Alberto Llorente

A refinement of Singer's bound for Liouvillian integration. The general case

Keywords: 2010 Mathematics Subject Classification: 20G20, 20C15 (primary), 12H05, 34M03, 68W30 (secondary) linear ordinary differential equations, Liouvillian solutions, complex primitive linear groups, Jordan bound, Singer bound

come

Introduction

Let K be an algebraically closed field of characteristic 0. The field of rational functions K = K(x) is closed for derivation and allows to define linear differential equations

y (r) = a 0 (x) y + a 1 (x) y + • • • + a r-1 (x) y (r-1) (1)
or explicit systems

y (r) = A 0 (x) y + A 1 (x) y + • • • + A r-1 (x) y (r-1) (2)
whose solutions may or may not be expressed in finite terms. By finite terms I mean a formula in the four field operations, derivatives, primitives, exponentials of primitives and roots of polynomials, which will be formalized in the concept of Liouvillian solution in §2.1. The computation of a non-null Liouvillian solution or the decision that they do not exist can be done algorithmically, as reviewed in §2.3, but all the known algorithms depend on a theorem of Singer (Theorem 13) that gives an explicit form of a first Liouvillian solution y 0 = 0 of (1), concretely having y 0 /y 0 algebraic of degree bounded by an arithmetic function I(r) that depends only on the order r of the equation. The optimal values of the function I(k) were only known up to k = 5, having to resort for higher order to a group-theoretical bound J of Jordan. The original article of Singer [START_REF] Michael | Liouvillian solutions of n-th order homogeneous linear differential equations[END_REF] took a rough bound J, giving a growth of log I(k) = O(k 2 log k). A history of the bounds J is told in §3.1. The optimal Jordan bound was given by Collins in [START_REF] Michael | On Jordan's theorem for complex linear groups[END_REF], yielding a growth of log J(k) = O(k log k). With this sharp Jordan bound, one needs to refine Singer's arguments, as done in §3.2, in order to achieve log I(k) = O(k log k).

Collins's bound is J(k) = (k + 1)! for k > 70, but we are also interested in the smaller values. In an auxiliary article [START_REF]Bounds for finite primitive complex linear groups[END_REF], Collins computes the optimal values J prim (k) restricting the group-theoretical considerations to primitive linear groups, which reaches the general value J prim (k) = (k + 1)! for k > 12. My refinement of Singer's argument allows us to take advantage of this restricted bound and afford I(k) = (k + 1)! for k 14, cf. §3.2. Although the present article focuses on the general case of the Singer bound, it is always important to keep the smallest amount of exceptions.

Jordan's group-theoretical consideration is the minimal index of an abelian normal subgroup of a linear group of degree k, while for the Singer bound we can drop the normality condition. This allows us to apply a reduction f k factor to J prim (k) in order to take account of the non-normal abelian subgroups, getting a better value for I(k). In this article I shall prove that the best reduction factor we can get using this refinement grows like f k = Θ(3 k/3 ), with 4 3 3 k/3 f k 3 √ 3 • 3 k/3 for k > 16, getting the reduced values of the restricted I prim (k) for these k and also the corresponding I(k) for k 19. In forthcoming articles I shall deal with the exceptions, refining I prim (k) for k 16 and computing the resulting I(k) for any k.

The classic algorithms for Liouvillian integration deal with differential equations like (1), having to convert the systems like (2) into scalar equations via Cyclic Vector Lemma, which is not very efficient. But Singer's theorem can be translated into systems (Theorem 14) and the algorithm proposed in my thesis [START_REF] Llorente | Métodos numérico-simbólicos para calcular soluciones liouvillianas de ecuaciones diferenciales lineales[END_REF] can handle systems directly.

In §2, I shall introduce the basic results on Liouvillian solutions, dealing with the bound I(k) in §3. The material up to §3.2 is based on the first chapter of my thesis, while the rest is new. In §4, I review part of Collins's work of [START_REF]Bounds for finite primitive complex linear groups[END_REF], where he uses strongly the Classification of Finite Simple Groups, for my further refinement in §6. Finally, I draw the conclusions of my refinement of Collins's work in §7, computing the resulting bound I with the new results.

