1 Introduction
The problem of solving polynomial equations in one variable, i.e., equations of the form
goes back to ancient times. Here by “solving” I mean finding a procedure or a formula which produces a solution for a given set of coefficients . 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 from . 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 which is expressible as a rational function of . For a general polynomial of degree , 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 th radical (or root) of is a solution to
Mathematicians attempted to solve polynomial equations this way for centuries, but only succeeded for , , and . It was shown by Ruffini, Abel and Galois in the early 19th century that a general polynomial of degree 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 , 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 in our collection. In 1786, Bring [16] showed that every polynomial equation of degree can be solved using these operations.
Note that the coefficients of (2) and (3) only depend on one parameter . 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 independent parameters . With this in mind, we define the resolvent degree of a polynomial in (1) as the smallest positive integer such that every root of can be obtained from 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 variables. Let us denote the largest possible value of by , as ranges over all polynomials of degree . The algebraic form of Hilbert’s 13th problem asks for the value of .
The actual wording of the 13th problem is a little different: Hilbert asked for the minimal integer one needs to solve every polynomial equation of degree , assuming that at each step one is allowed to perform the four arithmetic operations and apply any continuous (rather than algebraic) function in variables. Let us denote the maximal possible resolvent degree in this setting by , where “c” stands for “continuous”. Specifically, Hilbert asked whether or not . In this form, Hilbert’s 13th problem was solved by Kolmogorov [37] and Arnold [1] in 1957.1 They showed that, contrary to Hilbert’s expectation, for every . In other words, continuous functions in variable are enough to solve any polynomial equation of any degree. Moreover, any continuous function in 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 , not . They point to the body of work on going back centuries before Hilbert (see, e.g., [21]) and to Hilbert’s own 20th century writings, where only was considered (see, e.g., [31]). Arnold himself was a strong proponent of this point of view [13, pp. 45–46], [2].
Progress on the algebraic form of Hilbert’s 13th problem has been slow. From what I said above, when ; this was known before Hilbert and even before Galois. The value of remains open for every , and the possibility that for every has not been ruled out. The best known upper bounds on are of the form , where is an unbounded but very slow growing function of . 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], Sutherland [60] and Heberle–Sutherland [30].
There is another reading of the 13th problem, to the effect that Hilbert meant to allow global multi-valued continuous functions; see [2, p. 613]. These behave in many ways like algebraic functions. If we denote the resolvent degree in this sense by , where “C” stands for “global continuous”, then
As far as I am aware, nothing else is known about or for .
On the other hand, in recent decades, considerable progress has been made in studying a related but different invariant, the essential dimension.2 Joe Buhler and I [14] introduced this notion in the late 1990s. In special instances, it came up earlier, e.g., in the work of Kronecker [38], Klein [35], Chebotarev [15], Procesi [48]3 and Kawamata [34]4. Our focus in [14] 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 2–7. 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] but focusing on connected, rather than finite groups. The quantity defined above is recovered in this setting as . For more comprehensive surveys of essential dimension and resolvent degree, see [41, 51] and [25], respectively.
2 Essential dimension of a polynomial
Let be a base field, be a field containing and be a finite-dimensional -algebra (not necessarily commutative, associative or unital). We say that descends to an intermediate field if there exists a finite-dimensional -algebra such that . Equivalently, recall that, for any choice of an -vector space basis of , one can encode multiplication in into the structure constants given by . Then descends to if and only if there exists a basis such that all of the structure constants with respect to this basis lie in . The essential dimension is defined as the minimal value of the transcendence degree , where descends to . If the reference to the base field is clear from the context, we will write in place of .
If is a polynomial over , for some , as in (1), we define as , where . Note that if (or equivalently, ) is separable over , then descends to if and only if there exists an element which generates as an -algebra and such that the minimal polynomial of lies in .
