obtained from a1,,ana_{1},\ldots,a_{n} in a finite number of steps, assuming that at each step we are allowed to perform the four arithmetic operations and evaluate algebraic functions of rr variables. Let us denote the largest possible value of rd(f)\operatorname{rd}(f) by rd(n)\operatorname{rd}(n), as f(x)f(x) ranges over all polynomials of degree nn. The algebraic form of Hilbert’s 13th problem asks for the value of rd(n)\operatorname{rd}(n).

The actual wording of the 13th problem is a little different: Hilbert asked for the minimal integer rr one needs to solve every polynomial equation of degree nn, assuming that at each step one is allowed to perform the four arithmetic operations and apply any continuous (rather than algebraic) function in rr variables. Let us denote the maximal possible resolvent degree in this setting by crd(n)\operatorname{crd}(n), where “c” stands for “continuous”. Specifically, Hilbert asked whether or not crd(7)=3\operatorname{crd}(7)=3. In this form, Hilbert’s 13th problem was solved by Kolmogorov [37 A. N. Kolmogorov, On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables. Dokl. Akad. Nauk SSSR (N.S.)108, 179–182 (1956) ] and Arnold [1 V. I. Arnol'd, On functions of three variables. Dokl. Akad. Nauk SSSR114, 679–681 (1957) ] in 1957.1Arnold was a 19 year old undergraduate student in 1957. He later said that all of his numerous subsequent contributions to mathematics were, in one way or another, motivated by Hilbert’s 13th problem; see [2 V. I. Arnol'd, From Hilbert’s superposition problem to dynamical systems. Amer. Math. Monthly111, 608–624 (2004) ]. They showed that, contrary to Hilbert’s expectation, crd(n)=1\operatorname{crd}(n)=1 for every nn. In other words, continuous functions in 11 variable are enough to solve any polynomial equation of any degree. Moreover, any continuous function in nn variables can be expressed as a composition of functions of one variable and addition.

In spite of this achievement, Wikipedia lists the 13th problem as “unresolved”. While this designation is subjective, it reflects the view of many mathematicians that Hilbert’s true intention was to ask about rd(n)\operatorname{rd}(n), not crd(n)\operatorname{crd}(n). They point to the body of work on rd(n)\operatorname{rd}(n) going back centuries before Hilbert (see, e.g., [21 J. Dixmier, Histoire du 13e problème de Hilbert. In Analyse diophantienne et géométrie algébrique, Cahiers Sém. Hist. Math. Sér. 2, Univ. Paris VI, Paris, 85–94 (1993) ]) and to Hilbert’s own 20th century writings, where only rd(n)\operatorname{rd}(n) was considered (see, e.g., [31 D. Hilbert, Über die Gleichung neunten Grades. Math. Ann.97, 243–250 (1927) ]). Arnold himself was a strong proponent of this point of view [13 F. E. Browder (ed.), Mathematical developments arising from Hilbert problems, Proceedings of Symposia in Pure Mathematics, Vol. XXVIII, American Mathematical Society, Providence (1976) , pp. 45–46], [2 V. I. Arnol'd, From Hilbert’s superposition problem to dynamical systems. Amer. Math. Monthly111, 608–624 (2004) ].

Progress on the algebraic form of Hilbert’s 13th problem has been slow. From what I said above, rd(n)=1\operatorname{rd}(n)=1 when n5n\leqslant 5; this was known before Hilbert and even before Galois. The value of rd(n)\operatorname{rd}(n) remains open for every n6n\geqslant 6, and the possibility that rd(n)=1\operatorname{rd}(n)=1 for every nn has not been ruled out. The best known upper bounds on rd(n)\operatorname{rd}(n) are of the form rd(n)nα(n)\operatorname{rd}(n)\leqslant n-\alpha(n), where α(n)\alpha(n) is an unbounded but very slow growing function of nn. The list of people who have proved inequalities of this form includes some of the leading mathematicians of the past two centuries: Hamilton, Sylvester, Klein, Hilbert, Chebotarev, Segre, Brauer. Recently, their methods have been refined and their bounds sharpened by Wolfson [63 J. Wolfson, Tschirnhaus transformations after Hilbert. Enseign. Math.66, 489–540 (2020) ], Sutherland [60 A. J. Sutherland, Upper bounds on resolvent degree and its growth rate. arXiv:2107.08139 (2021) ] and Heberle–Sutherland [30 C. Heberle and A. J. Sutherland, Upper bounds on resolvent degree via sylvester’s obliteration algorithm. arXiv:2110.08670 (2021) ].

There is another reading of the 13th problem, to the effect that Hilbert meant to allow global multi-valued continuous functions; see [2 V. I. Arnol'd, From Hilbert’s superposition problem to dynamical systems. Amer. Math. Monthly111, 608–624 (2004) , p. 613]. These behave in many ways like algebraic functions. If we denote the resolvent degree in this sense by Crd(n)\operatorname{Crd}(n), where “C” stands for “global continuous”, then

1=crd(n)Crd(n)rd(n)nα(n).1=\operatorname{crd}(n)\leqslant\operatorname{Crd}(n)\leqslant\operatorname{rd}(n)\leqslant n-\alpha(n).

As far as I am aware, nothing else is known about Crd(n)\operatorname{Crd}(n) or rd(n)\operatorname{rd}(n) for n6n\geqslant 6.