Singer's bound for Liouvillian integration

Preliminary definitions

Let me introduce the basic definitions we will need from Differential Algebra. Definition 1. A differential field is a field K endowed with a unary operation , the derivation, that is additive and satisfies the Leibniz rule (a • b) = a • b + a • b for any a, b ∈ K. The field of constants of K is the subfield {a ∈ K : a = 0}. Definition 2. A differential subfield K of a differential field F is a subfield of F closed for the derivation. In this case we speak of the extension of differential fields F/K. Definition 3. An extension of differential fields F/K is Liouvillian if there exists a tower of differential fields

K = K 0 ⊂ K 1 ⊂ • • • ⊂ K m = F
such that the field of constants is the same in each step and each extension K i+1 /K i is K i+1 = K i (a i ) with one of the following options: a i ∈ K i , a i /a i ∈ K i or a i is algebraic over K i .

Remark 4. Informally speaking, a Liouvillian extension is made up by finitely many iterated steps of the four field operations, derivatives, primitives, exponentials of primitives and roots of polynomials.

Definition 5. A Liouvillian solution of a differential equation or system is a solution over a Liouvillian extension of the differential field where it is defined.

Definition 6. A Picard-Vessiot extension of an explicit system of linear differential equations ∆ of order r and dimension n over a differential field K is an extension of differential fields F/K with the same field of constants C and such that the C-vector space V ⊂ F n of solutions of ∆ has the expected dimension nr and F is generated as a differential field by K and the entries of V .

Remark 7. Picard-Vessiot extensions are granted to exist when the field of constants is algebraically closed, cf. [START_REF] Magid | Lectures on Differential Galois Theory[END_REF]thm. 3.4]. They are also granted to be unique up to isomorphism of differential fields, cf. [Mag97, thm. 3.5].

Definition 8. An extension of differential fields F/K has an associated Galois group Gal(F/K) of the automorphisms of F that commute with the derivation and keep K fixed. An explicit system of linear differential equations ∆ over a differential field K has the Galois group Gal(∆) of any Picard-Vessiot extension thereof.

Remark 9. The Galois group of a linear differential equation or system is a Zariskiclosed subgroup of GL(V ), where V is the vector space (over the field of constants) of the solutions.

Liouvillian solutions

Liouvillian integrability is the differential counterpart of solvability by radicals, as the following theorem states.

Theorem 10 (Kolchin). Let K be a differential field with algebraically closed field of constants K. Let ∆ be a linear differential equation with coefficients in K. All the solutions of ∆ are Liouvillian over K if and only if Gal(∆) • is solvable, where Gal(∆) • is the connected component of Gal(∆) containing the identity, which is a subgroup. [Kol48] [vdPS03,thm. 1.43] According to Lie-Kolchin Theorem, stated below, the solvability of Gal(∆) • is equivalent to its triangularizability.

Theorem 11 (Lie-Kolchin). A solvable connected linear algebraic group is triangularizable. In particular, it admits an invariant line. [START_REF] Kaplansky | An Introduction to Differential Algebra[END_REF]thm. 4.11] So, if all the solutions of ∆ are Liouvillian over K, then Gal(∆) • has an invariant line of solutions Ky, for certain solution y = 0 of ∆. In this case, y /y is algebraic over K, so y is a Liouvillian solution of a very special kind. What happens if ∆ has some non-zero Liouvillian solutions, but not necessarily all of them? After some technicalities, this case can be reduced to the previous one, proving the following result.

Theorem 12 (Vessiot-Kolchin). Let K be a differential field with algebraically closed field of constants. Let ∆ be a homogeneous linear differential equation with coefficients in K. Let F be a Picard-Vessiot extension of K for ∆. If ∆ admits a non-zero solution Liouvillian over K, then it has a non-zero solution y ∈ F such that y /y is algebraic over K.

This result is assumed as known by M. F. Singer in his proof of the next theorem, which is a stronger result that bounds the degree of y /y over K in terms of the order of ∆ alone.

Theorem 13 (Singer). Let K be a differential field with algebraically closed field of constants. Let ∆ be a homogeneous linear differential equation of order r with coefficients in K. Let F be a Picard-Vessiot extension of K for ∆. If ∆ admits a non-zero solution Liouvillian over K, then it has a non-zero solution y ∈ F such that y /y is algebraic over K of degree I(r) at most, for the function I of Theorem 15. [Sin81, thm. 2.4] Theorem 13 is stated for scalar equations, but it is valid for differential systems in the following form, though the classic algorithms are stated only for scalar equations. As I will need a precise form of this result for differential systems, I present here a complete proof.