In classical language, the passage from to is called a Tschirnhaus transformation. Note that
for some . Here is modulo . Tschirnhaus’ strategy for solving polynomial equations in radicals by induction on degree was to transform to a simpler polynomial , find a root of and then recover a root of from (4) by solving a polynomial equation of degree . In his 1683 paper [62], Tschirnhaus successfully implemented this strategy for but made a mistake in implementing it for higher . Tschirnhaus did not know that a general polynomial of degree 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 taken over all field extensions and all separable polynomials of degree by . Kronecker [38] and Klein [35] showed that
This classical result is strengthened in [14] as follows.
Assume . Then ,
and for every . In particular,
for every .
I recently learned that a variant of the inequality was known to Chebotarev [15] as far back as 1943.
The problem of finding the exact value of may be viewed as being analogous to Hilbert’s 13th problem with , or replaced by . Since Hilbert specifically asked about , the case where is of particular interest.
If , then .
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 from Theorem 1, we can slightly strengthen (6) in characteristic as follows:
Beyond (7), nothing is known about for any . I will explain where I think the difficulty lies in Section 5.
Analogous questions can be asked about polynomials that are not separable, assuming . In this setting, the role of the degree is played by the “generalized degree” . Here , where is the separable closure of in and is the so-called type of the purely inseparable algebra defined as follows. Given , let us define the exponent to be the smallest integer such that . Then is the largest value of as ranges over . Choose an of exponent , and define as the largest value of . Now choose of exponent , and define as the largest value of , etc. We stop when . By a theorem of Pickert, the resulting integer sequence satisfies and does not depend on the choice of the elements . One can now define by analogy with : is the maximal value of , as ranges over all field extension of and ranges over all polynomials of generalized degree . Surprisingly, the case where (i.e., the polynomials in question are not separable) turns out to be easier. We refer the reader to [53], where an exact formula for is obtained.
3 Essential dimension of a functor
Following Merkurjev [6], we will now define essential dimension for a broader class of objects, beyond polynomials or finite-dimensional algebras. Let be a base field, which we assume to be fixed throughout, and be a covariant functor from the category of field extensions to the category of sets. Any object in the image of the natural (“base change”) map is said to descend to . The essential dimension is defined as the minimal value of , where the minimum is taken over all intermediate fields such that descends to .
For example, consider the functor of -dimensional associative algebras given by
For , the new definition of is the same as the definition in the previous section. Recall that, after choosing a -basis for , we can describe completely in terms of the structure constants . In particular, descends to the subfield of , and consequently, .
Another interesting example is the functor of non-degenerate -dimensional quadratic forms,
For simplicity, let us assume that the base field is of characteristic different from . Under this assumption, a quadratic form on is the same thing as a symmetric bilinear form . One passes back and forth between and using the formulas
for any . The form (or equivalently, ) is called degenerate if the linear form is identically zero for some . A variant of the Gram–Schmidt process shows that there exists an orthogonal basis of with respect to . In other words, in some basis of , can be written as
for some in . In particular, we have that descends to , and thus . Note that is non-degenerate if and only if .
Yet another interesting example is provided by the functor of elliptic curves,
For simplicity, assume that or . Then every elliptic curve over is isomorphic to the plane curve cut out by a Weierstrass equation for some . Hence, descends to and .