On the other hand, in recent decades, considerable progress has been made in studying a related but different invariant, the essential dimension.2The term “essential dimension” was coined by Joe Buhler. The term “resolvent degree” was introduced by Richard Brauer in [8 R. Brauer, On the resolvent problem. Ann. Mat. Pura Appl. (4)102, 45–55 (1975) ]. Joe Buhler and I [14 J. Buhler and Z. Reichstein, On the essential dimension of a finite group. Compositio Math.106, 159–179 (1997) ] introduced this notion in the late 1990s. In special instances, it came up earlier, e.g., in the work of Kronecker [38 L. Kronecker, Ueber die Gleichungen fünften Grades. J. Reine Angew. Math.59, 306–310 (1861) ], Klein [35 F. Klein, Lectures on the icosahedron and the solution of equations of the fifth degree. Revised ed., Dover Publications, New York (1956) ], Chebotarev [15 N. G. Chebotarev, The problem of resolvents and critical manifolds. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR]7, 123–146 (1943) ], Procesi [48 C. Procesi, Non-commutative affine rings. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8)8, 237–255 (1967) ]3Procesi asked about the minimal number of independent parameters required to define a generic division algebra of degree nn. In modern terminology, this number is the essential dimension of the projective linear group PGLn\mathrm{PGL}_{n}. and Kawamata [34 Y. Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math.363, 1–46 (1985) ]4Kawamata defined an invariant Var(f)\operatorname{Var}(f) of an algebraic fiber space f ⁣:XSf\colon X\to S, which he informally described as “the number of moduli of fibers of ff in the sense of birational geometry”. In modern terminology, Var(f)\operatorname{Var}(f) is the essential dimension of ff.. Our focus in [14 J. Buhler and Z. Reichstein, On the essential dimension of a finite group. Compositio Math.106, 159–179 (1997) ] was on polynomials and field extensions. It later became clear that the notion of essential dimension is of interest in other contexts: quadratic forms, central simple algebras, torsors, moduli stacks, representations of groups and algebras, etc. In each case, it poses new questions about the underlying objects and occasionally leads to solutions of pre-existing open problems.

This paper has two goals. The first is to survey some of the research on essential dimension in Sections 27. This survey is not comprehensive; it is only intended to convey the flavor of the subject and sample some of its highlights. My second goal for this paper is to define the notion of resolvent degree of an algebraic group in Section 8, building on the work of Farb and Wolfson [25 B. Farb and J. Wolfson, Resolvent degree, Hilbert’s 13th problem and geometry. Enseign. Math.65, 303–376 (2019) ] but focusing on connected, rather than finite groups. The quantity rd(n)\operatorname{rd}(n) defined above is recovered in this setting as rd(Sn)\operatorname{rd}(\mathrm{S}_{n}). For more comprehensive surveys of essential dimension and resolvent degree, see [41 A. S. Merkurjev, Essential dimension: a survey. Transform. Groups18, 415–481 (2013) , 51 Z. Reichstein, Essential dimension. In Proceedings of the International Congress of Mathematicians, Vol. II (Hyderabad, India, 2010), Hindustan Book Agency, New Delhi, 162–188 (2011) ] and [25 B. Farb and J. Wolfson, Resolvent degree, Hilbert’s 13th problem and geometry. Enseign. Math.65, 303–376 (2019) ], respectively.

2 Essential dimension of a polynomial

Let kk be a base field, KK be a field containing kk and LL be a finite-dimensional KK-algebra (not necessarily commutative, associative or unital). We say that LL descends to an intermediate field kK0Kk\subset\nobreak K_{0}\subset K if there exists a finite-dimensional K0K_{0}-algebra L0L_{0} such that L=L0K0KL=L_{0}\otimes_{K_{0}}K. Equivalently, recall that, for any choice of an KK-vector space basis e1,,ene_{1},\ldots,e_{n} of LL, one can encode multiplication in LL into the n3n^{3} structure constants cijhKc_{ij}^{h}\in K given by eiej=h=1ncijhehe_{i}e_{j}=\sum_{h=1}^{n}c_{ij}^{h}e_{h}. Then LL descends to K0KK_{0}\subset K if and only if there exists a basis e1,,ene_{1},\ldots,e_{n} such that all of the structure constants eijhe_{ij}^{h} with respect to this basis lie in K0K_{0}. The essential dimension edk(L/K)\operatorname{ed}_{k}(L/K) is defined as the minimal value of the transcendence degree trdegk(K0)\operatorname{trdeg}_{k}(K_{0}), where LL descends to K0K_{0}. If the reference to the base field kk is clear from the context, we will write ed\operatorname{ed} in place of edk\operatorname{ed}_{k}.

If f(x)=xn+a1xn1++anf(x)=x^{n}+a_{1}x^{n-1}+\cdots+a_{n} is a polynomial over KK, for some a1,,ana_{1},\ldots,a_{n}, as in (1), we define edk(f)\operatorname{ed}_{k}(f) as edk(L/K)\operatorname{ed}_{k}(L/K), where L=K[x]/(f(x))L=K[x]/(f(x)). Note that if f(x)f(x) (or equivalently, LL) is separable over KK, then LL descends to K0K_{0} if and only if there exists an element yL\overline{y}\in L which generates LL as an KK-algebra and such that the minimal polynomial g(y)=yn+b1yn1++bng(y)=y^{n}+b_{1}y^{n-1}+\cdots+b_{n} of y\overline{y} lies in K0[y]K_{0}[y].

In classical language, the passage from f(x)f(x) to g(y)g(y) is called a Tschirnhaus transformation. Note that


for some c0,c1,,cn1Kc_{0},c_{1},\ldots,c_{n-1}\in K. Here xL\overline{x}\in L is xx modulo (f(x))(f(x)). Tschirnhaus’ strategy for solving polynomial equations in radicals by induction on degree was to transform f(x)f(x) to a simpler polynomial g(y)g(y), find a root of g(y)g(y) and then recover a root of f(x)f(x) from (4) by solving a polynomial equation of degree n1\leqslant n-1. In his 1683 paper [62 E. W. Tschirnhaus, A method for removing all intermediate terms from a given equation. ACM SIGSAM Bulletin, 37, 1–3 (2003) ], Tschirnhaus successfully implemented this strategy for n=3n=3 but made a mistake in implementing it for higher nn. Tschirnhaus did not know that a general polynomial of degree 5\geqslant 5 cannot be solved in radicals or that his method for solving cubic polynomials had been discovered by Cardano a century earlier.

Let us denote the maximal value of ed(f)\operatorname{ed}(f) taken over all field extensions K/kK/k and all separable polynomials f(x)K[x]f(x)\in K[x] of degree nn by edk(n)\operatorname{ed}_{k}(n). Kronecker [38 L. Kronecker, Ueber die Gleichungen fünften Grades. J. Reine Angew. Math.59, 306–310 (1861) ] and Klein [35 F. Klein, Lectures on the icosahedron and the solution of equations of the fifth degree. Revised ed., Dover Publications, New York (1956) ] showed that


This classical result is strengthened in [14 J. Buhler and Z. Reichstein, On the essential dimension of a finite group. Compositio Math.106, 159–179 (1997) ] as follows.

Theorem 1.

Assume char(k)2\operatorname{char}(k)\neq 2. Then edk(1)=0\operatorname{ed}_{k}(1)=0,


and edk(n+2)edk(n)+1\operatorname{ed}_{k}(n+2)\geqslant\operatorname{ed}_{k}(n)+1 for every n1n\geqslant 1. In particular,

n2edk(n)n3\smash[b]{\Bigl\lfloor\frac{n}{2}\Bigr\rfloor}\leqslant\operatorname{ed}_{k}(n)\leqslant n-3

for every n5n\geqslant 5.

I recently learned that a variant of the inequality edC(n)n2\operatorname{ed}_{\mathbb{C}}(n)\geqslant\bigl\lfloor\frac{n}{2}\bigr\rfloor was known to Chebotarev [15 N. G. Chebotarev, The problem of resolvents and critical manifolds. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR]7, 123–146 (1943) ] as far back as 1943.

The problem of finding the exact value of ed(n)\operatorname{ed}(n) may be viewed as being analogous to Hilbert’s 13th problem with rd(n)\operatorname{rd}(n), crd(n)\operatorname{crd}(n) or Crd(n)\operatorname{Crd}(n) replaced by ed(n)\operatorname{ed}(n). Since Hilbert specifically asked about rd(7)\operatorname{rd}(7), the case where n=7n=7 is of particular interest.

Theorem 2 (Duncan [23 A. Duncan, Essential dimensions of A7 and S7. Math. Res. Lett.17, 263–266 (2010) ]).

If char(k)=0\operatorname{char}(k)=0, then edk(7)=4\operatorname{ed}_{k}(7)=4.

The proof of Theorem 2 relies on the same general strategy as Klein’s proof of (5); I will discuss it further it in Section 6. Combining Theorem 2 with the inequality edk(n+2)edk(n)+1\operatorname{ed}_{k}(n+2)\geqslant\operatorname{ed}_{k}(n)+1 from Theorem 1, we can slightly strengthen (6) in characteristic 00 as follows:

n+12ed(n)n3for every n7.\smash[b]{\Bigl\lfloor\frac{n+1}{2}\Bigr\rfloor}\leqslant\operatorname{ed}(n)\leqslant n-3\quad\textrm{for every}\ n\geqslant 7.

Beyond (7), nothing is known about edC(n)\operatorname{ed}_{\mathbb{C}}(n) for any n8n\geqslant 8. I will explain where I think the difficulty lies in Section 5.

Analogous questions can be asked about polynomials that are not separable, assuming char(k)=p>0\operatorname{char}(k)=p>0. In this setting, the role of the degree is played by the “generalized degree” (n,e)(n,\mathbf{e}). Here n=[S:K]n=[S:K], where SS is the separable closure of KK in L=K[x]/(f(x))L=K[x]/(f(x)) and e=(e1,,er)\mathbf{e}=(e_{1},\dots,e_{r}) is the so-called type of the purely inseparable algebra L/SL/S defined as follows. Given xLx\in L, let us define the exponent exp(x,S)\exp(x,S) to be the smallest integer ee such that xpeSx^{p^{e}}\in S. Then e1e_{1} is the largest value of exp(x,S)\exp(x,S) as xx ranges over LL. Choose an x1Lx_{1}\in L of exponent e1e_{1}, and define e2e_{2} as the largest value of exp(x,S[x1])\exp(x,S[x_{1}]). Now choose x2Lx_{2}\in L of exponent e2e_{2}, and define e3e_{3} as the largest value of exp(x,S[x1,x2])\exp(x,S[x_{1},x_{2}]), etc. We stop when S[x1,,xr]=LS[x_{1},\ldots,x_{r}]=L. By a theorem of Pickert, the resulting integer sequence e1,,ere_{1},\ldots,e_{r} satisfies e1er1e_{1}\geqslant\cdots\geqslant e_{r}\geqslant 1 and does not depend on the choice of the elements x1,,xrx_{1},\ldots,x_{r}. One can now define edk(n,e)\operatorname{ed}_{k}(n,\mathbf{e}) by analogy with edk(n)\operatorname{ed}_{k}(n): edk(n,e)\operatorname{ed}_{k}(n,\mathbf{e}) is the maximal value of edk(f)\operatorname{ed}_{k}(f), as KK ranges over all field extension of kk and f(x)K[x]f(x)\in K[x] ranges over all polynomials of generalized degree (n,e)(n,\mathbf{e}). Surprisingly, the case where e\mathbf{e}\neq\emptyset (i.e., the polynomials f(x)f(x) in question are not separable) turns out to be easier. We refer the reader to [53 Z. Reichstein and A. K. Shukla, Essential dimension of inseparable field extensions. Algebra Number Theory13, 513–530 (2019) ], where an exact formula for ed(n,e)\operatorname{ed}(n,\mathbf{e}) is obtained.