Informally, we think of as specifying the type of algebraic object under consideration (e.g., algebras or quadratic forms or elliptic curves), as the set of objects of this type defined over , and as the minimal number of parameters required to define . 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 . That is,
as varies over all fields containing and varies over . Note that can be either a non-negative integer or . In particular, the arguments above yield
One can show that the last two of these inequalities are, in fact, sharp. The exact value of is unknown for most ; however, for large ,
Similarly,
where and are the functors of -dimensional Lie algebras and commutative algebras, respectively. These formulas are deduced from the formulas for the dimensions of the varieties of structure constants for -dimensional associative, Lie and commutative algebras due to Neretin [44].5
This brings us to the functor , where is an algebraic group defined over . The essential dimension of this functor is a numerical invariant of . This invariant has been extensively studied; it will be our main focus in the next section. The functor associates to a field , the set of isomorphism classes of -torsors over . Recall that a -torsor over over is an algebraic variety with a -action defined over such that, over the algebraic closure , becomes equivariantly isomorphic to acting on itself by left translations. If has a -point , then taking to is, in fact, an isomorphism over . In this case, the torsor is called “trivial” or “split”. The interesting (non-trivial) torsors over have no -points. For example, if is a cyclic group of order and , then every -torsor is of the form , where is the subvariety of cut out by the quadratic equation for some . Informally, is a pair of points (roots of this equation) permuted by ; it is split if and only if these points are defined over (i.e., is a complete square in ). In fact, is in bijective correspondence with given by , where is the multiplicative group of . Note that, in this example, is, in fact, a group. This is the case whenever is abelian. For a non-abelian algebraic group , carries no natural group structure; it is only a set with a marked element (the trivial torsor).
For many linear algebraic groups , the functor parametrizes interesting algebraic objects. For example, when is the orthogonal group , is the functor we considered above. When is the projective linear group , is the set of isomorphism classes of central simple algebras of degree over . When is the exceptional group of type , is the set of isomorphism classes of octonion algebras over .
4 Essential dimension of an algebraic group
The essential dimension of the functor is abbreviated as . Here is an algebraic group defined over . This number is always finite if is linear but may be infinite if is an abelian variety [12]. If is the symmetric group , then
where is the quantity we defined and studied in Section 2. Indeed, is the set of étale algebras of degree . Étale algebras of degree are precisely the algebras of the form , where is a separable (but not necessarily irreducible) polynomial of degree over . Thus (8) is just a restatement of the definition of .
Another interesting example is the general linear group . Elements of are the -dimensional vector spaces over . Since there is only one -dimensional -vector space up to -isomorphism, we see that . In particular, every object of descends to , and we conclude that . I will now give a brief summary of three methods for proving lower bounds on for various linear algebraic groups .
4.1 Cohomological invariants
Let be a covariant functor from the category of field extensions to the category of sets, as in the previous section. A cohomological invariant of degree for is a morphism of functors
for some discrete -module . In many interesting examples, is the module of th roots of unity with a natural -action (trivial if contains a primitive -th root of unity). The following observation is due to J.-P. Serre.
Assume that the base field is algebraically closed. If has a non-trivial cohomological invariant , then .
The proof is an immediate consequence of the Serre vanishing theorem. Cohomological invariants of an algebraic group (or equivalently, of the functor ) were introduced by Serre and Rost in the early 1990s, and have been extensively studied since then; see [57]. These invariants give rise to a number of interesting lower bounds on for various groups ; in particular,
,
for every ,
,
,
.
Inequalities (1), (2) and (3) turn out to be exact; (4) is best known, and (5) is best known for even ; see (7).
4.2 Finite abelian subgroups
Let be a reductive group over and be a finite abelian subgroup of of rank .
Note that both parts are vacuous if lies in a maximal torus of . Indeed, in this case, the centralizer contains , so . In other words, only non-toral finite abelian subgroups 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 in many cases, such as and , where denotes the split simply connected exceptional group of type and similarly for .
4.3 The Brauer class bound
Consider a linear algebraic group defined over our base field . Suppose fits into a central exact sequence of algebraic groups (again, defined over )
where is diagonalizable over . For every field extension , this sequence gives rise to the exact sequence of pointed sets
Every element has an index, , defined as follows. If , then is a Brauer class over , and denotes the Schur index of , as usual. In general, we consider the character group whose elements are homomorphisms . Note that is a finitely generated abelian group and each character induces a homomorphism
The index of is defined as the minimal value of
as ranges over generating sets of . Here each lies in , and denotes its Schur index, as above. We now define as the maximal index of , where the maximum is taken over all field extensions , as ranges over the image in .
is the greatest common divisor of , where ranges over the linear representations of over such that the restriction is faithful.