3 Essential dimension of a functor

Following Merkurjev [6 G. Berhuy and G. Favi, Essential dimension: a functorial point of view (after A. Merkurjev). Doc. Math.8, 279–330 (2003) ], we will now define essential dimension for a broader class of objects, beyond polynomials or finite-dimensional algebras. Let kk be a base field, which we assume to be fixed throughout, and F\mathcal{F} be a covariant functor from the category of field extensions K/kK/k to the category of sets. Any object αF(K)\alpha\in\mathcal{F}(K) in the image of the natural (“base change”) map F(K0)F(K)\mathcal{F}(K_{0})\to\mathcal{F}(K) is said to descend to K0K_{0}. The essential dimension edk(α)\operatorname{ed}_{k}(\alpha) is defined as the minimal value of trdegk(K0)\operatorname{trdeg}_{k}(K_{0}), where the minimum is taken over all intermediate fields kK0Kk\subset K_{0}\subset K such that α\alpha descends to K0K_{0}.

For example, consider the functor Assn\mathrm{Ass}_{n} of nn-dimensional associative algebras given by

Assn(K)={n-dimensional associative K-algebras,up to K-isomorphism}.\begin{split}\mathrm{Ass}_{n}(K)=\{&n\textrm{-dimensional associative $K$-algebras,}\\[-3.0pt] &\textrm{up to}\ K\textrm{-isomorphism}\}.\end{split}

For AAssn(K)A\in\mathrm{Ass}_{n}(K), the new definition of edk(A)\operatorname{ed}_{k}(A) is the same as the definition in the previous section. Recall that, after choosing a KK-basis for AA, we can describe AA completely in terms of the n3n^{3} structure constants cijhc_{ij}^{h}. In particular, AA descends to the subfield K0=k(cijh)K_{0}=k(c_{ij}^{h}) of KK, and consequently, edk(A)n3\operatorname{ed}_{k}(A)\leqslant n^{3}.

Another interesting example is the functor of non-degenerate nn-dimensional quadratic forms,

Quadn(K)={non-degenerate quadratic forms on Kn,up to K-isomorphism}.\begin{split}\mathrm{Quad}_{n}(K)=\{&\textrm{non-degenerate quadratic forms on}\ K^{n},\\[-3.0pt] &\textrm{up to}\ K\textrm{-isomorphism}\}.\end{split}

For simplicity, let us assume that the base field kk is of characteristic different from 22. Under this assumption, a quadratic form qq on KnK^{n} is the same thing as a symmetric bilinear form bb. One passes back and forth between qq and bb using the formulas

q(v)=b(v,v)andb(v,w)=11q(v+w)q(v)q(w)2q(v)=b(v,v)\quad\textrm{and}\quad b(v,w)=\frac{\vphantom{1^{1}}q(v+w)-q(v)-q(w)}{2}

for any v,wKnv,w\in K^{n}. The form qq (or equivalently, bb) is called degenerate if the linear form b(v,)b(v,*) is identically zero for some 0vKn0\neq v\in K^{n}. A variant of the Gram–Schmidt process shows that there exists an orthogonal basis of KnK^{n} with respect to bb. In other words, in some basis e1,,ene_{1},\ldots,e_{n} of KnK^{n}, qq can be written as


for some a1,,ana_{1},\ldots,a_{n} in KK. In particular, we have that qq descends to K0=k(a1,,an)K_{0}=k(a_{1},\ldots,a_{n}), and thus edk(q)n\operatorname{ed}_{k}(q)\leqslant n. Note that qq is non-degenerate if and only if a1,,an0a_{1},\ldots,a_{n}\neq 0.

Yet another interesting example is provided by the functor of elliptic curves,

Ell(K)={elliptic curves over K,up to K-isomorphism}.\operatorname{Ell}(K)=\{\textrm{elliptic curves over}\ K,\,\textrm{up to}\ K\textrm{-isomorphism}\}.

For simplicity, assume that char(k)2\operatorname{char}(k)\neq 2 or 33. Then every elliptic curve XX over KK is isomorphic to the plane curve cut out by a Weierstrass equation y2=x3+ax+by^{2}=x^{3}+ax+b for some a,bKa,b\in K. Hence, XX descends to K0=k(a,b)K_{0}=k(a,b) and ed(X)2\operatorname{ed}(X)\leqslant 2.

Informally, we think of F\mathcal{F} as specifying the type of algebraic object under consideration (e.g., algebras or quadratic forms or elliptic curves), F(K)\mathcal{F}(K) as the set of objects of this type defined over KK, and edk(α)\operatorname{ed}_{k}(\alpha) as the minimal number of parameters required to define α\alpha. In most cases, essential dimension varies from object to object, and it is natural to consider what happens under a “worst case scenario”, i.e., how many parameters are needed to define the most general object of a given type. This number is called the essential dimension of the functor F\mathcal{F}. That is,


as KK varies over all fields containing kk and α\alpha varies over F(K)\mathcal{F}(K). Note that edk(F)\operatorname{ed}_{k}(\mathcal{F}) can be either a non-negative integer or \infty. In particular, the arguments above yield

ed(Assn)n3,ed(Quadn)nanded(Ell)2.\operatorname{ed}(\mathrm{Ass}_{n})\leqslant n^{3},\quad\operatorname{ed}(\mathrm{Quad}_{n})\leqslant n\quad\textrm{and}\quad\operatorname{ed}(\operatorname{Ell})\leqslant 2.

One can show that the last two of these inequalities are, in fact, sharp. The exact value of ed(Assn)\operatorname{ed}(\mathrm{Ass}_{n}) is unknown for most nn; however, for large nn,




where Lien\mathrm{Lie}_{n} and Commn\operatorname{Comm}_{n} are the functors of nn-dimensional Lie algebras and commutative algebras, respectively. These formulas are deduced from the formulas for the dimensions of the varieties of structure constants for nn-dimensional associative, Lie and commutative algebras due to Neretin [44 Y. A. Neretin, An estimate for the number of parameters defining an n-dimensional algebra. Izv. Akad. Nauk SSSR Ser. Mat.51, 306–318, 447 (1987) ].5Note the resemblance of these asymptotic formulas to the classical theorem of Higman and Sims, which assert that the number of finite pp-groups of order pnp^{n} (up to isomorphism) is asymptotically p2n3/27+O(n8/3)p^{\smash[b]{2n^{3}/27+O(n^{8/3})}}. This is not an accident; see [45 B. Poonen, The moduli space of commutative algebras of finite rank. J. Eur. Math. Soc. (JEMS)10, 817–836 (2008) ].