Let be a prime different from . Assume that the exponent of every element of in the image of
is a power of for every field extension . (This is automatic if is a -group.) Then .
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] in the special case where or and in [26] in an even more special case, where . It was proved in full generality in [33].
Theorem 5 is responsible for some of the strongest results in this theory, including the exact formulas for the essential dimension of a finite -group (Theorem 6 below), the essential -dimension of an algebraic torus, and the essential dimension of spinor groups . The latter turned out to increase exponentially in :
This inequality was first proved in [9]. The exact value of subsequently got pinned down in [10, 18] in characteristic , [28] in characteristic and [61] in characteristic . When , inequality (9) is sharp for modulo , and is off by otherwise. Here is the largest power of dividing .
The exponential growth of came as a surprise. Prior to [9], the best known lower bounds on were linear (see [19, Section 7]), on the order of . Moreover, the exact values of for obtained by Rost and Garibaldi [27] appeared to suggest that these linear bounds should be sharp. The fact that increases exponentially in has found interesting applications in the theory of quadratic forms. For details, see [10, 18].
5 Essential dimension at
Once again, fix a base field , and let be a covariant functor from the category of field extensions to the category of sets. The essential dimension of an object at a prime is defined as the minimal value of , where the minimum ranges over all finite field extensions of degree prime to and denotes the image of under the natural map . Finally, the essential dimension of at is the maximal value of , as ranges over all fields containing and ranges over . When for an algebraic group , we write in place of . Once again, if the reference to the base field is clear from the context, we will abbreviate as . By definition, and .
The reason to consider in place of is that the former is often more accessible. In fact, most of the methods we have for proving a lower bound on (respectively, ) turn out to produce a lower bound on (respectively, ) for some prime . For example, the lower bound in Theorem 5 (b) is really . In Theorem 4, one can usually choose to be a -group, in which case the conclusion can be strengthened to in part (a) and in part (b). In Theorem 3, if is -torsion (which can often be arranged), then .
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 can be subdivided into two types. In type 1 problems, we are allowed to pass from to a finite extension of degree prime to , for one prime , whereas in type 2 problems this is not allowed. For example, given an algebraic variety defined over , deciding whether or not has a -cycle of degree 1 is a type 1 problem (it is equivalent to showing that there is a -cycle of degree prime to , for every prime ), whereas deciding whether or not has a -point is a type 2 problem. As I observed in [51, 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 for a given algebraic group and a given prime is of type 1, whereas the problem of computing is of type 2. For simplicity, let us assume that is a finite group. In this case, , where is the Sylow -subgroup of . In other words, the problem of computing reduces to the case where is a -group. In this case, we have the following remarkable theorem of Karpenko and Merkurjev [32].
Let be a prime and be a field containing a primitive th root of unity. Then, for any finite -group ,
where denotes the minimal dimension of a faithful representation of defined over .
Theorem 6 reduces the computation of to . For a given finite -group , one can often (though not always) compute in closed form using the machinery of character theory; see, e.g., [3, 36, 42, 43].
The situation is quite different when computing for an arbitrary finite group . Clearly, , where ranges over the prime integers. In those cases, where is strictly larger than , the exact value of 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 . For simplicity, assume that is the field of complex numbers. Here the type 1 problem is solved completely: for every prime . Thus , and (7) tells us that
The remaining type 2 problem is to bridge the gap between and the true value of . This problem has only been solved for ; 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
for any prime , every field and every separable polynomial .6 Indeed, denote the Galois group of by . Then, after passing from to a finite extension whose degree is prime to , we may replace by its -Sylow subgroup . Since every -group is solvable, this means that becomes solvable in radicals over , and hence its resolvent degree becomes , as desired.
Inequality (10) accounts, at least in part, for the difficulty of showing that for any . The methods used to prove lower bounds on the essential dimension of algebraic groups in Section 4, and anything resembling these methods, cannot possibly work here; otherwise, we would also be able to prove that for some prime , contradicting (10).
A similar situation arises in computing the essential dimension of a finite -group over a field of characteristic . Superficially this problem looks very different from Hilbert’s 13th problem (where one usually works over ); the common feature is that both are type 2 problems. Indeed, it is shown in [54] that for every non-trivial -group . Using the method described in the next section, one can often show that , but we are not able to prove that for any -group and any field of characteristic . On the other hand, Ledet [39] conjectured that
for any prime and any infinite field of characteristic . Here denotes the cyclic group of order . Ledet showed that for every and that equality holds when .
My general feeling is that type 2 problems arising in different contexts are linked in some way, and that solving one of them (e.g., proving Ledet’s conjecture) can shed light on the others (e.g., Hilbert’s 13th problem). The only bit of evidence I have in this direction is the following theorem from [11] linking a priori unrelated type 2 problems in characteristic and in characteristic .
Let be a prime and be a finite group satisfying the following conditions:
does not have a non-trivial normal -subgroup, and
has an element of order .
If Ledet’s conjecture (11) holds, then .
The following family of examples is particularly striking. Let be a prime and a positive integer. Choose a positive integer such that is a prime. Note that, by Dirichlet’s theorem on primes in arithmetic progressions, there are infinitely many such . Let be a cyclic group of order . Then is cyclic of order ; let denote the unique subgroup of order . Applying Theorem 7 to , we obtain the following.
Corollary 8. If Ledet’s conjecture (11) holds, then
Note that, since the Sylow subgroups of are all cyclic,
for every prime , so the inequality of Corollary 8 is a type 2 result. An unconditional proof of this inequality or even of the weaker inequality is currently out of reach for any specific choice of and .
6 Essential dimension and the Jordan property
An alternative (equivalent) definition of essential dimension of a finite group is as follows. An action of on an algebraic variety is said to be linearizable if there exists a -equivariant dominant rational map for some linear representation . Then is the minimal value of , as ranges over all linearizable varieties with a faithful -action defined over . In particular, , where is the minimal dimension of a faithful linear representation of over , as in Theorem 6.
This geometric interpretation of can sometimes be used to prove lower bounds on by narrowing the possibilities for and ruling them out one by one using Theorem 4 (a). For the remainder of this section, I will assume that is a finite group and the base field is the field of complex numbers and will write in place of .
Suppose . Then is a single point, and only the trivial group can act faithfully on a point. Thus if and only if is the trivial group.
Now suppose . Then is a curve with a dominant map . By Lüroth’s theorem, is birationally isomorphic to and thus is a subgroup of . Finite subgroups of were classified by Klein [35]. Here is a complete list: cyclic groups and dihedral groups for every , , and . Theorem 4 (a) rules out the groups on this list which contain . We thus obtain the following.
Let be a finite group. Then if and only if is either cyclic or odd dihedral.
To classify groups of essential dimension (or more realistically, show that for a particular finite group ) in a similar manner, we need a classification of finite subgroups of , extending Klein’s classification of finite subgroups in . Here ranges over the unirational complex varieties of dimension , and denotes the groups of birational automorphisms of . In dimension , every unirational variety is rational, so we are talking about classifying finite subgroups of the Cremona group . Such a classification exists, though it is rather complicated; see [22]. Serre used this approach to show that (see [59, Theorem 3.6]). Again, this is a type 2 phenomenon since . Duncan [24] subsequently extended Serre’s argument to a full classification of finite groups of essential dimension .