This brings us to the functor H1(,G)H^{1}(*,G), where GG is an algebraic group defined over kk. The essential dimension of this functor is a numerical invariant of GG. This invariant has been extensively studied; it will be our main focus in the next section. The functor H1(,G)H^{1}(*,G) associates to a field K/kK/k, the set H1(K,G)H^{1}(K,G) of isomorphism classes of GG-torsors TT over KK. Recall that a GG-torsor over TT over KK is an algebraic variety with a GG-action defined over KK such that, over the algebraic closure K\overline{K}, TT becomes equivariantly isomorphic to GG acting on itself by left translations. If TT has a KK-point xx, then GTG\to T taking gg to gxg\cdot x is, in fact, an isomorphism over KK. In this case, the torsor TT is called “trivial” or “split”. The interesting (non-trivial) torsors over KK have no KK-points. For example, if G=C2G=C_{2} is a cyclic group of order 22 and char(k)2\operatorname{char}(k)\neq 2, then every C2C_{2}-torsor is of the form TaT_{a}, where TaT_{a} is the subvariety of A1\mathbb{A}^{1} cut out by the quadratic equation x2a=0x^{2}-a=0 for some aKa\in K. Informally, TaT_{a} is a pair of points (roots of this equation) permuted by C2C_{2}; it is split if and only if these points are defined over KK (i.e., aa is a complete square in KK). In fact, H1(K,C2)H^{1}(K,C_{2}) is in bijective correspondence with K/(K)2K^{*}/(K^{*})^{2} given by Taamod(K)2T_{a}\mapsto a\bmod(K^{*})^{2}, where KK^{*} is the multiplicative group of KK. Note that, in this example, H1(K,G)H^{1}(K,G) is, in fact, a group. This is the case whenever GG is abelian. For a non-abelian algebraic group GG, H1(K,G)H^{1}(K,G) carries no natural group structure; it is only a set with a marked element (the trivial torsor).

For many linear algebraic groups GG, the functor H1(,G)H^{1}(\ast,G) parametrizes interesting algebraic objects. For example, when GG is the orthogonal group On\mathrm{O}_{n}H1(,On)H^{1}(\ast,\mathrm{O}_{n}) is the functor Quadn\mathrm{Quad}_{n} we considered above. When GG is the projective linear group PGLn\mathrm{PGL}_{n}, H1(K,PGLn)H^{1}(K,\mathrm{PGL}_{n}) is the set of isomorphism classes of central simple algebras of degree nn over KK. When GG is the exceptional group of type G2G_{2}, H1(K,G2)H^{1}(K,G_{2}) is the set of isomorphism classes of octonion algebras over KK.

4 Essential dimension of an algebraic group

The essential dimension of the functor H1(,G)H^{1}(*,G) is abbreviated as edk(G)\operatorname{ed}_{k}(G). Here GG is an algebraic group defined over kk. This number is always finite if GG is linear but may be infinite if GG is an abelian variety [12 P. Brosnan and R. Sreekantan, Essential dimension of abelian varieties over number fields. C. R. Math. Acad. Sci. Paris346, 417–420 (2008) ]. If GG is the symmetric group Sn\mathrm{S}_{n}, then


where edk(n)\operatorname{ed}_{k}(n) is the quantity we defined and studied in Section 2. Indeed, H1(K,Sn)H^{1}(K,\mathrm{S}_{n}) is the set of étale algebras L/KL/K of degree nn. Étale algebras of degree nn are precisely the algebras of the form K[x]/(f(x))K[x]/(f(x)), where f(x)f(x) is a separable (but not necessarily irreducible) polynomial of degree nn over KK. Thus (8) is just a restatement of the definition of edk(n)\operatorname{ed}_{k}(n).

Another interesting example is the general linear group G=GLnG=\mathrm{GL}_{n}. Elements of H1(K,GLn)H^{1}(K,\mathrm{GL}_{n}) are the nn-dimensional vector spaces over KK. Since there is only one nn-dimensional KK-vector space up to KK-isomorphism, we see that H1(K,GLn)={1}H^{1}(K,\mathrm{GL}_{n})=\{1\}. In particular, every object of H1(K,GLn)H^{1}(K,\mathrm{GL}_{n}) descends to kk, and we conclude that edk(GLn)=0\operatorname{ed}_{k}(\mathrm{GL}_{n})=0. I will now give a brief summary of three methods for proving lower bounds on edk(G)\operatorname{ed}_{k}(G) for various linear algebraic groups GG.

4.1 Cohomological invariants

Let F\mathcal{F} be a covariant functor from the category of field extensions K/kK/k to the category of sets, as in the previous section. A cohomological invariant of degree dd for F\mathcal{F} is a morphism of functors

FHd(,M)\mathcal{F}\to H^{d}(*,M)

for some discrete Gal(k)\operatorname{Gal}(k)-module MM. In many interesting examples, M=μmM=\mu_{m} is the module of mmth roots of unity with a natural Gal(k)\operatorname{Gal}(k)-action (trivial if kk contains a primitive mm-th root of unity). The following observation is due to J.-P. Serre.

Theorem 3.

Assume that the base field kk is algebraically closed. If F\mathcal{F} has a non-trivial cohomological invariant FHd(,M)\mathcal{F}\to H^{d}(*,M), then edk(F)d\operatorname{ed}_{k}(\mathcal{F})\geqslant d.

The proof is an immediate consequence of the Serre vanishing theorem. Cohomological invariants of an algebraic group GG (or equivalently, of the functor H1(,G)H^{1}(\ast,G)) were introduced by Serre and Rost in the early 1990s, and have been extensively studied since then; see [57 J.-P. Serre, Cohomological invariants, Witt invariants, and trace forms. In Cohomological invariants in Galois cohomology, Univ. Lecture Ser. 28, American Mathematical Society, Providence, 1–100 (2003) ]. These invariants give rise to a number of interesting lower bounds on edk(G)\operatorname{ed}_{k}(G) for various groups GG; in particular,

  1. ed(On)n\operatorname{ed}(\mathrm{O}_{n})\geqslant n,

  2. ed(SOn)n1\operatorname{ed}(\mathrm{SO}_{n})\geqslant n-1 for every n3n\geqslant 3,

  3. ed(G2)3\operatorname{ed}(G_{2})\geqslant 3,

  4. ed(F4)5\operatorname{ed}(F_{4})\geqslant 5,

  5. ed(Sn)n2\operatorname{ed}(\mathrm{S}_{n})\geqslant\bigl\lfloor\frac{n}{2}\bigr\rfloor.

Inequalities (1), (2) and (3) turn out to be exact; (4) is best known, and (5) is best known for even nn; see (7).

4.2 Finite abelian subgroups

Theorem 4.

Let GG be a reductive group over kk and AA be a finite abelian subgroup of GG of rank rr.

  1. [55 Z. Reichstein and B. Youssin, Essential dimensions of algebraic groups and a resolution theorem for G-varieties. Canad. J. Math.52, 1018–1056 (2000) ] Assume char(k)=0\operatorname{char}(k)=0. If the centralizer CG(A)C_{G}(A) is finite, then ed(G)r\operatorname{ed}(G)\geqslant r.

  2. [29 P. Gille and Z. Reichstein, A lower bound on the essential dimension of a connected linear group. Comment. Math. Helv.84, 189–212 (2009) ] Assume char(k)\operatorname{char}(k) does not divide A\lvert A\rvert. If GG is connected and the dimension of the maximal torus of CG(A)C_{G}(A) is dd, then ed(G)rd\operatorname{ed}(G)\geqslant r-d.

Note that both parts are vacuous if AA lies in a maximal torus TT of GG. Indeed, in this case, the centralizer CG(A)C_{G}(A) contains TT, so drd\geqslant r. In other words, only non-toral finite abelian subgroups AA of linear algebraic groups are of interest here. These have been much studied and catalogued, starting with the work of Borel in the 1950s. Theorem 4 yields the best known lower bound on ed(G)\operatorname{ed}(G) in many cases, such as ed(E7)7\operatorname{ed}(E_{7})\geqslant 7 and ed(E8)9\operatorname{ed}(E_{8})\geqslant 9, where E7E_{7} denotes the split simply connected exceptional group of type E7E_{7} and similarly for E8E_{8}.

4.3 The Brauer class bound

Consider a linear algebraic group GG defined over our base field kk. Suppose GG fits into a central exact sequence of algebraic groups (again, defined over kk)

1DGG1,1\to D\to G\to\overline{G}\to 1,

where DD is diagonalizable over kk. For every field extension K/kK/k, this sequence gives rise to the exact sequence of pointed sets

H1(K,G)H1(K,G)H2(K,D).H^{1}(K,G)\to H^{1}(K,\overline{G})\overset{\partial}{\to}H^{2}(K,D).

Every element αH2(K,D)\alpha\in H^{2}(K,D) has an index, ind(α)\operatorname{ind}(\alpha), defined as follows. If DGmD\simeq\mathbb{G}_{m}, then α\alpha is a Brauer class over KK, and ind(α)\operatorname{ind}(\alpha) denotes the Schur index of α\alpha, as usual. In general, we consider the character group X(D)X(D) whose elements are homomorphisms x ⁣:DGmx\colon D\to\mathbb{G}_{m}. Note that X(D)X(D) is a finitely generated abelian group and each character xX(D)x\in X(D) induces a homomorphism

x ⁣:H2(K,D)H2(K,Gm).x_{*}\colon H^{2}(K,D)\to H^{2}(K,\mathbb{G}_{m}).

The index of αH2(K,D)\alpha\in H^{2}(K,D) is defined as the minimal value of


as {x1,,xr}\{x_{1},\ldots,x_{r}\} ranges over generating sets of X(D)X(D). Here each (xi)(α)(x_{i})_{*}(\alpha) lies in H1(K,Gm)H^{1}(K,\mathbb{G}_{m}), and ind(xi)(α)\operatorname{ind}(x_{i})_{*}(\alpha) denotes its Schur index, as above. We now define ind(G,D)\operatorname{ind}(G,D) as the maximal index of α(H1(K,G))H2(K,D)\alpha\in\nobreak\partial(H^{1}(K,\overline{G}))\subset H^{2}(K,D), where the maximum is taken over all field extensions K/kK/k, as α\alpha ranges over the image H1(K,G)H^{1}(K,\overline{G}) in H2(K,D)H^{2}(K,D).

Theorem 5.

  1. ind(G,D)\operatorname{ind}(G,D) is the greatest common divisor of dim(ρ)\dim(\rho), where ρ\rho ranges over the linear representations of GG over kk such that the restriction ρD\rho_{|D} is faithful.

  2. Let pp be a prime different from char(k)\operatorname{char}(k). Assume that the exponent of every element of H2(K,D)H^{2}(K,D) in the image of

     ⁣:H1(K,G)H2(K,D)\partial\colon H^{1}(K,\overline{G})\to H^{2}(K,D)

    is a power of pp for every field extension K/kK/k. (This is automatic if DD is a pp-group.) Then edk(G)ind(G,D)dim(G)\operatorname{ed}_{k}(G)\geqslant\operatorname{ind}(G,D)-\dim(G).