In dimension , there is the additional complication that unirational complex varieties do not need to be rational. Here only a partial analogue of Klein’s classification exists, namely the classification of rationally connected -folds with the action of a finite simple group , due to Prokhorov [49]. Duncan used this classification to prove Theorem 2. More specifically, he showed that ; see (8). Subsequently, Beauville [4] showed that the only finite simple groups of essential dimension are and possibly .7
In dimension , even a partial analogue of Klein’s classification of finite subgroups of is out of reach. However, a recent break-through in Mori theory gives us a new insight into the asymptotic behavior of for certain infinite sequences of finite groups. Recall that an abstract group is called Jordan if there exists an integer (called a Jordan constant of ) such that every finite subgroup has a normal abelian subgroup of index . This definition, due to Popov [46], was motivated by the classical theorem of Camille Jordan which asserts that is Jordan, and by a theorem of Serre [58] which asserts that the Cremona group is also Jordan. The following result, due to Prokhorov, Shramov and Birkar8, is a far-reaching generalization of Serre’s theorem.
I will say that a collection of abstract groups is uniformly Jordan if they are all Jordan with the same constant.
Fix . Then the groups are uniformly Jordan, as ranges over -dimensional rationally connected complex varieties.
Unirational varieties are rationally connected. The converse is not known, though it is generally believed to be false. Rationally connected varieties naturally arise in the context of Mori theory, and we are forced to consider them even if we are only really interested in unirational varieties. Note that Theorem 10 does not become any easier to prove if one requires to be unirational. In fact, prior to Birkar [7] (respectively, prior to Prokhorov–Shramov [49]), it was an open question, due to Serre [58, Section 6], whether for each (respectively, for ) there exists even a single finite group which does not embed into .9
Now observe that, while every finite group is obviously Jordan, being uniformly Jordan is a strong condition on a sequence of finite groups
Suppose sequence (12) is chosen so that no infinite subsequence is uniformly Jordan. Then we claim that
Indeed, if , then there exists a -dimensional linearizable variety with a faithful -action. In particular, is contained in . Since is linearizable, it is unirational and hence rationally connected. On the other hand, since no infinite subsequence of (12) is uniformly Jordan, Theorem 10 tells us that there are at most finitely many groups with , and (13) follows. Here is an interesting family of examples.
For each positive integer , let be a cyclic group of order and be a subgroup of . If , then .
Note that this method does not give us any information about for any particular choice of and of . For example, while Theorem 11 tells us that
for all but finitely many primes , it does not allow us to exhibit a specific prime for which this inequality holds. The reason is that, when , a specific Jordan constant for the family of groups in Theorem 10 is out of reach. In particular, an unconditional proof of Corollary 8 along these lines does not appear feasible. Nevertheless, Theorem 11 represents a big step forward: previously, it was not even known that for any prime .
A classification of the subgroups of , as ranges over the unirational varieties of dimension is a rather blunt instrument. It would be preferable to find some topological or algebro-geometric obstruction to the existence of a linearization map , which can be read off from the -variety without enumerating all the possibilities for . Unfortunately, all known obstructions of this sort are of type 1: they do not distinguish between dominant rational maps and correspondences of degree prime to , for a suitable prime and thus cannot help us if .
Another draw-back of this method is that, as we mentioned in the previous section, beyond dimension ,10 none of the classification theorems we need are available in prime characteristic.
7 Essential dimension of a representation
7.1 Representations of finite groups in characteristic 0
Let be a finite group of exponent , be a field of characteristic , be a field extension, be a representation of , and be the character of . Can we realize over ? In other words, is there a representation such that and are equivalent over ? A celebrated theorem of Richard Brauer asserts that the answer is “yes” as long as contains a primitive root of unity of degree . If it does not, there is a classical way to quantify how far is from being definable over via the Schur index, at least in the case where is absolutely irreducible and the character value lies in for every . The Schur index of is defined as the index of the envelope
which, under our assumptions on , is a central simple algebra of degree over . The Schur index of is equal to the minimal degree of a field extension such that can be realized over .