Part (1) is known as Merkurjev’s index formula. The inequality of part (2) is based on Karpenko’s incompressibility theorem. Part (b) first appeared in [9 P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension and algebraic stacks. arXiv:math/0701903 (2007) ] in the special case where D=GmD=\mathbb{G}_{m} or μpr\mu_{p^{r}} and in [26 M. Florence, On the essential dimension of cyclic p-groups. Invent. Math.171, 175–189 (2008) ] in an even more special case, where D=μpD=\mu_{p}. It was proved in full generality in [33 N. A. Karpenko and A. S. Merkurjev, Essential dimension of finite p-groups. Invent. Math.172, 491–508 (2008) ].

Theorem 5 is responsible for some of the strongest results in this theory, including the exact formulas for the essential dimension of a finite pp-group (Theorem 6 below), the essential pp-dimension of an algebraic torus, and the essential dimension of spinor groups Spinn\mathrm{Spin}_{n}. The latter turned out to increase exponentially in nn:

ed(Spinn)2(n1)/2n(n1)2.\operatorname{ed}(\mathrm{Spin}_{n})\geqslant 2^{\lfloor(n-1)/2\rfloor}-\frac{n(n-1)}{2}.

This inequality was first proved in [9 P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension and algebraic stacks. arXiv:math/0701903 (2007) ]. The exact value of ed(Spinn)\operatorname{ed}(\mathrm{Spin}_{n}) subsequently got pinned down in [10 P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension, spinor groups, and quadratic forms. Ann. of Math. (2)171, 533–544 (2010) , 18 V. Chernousov and A. Merkurjev, Essential dimension of spinor and Clifford groups. Algebra Number Theory8, 457–472 (2014) ] in characteristic 00, [28 S. Garibaldi and R. M. Guralnick, Spinors and essential dimension. Compos. Math.153, 535–556 (2017) ] in characteristic p2p\neq 2 and [61 B. Totaro, Essential dimension of the spin groups in characteristic 2. Comment. Math. Helv.94, 1–20 (2019) ] in characteristic 22. When n15n\geqslant 15, inequality (9) is sharp for n≢0n\not\equiv 0 modulo 44, and is off by 2ν2(n)2^{\nu_{2}(n)} otherwise. Here 2ν2(n)2^{\nu_{2}(n)} is the largest power of 22 dividing nn.

The exponential growth of ed(Spinn)\operatorname{ed}(\mathrm{Spin}_{n}) came as a surprise. Prior to [9 P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension and algebraic stacks. arXiv:math/0701903 (2007) ], the best known lower bounds on ed(Spinn)\operatorname{ed}(\mathrm{Spin}_{n}) were linear (see [19 V. Chernousov and J.-P. Serre, Lower bounds for essential dimensions via orthogonal representations. J. Algebra305, 1055–1070 (2006) , Section 7]), on the order of n2\frac{n}{2}. Moreover, the exact values of ed(Spinn)\operatorname{ed}(\mathrm{Spin}_{n}) for n14n\leqslant 14 obtained by Rost and Garibaldi [27 S. Garibaldi, Cohomological invariants: exceptional groups and spin groups. Mem. Amer. Math. Soc.200, xii+81 (2009) ] appeared to suggest that these linear bounds should be sharp. The fact that ed(Spinn)\operatorname{ed}(\mathrm{Spin}_{n}) increases exponentially in nn has found interesting applications in the theory of quadratic forms. For details, see [10 P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension, spinor groups, and quadratic forms. Ann. of Math. (2)171, 533–544 (2010) , 18 V. Chernousov and A. Merkurjev, Essential dimension of spinor and Clifford groups. Algebra Number Theory8, 457–472 (2014) ].

5 Essential dimension at pp

Once again, fix a base field kk, and let F\mathcal{F} be a covariant functor from the category of field extensions K/kK/k to the category of sets. The essential dimension edk(α;p)\operatorname{ed}_{k}(\alpha;p) of an object αF(K)\alpha\in\mathcal{F}(K) at a prime pp is defined as the minimal value of edk(α;p)\operatorname{ed}_{k}(\alpha^{\prime};p), where the minimum ranges over all finite field extensions K/KK^{\prime}/K of degree prime to pp and α\alpha^{\prime} denotes the image of α\alpha under the natural map F(K)F(K)\mathcal{F}(K)\to\mathcal{F}(K^{\prime}). Finally, the essential dimension edk(F;p)\operatorname{ed}_{k}(\mathcal{F};p) of F\mathcal{F} at pp is the maximal value of edk(α)\operatorname{ed}_{k}(\alpha), as KK ranges over all fields containing kk and α\alpha ranges over F(K)\mathcal{F}(K). When F=H1(,G)\mathcal{F}=H^{1}(*,G) for an algebraic group GG, we write edk(G;p)\operatorname{ed}_{k}(G;p) in place of edk(F;p)\operatorname{ed}_{k}(\mathcal{F};p). Once again, if the reference to the base field is clear from the context, we will abbreviate edk\operatorname{ed}_{k} as ed\operatorname{ed}. By definition, ed(α;p)ed(α)\operatorname{ed}(\alpha;p)\leqslant\operatorname{ed}(\alpha) and ed(F;p)ed(F)\operatorname{ed}(\mathcal{F};p)\leqslant\operatorname{ed}(\mathcal{F}).

The reason to consider ed(F;p)\operatorname{ed}(\mathcal{F};p) in place of ed(F)\operatorname{ed}(\mathcal{F}) is that the former is often more accessible. In fact, most of the methods we have for proving a lower bound on edk(α)\operatorname{ed}_{k}(\alpha) (respectively, edk(F)\operatorname{ed}_{k}(\mathcal{F})) turn out to produce a lower bound on edk(α;p)\operatorname{ed}_{k}(\alpha;p) (respectively, edk(F;p)\operatorname{ed}_{k}(\mathcal{F};p)) for some prime pp. For example, the lower bound in Theorem 5 (b) is really edk(G;p)ind(G,D)dim(G)\operatorname{ed}_{k}(G;p)\geqslant\operatorname{ind}(G,D)-\dim(G). In Theorem 4, one can usually choose AA to be a pp-group, in which case the conclusion can be strengthened to ed(G;p)r\operatorname{ed}(G;p)\geqslant r in part (a) and ed(G;p)rd\operatorname{ed}(G;p)\geqslant r-d in part (b). In Theorem 3, if MM is pp-torsion (which can often be arranged), then ed(G;p)d\operatorname{ed}(G;p)\geqslant d.