The essential dimension gives us a different way to quantify how far is from being definable over . Here we do not need to assume that is irreducible or that its character values lie in . We simply think of as an object of the functor
The naive upper bound on is , where is the dimension of and is the minimal number of generators of . Indeed, if is generated by elements and is the matrix , then descends to the field
of transcendence degree at most over . It is shown in [32] that, in fact, and, moreover, . We have also proved lower bounds on in many cases (for details, see [32]). Note that these are quite delicate: by Brauer’s theorem, as long as contains suitable roots of unity.
7.2 Representations of finite groups in positive characteristic
Here the situation is entirely different.
Let be a finite group, be a field of characteristic and be the Sylow -subgroup of . Then
Note that, by a theorem of Higman, in characteristic , is cyclic if and only if the group algebra is of finite representation type, i.e., if and only if (or equivalently, ) has only finitely many indecomposable representations. Since is always of finite representation type in characteristic , we obtain the following.
Corollary 13. Let be a finite group and be a field of arbitrary characteristic. Then
if is of finite representation type, and
otherwise.
7.3 Representations of algebras
For simplicity, let us assume that the base field is algebraically closed. A celebrated theorem of Drozd asserts that every finite-dimensional -algebra falls into one of three categories: (a) finite representation type, (b) tame and (c) wild.
Informally speaking, is of tame representation type if, for every positive integer , the -dimensional indecomposable -modules occur in (at most) a finite number of one-parameter families. On the other hand, is of wild representation type if the representation theory of contains that of the free -algebra on two generators.
We can define the functor of representations in the same way as before: to a field , it associates isomorphism classes of finite-dimensional -modules. Corollary 13 tells us that, when is a group ring, the essential dimension of the functor distinguishes between algebras of finite representation type and algebras of the other two types. It does not distinguish between tame and wild representations types since in both cases. Benson suggested that it may be possible to distinguish between these two types of algebras by considering the rate of growth of , where is the set of isomorphism classes of -representations of of dimension . This is confirmed by the following theorem of Scavia [56].
If is of finite representation type, then is bounded from above as .
If is tame, then there exists a constant such that for every .
If is wild, then there exist constants such that for every .
This gives us three new invariants of finite-dimensional algebras, for . Informally, (respectively, ) quantifies “how wild” (respectively, “how tame”) is. Scavia [56] computed and explicitly in combinatorial terms in the case, where is a quiver algebra.
8 Back to resolvent degree
8.1 The level of a field extension
Let be a base field, be a field containing , and be a field extension of finite degree. I will say that is of level if there exists a finite tower of subfields
such that and for every . The level of is the smallest such ; I will denote it by . Clearly,
If is a field of rational functions on some algebraic variety defined over , then it is natural to think of elements of as algebraic (multi-valued) functions on in at most variables, and elements of as compositions of algebraic functions in at most variables.
Example 15. If the field extension is solvable, then we claim that . Indeed, here we can choose the tower (14) so that each is obtained from by adjoining a single radical. Then for each , and hence, , as claimed.
8.2 The resolvent degree of a functor
Let be a functor from the category of field extensions to the category of sets with a marked element. We will denote the marked element in by and will refer to it as being “split”. We will say that a field extension splits an object if . Here, as usual, denotes the image of under the natural map . Let us assume that
This is a strong condition of ; in particular, it implies that whenever is algebraically closed.
I will now define the resolvent degrees of and of the functor satisfying condition (15) by analogy with the definitions of and in Section 3. The resolvent degree is the minimal integer such that is split by a field extension of level (or equivalently, of level ). The resolvent degree is the maximal value of , as ranges over all fields containing and ranges over .
Example 16. Let be an integer not divisible by . Then the functor satisfies condition (15). I claim that this functor has resolvent degree . Indeed, let , and let be a primitive th root of unity in . By the Merkurjev–Suslin theorem, over , we can write
for some . Now splits . By our construction, is solvable over . Thus, as we saw in Example 15, . This shows that , as claimed. Using the norm residue isomorphism theorem (formerly known as the Bloch–Kato conjecture) in place of Merkurjev–Suslin, one shows in the same manner that has resolvent degree for every .
The resolvent degrees and at a prime are defined in the same way as and . Here is a functor satisfying (15), is an object of . That is, is the minimal value of , as ranges over all field extension of such that is finite and prime to , and is the maximal value of , where ranges over all field containing , and ranges over . A variant of the argument we used to prove (10) shows that for every base field , every functor satisfying (15) and every prime .
8.3 The resolvent degree of an algebraic group
The functor whose objects over are -torsors over satisfies condition (15) for every algebraic group defined over . I will write for the resolvent degree of this functor. For simplicity, let us assume that for the remainder of this section. I will write in place of .
Note that the quantity we defined in the introduction can be recovered in this setting as ; cf. (8). Moreover, for a finite group , our definition of coincides with the definition given by Farb and Wolfson in [25].
Recall that, for a polynomial , our definition of was motivated by wanting to express a root of as a composition of algebraic functions in variables applied to the coefficients. Equivalently, we wanted to find the smallest such that the -cycle in given by has an -point for some field extension of level . If is a linear algebraic group and is a -torsor, then our more general definition of retains this flavor. Indeed, is an affine variety defined over , and saying that is split by is the same as saying that has an -point.
While little is known about , it is natural to ask what is for other algebraic groups . Such questions can be thought of as variants of Hilbert’s 13th problem. Let us now take a closer look at the case where is linear and connected. The following folklore conjecture is implicit in the work of Tits.
Conjecture 17. Let be a connected complex linear algebraic group and be a field containing . Then every is split by some solvable field extension .
Since solvable extensions are of level , this conjecture implies that for every connected linear algebraic group .11 I can prove the following weaker inequality unconditionally [52].
Let be a connected complex linear algebraic group. Then .
Note that if we knew that for every , we would be able to conclude that for every field extension of finite degree. This would, in turn, imply that for every functor satisfying (15). Setting , we obtain for every algebraic group . In particular, if we were able to show that for some functor satisfying (15), we would be able to conclude that for some . This would constitute major progress on Hilbert’s 13th problem. I do not see how to reverse this implication though: an upper bound on for every connected group (such as the inequality of Theorem 18) does not appear to tell us anything about . However, Conjecture 17 and Theorem 18 make me take more seriously the possibility that may be identically or at least bounded as .
Acknowledgements I am grateful to EMS Magazine’s editor Vladimir L. Popov for asking me to write a survey on essential dimension and resolvent degree, and for his help with the exposition. I would also like to thank Mikhail Borovoi for stimulating discussions concerning Conjecture 17, and to Patrick Brosnan, Danny Ofek, Federico Scavia and Jesse Wolfson for constructive comments.
- 1
Arnold 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].
- 2
The term “essential dimension” was coined by Joe Buhler. The term “resolvent degree” was introduced by Richard Brauer in [8].
- 3
Procesi asked about the minimal number of independent parameters required to define a generic division algebra of degree . In modern terminology, this number is the essential dimension of the projective linear group .
- 4
Kawamata defined an invariant of an algebraic fiber space , which he informally described as “the number of moduli of fibers of in the sense of birational geometry”. In modern terminology, is the essential dimension of .
- 5
Note the resemblance of these asymptotic formulas to the classical theorem of Higman and Sims, which assert that the number of finite -groups of order (up to isomorphism) is asymptotically . This is not an accident; see [45].
- 6
For the precise definitions of and , see Section 8.
- 7
It is not known whether the essential dimension of is or .
- 8
Prokhorov and Shramov [50] proved this theorem assuming the Borisov–Alexeev–Borisov (BAB) conjecture. The BAB conjecture was subsequently proved by Birkar [7].
- 9
For the current status of Serre’s questions from [58, Section 6], see [47, Section 3].
- 10
Groups of essential dimension have been classified over an arbitrary field ; see [40, 20]. Recall that Theorem 9 assumes that .
- 11
Other interesting consequences of Conjecture 17 are discussed in [17].
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