This is a special case of a general meta-mathematical phenomenon: many problems concerning algebraic objects (such as finite-dimensional algebras or polynomials or algebraic varieties) over fields KK can be subdivided into two types. In type 1 problems, we are allowed to pass from KK to a finite extension K/KK^{\prime}/K of degree prime to pp, for one prime pp, whereas in type 2 problems this is not allowed. For example, given an algebraic variety XX defined over KK, deciding whether or not XX has a 00-cycle of degree 1 is a type 1 problem (it is equivalent to showing that there is a 00-cycle of degree prime to pp, for every prime pp), whereas deciding whether or not XX has a KK-point is a type 2 problem. As I observed in [51 Z. Reichstein, Essential dimension. In Proceedings of the International Congress of Mathematicians, Vol. II (Hyderabad, India, 2010), Hindustan Book Agency, New Delhi, 162–188 (2011) , Section 5], most of the technical tools we have are tailor-made for type 1 problems, whereas many long-standing open questions across several areas of algebra and algebraic geometry are of type 2.

In the context of essential dimension, the problem of computing ed(G;p)\operatorname{ed}(G;p) for a given algebraic group GG and a given prime pp is of type 1, whereas the problem of computing ed(G)\operatorname{ed}(G) is of type 2. For simplicity, let us assume that GG is a finite group. In this case, edk(G;p)=edk(Gp;p)\operatorname{ed}_{k}(G;p)=\operatorname{ed}_{k}(G_{p};p), where GpG_{p} is the Sylow pp-subgroup of GG. In other words, the problem of computing edk(G;p)\operatorname{ed}_{k}(G;p) reduces to the case where GG is a pp-group. In this case, we have the following remarkable theorem of Karpenko and Merkurjev [32 N. Karpenko and Z. Reichstein, A numerical invariant for linear representations of finite groups. Comment. Math. Helv.90, 667–701 (2015) ].

Theorem 6.

Let pp be a prime and kk be a field containing a primitive ppth root of unity. Then, for any finite pp-group PP,


where rdimk(P)\operatorname{rdim}_{k}(P) denotes the minimal dimension of a faithful representation of PP defined over kk.

Theorem 6 reduces the computation of edk(G;p)\operatorname{ed}_{k}(G;p) to rdimk(Gp)\operatorname{rdim}_{k}(G_{p}). For a given finite pp-group PP, one can often (though not always) compute rdimk(P)\operatorname{rdim}_{k}(P) in closed form using the machinery of character theory; see, e.g., [3 M. Bardestani, K. Mallahi-Karai and H. Salmasian, Kirillov’s orbit method and polynomiality of the faithful dimension of p-groups. Compos. Math.155, 1618–1654 (2019) , 36 H. Knight, The essential p-dimension of finite simple groups of Lie type. arXiv:2109.02698 (2021) , 42 A. Meyer and Z. Reichstein, Some consequences of the Karpenko–Merkurjev theorem. Doc. Math. 445–457 (2010) , 43 A. Moreto, On the minimal dimension of a faithful linear representation of a finite group. arXiv:2102.01463 (2021) ].

The situation is quite different when computing edk(G)\operatorname{ed}_{k}(G) for an arbitrary finite group GG. Clearly, edk(G)maxpedk(G;p)\operatorname{ed}_{k}(G)\geqslant\max_{p}\operatorname{ed}_{k}(G;p), where pp ranges over the prime integers. In those cases, where edk(G)\operatorname{ed}_{k}(G) is strictly larger than maxpedk(G;p)\max_{p}\operatorname{ed}_{k}(G;p), the exact value of edk(G)\operatorname{ed}_{k}(G) is usually difficult to establish. The only approach that has been successful to date relies on classification results in algebraic geometry, which are currently only available in low dimensions. I will return to this topic in the next section.

To illustrate the distinction between type 1 and type 2 problems, consider the symmetric group G=SnG=\mathrm{S}_{n}. For simplicity, assume that k=Ck=\mathbb{C} is the field of complex numbers. Here the type 1 problem is solved completely: edC(Sn;p)=np\operatorname{ed}_{\mathbb{C}}(\mathrm{S}_{n};p)=\bigl\lfloor\frac{n}{p}\bigr\rfloor for every prime pp. Thus maxpedC(Sn;p)=n2\max_{p}\operatorname{ed}_{\mathbb{C}}(\mathrm{S}_{n};p)=\bigl\lfloor\frac{n}{2}\bigr\rfloor, and (7) tells us that

edC(Sn)>maxpedC(G;p)for every odd n7.\operatorname{ed}_{\mathbb{C}}(\mathrm{S}_{n})>\max_{p}\operatorname{ed}_{\mathbb{C}}(G;p)\quad\textrm{for every odd}\ n\geqslant 7.

The remaining type 2 problem is to bridge the gap between n2\bigl\lfloor\frac{n}{2}\bigr\rfloor and the true value of edC(Sn)\operatorname{ed}_{\mathbb{C}}(\mathrm{S}_{n}). This problem has only been solved for n7n\leqslant 7; see Theorems 1, 2 and (8).

Note that the algebraic form of Hilbert’s 13th problem is also of type 2 in the sense that

rd(f;p)1\operatorname{rd}(f;p)\leqslant 1

for any prime pp, every field KK and every separable polynomial f(x)K[x]f(x)\in K[x].6For the precise definitions of rd(f)\operatorname{rd}(f) and rd(f;p)\operatorname{rd}(f;p), see Section 8