Theorem 14. Let K be a differential field with algebraically closed field of constants. Let ∆ be an n × n explicit differential system of order r with coefficients in K. Let F be a Picard-Vessiot extension of K for ∆. If ∆ has a non-zero solution Liouvillian over K, then there exist an intermediate differential field F 0 and a non-zero solution (y 1 , y 2 , . . . , y n ) ∈ F n of ∆ such that F 0 /K is an algebraic extension of degree I(rn) at most (for the function I of Theorem 15) and, for each i and j with y i = 0, y i /y i ∈ F 0 and y j /y i ∈ F 0 .

Proof. Write

∆ :

y (r) = A 0 y + A 1 y + • • • + A r-1 y (r-1) , with A 0 , A 1 , . . . , A r-1 ∈ K n×n .
Let u = Bu be the companion system of ∆, with 1 u = y, y , . . . , y (r-1) . By virtue of Cyclic Vector Lemma, u = Bu is equivalent to a scalar equation

∆ 0 : v (rn) = a 0 v + a 1 v + • • • + a rn-1 v (rn-1) ,
with a 0 , a 1 , . . . , a rn-1 ∈ K and Pu = (v, v , . . . , v (rn-1) ) .

If ∆ has a non-zero solution Liouvillian over K, the associated solution of ∆ 0 is also non-zero and Liouvillian over K. According to Theorem 13, ∆ 0 has a non-zero solution v 0 ∈ F such that v 0 /v 0 is algebraic over K of degree I(rn) at most. The differential field F 0 = K(v 0 /v 0 ) is an intermediate field of F/K. Moreover F 0 /K is an algebraic extension of degree I(rn) at most. By induction one can prove that v

(k) 0 ∈ v 0 F 0 , hence the associated solution of u = Bu is u 0 = P -1 (v 0 , v 0 , . . . , v (rn-1) 0 ) ∈ v 0 F rn 0 1
Here, an underlined column vector is the (horizontal) list of its entries. and the associated solution y 0 = (y 1 , y 2 , . . . , y n ) of ∆ belongs to v 0 F n 0 . Clearly y 0 = 0. It is easy to prove that y 0 belongs to v 0 F n 0 . Therefore, for each i and j with y i = 0, y i /y i and y j /y i belong to F 0 .

It suffices to find a single non-zero Liouvillian solution because the classic d'Alembert reduction method reduces the problem to lower order. For a solution f = 0 of a scalar equation a r y (r) + a r-1 y (r-1) + • • • + a 0 y = 0, we apply the change of variable y = f u and get

b r-1 u (r-1) + b r-2 u (r-2) + • • • + b 0 u = 0.
Notice that, if a 0 , a 1 , . . . , a r lie in a differential field K,

then b 0 , b 1 , . . . , b r-1 lie in F = K(f /f ).
If f is a solution given by Theorem 13, then F/K is an algebraic extension of degree I(r) at most. A generalization for differential systems is found in [CL72, pp. 71-73]. For a system y = Ay, with A ∈ K n×n , and a particular solution f = (f 1 , f 2 , . . . , f n ) , with2 f 1 = 0, the reduced system u = Bu has B ∈ F (n-1)×(n-1) with

F = K(f 2 /f 1 , f 3 /f 1 , . . . , f n /f 1 ).
If f is a solution of Theorem 14, then F/K is an algebraic extension of degree I(n) at most.

Review of methods on Liouvillian solutions

The first complete algorithm for computing the Liouvillian solutions of a differential equation is Kovacic's, published in [START_REF] Kovacic | An algorithm for solving second order linear homogeneous differential equations[END_REF], valid for second-order equations over the Riemann sphere. Kovacic algorithm relies on the classification of the algebraic subgroups of SL(2, C), which is finer than the value I(2) = 12. If a second-order equation has non-zero Liouvillian solutions, it has a solution y = 0 with y /y an algebraic function of degree 1, 2, 4, 6 or 12. The Kovacic algorithm tries sequentially these options. It computes all the possible principal parts of the coefficients of the minimal polynomial of y /y at all the singularities of the equation and tries to glue them into a minimal polynomial. An alternative to Kovacic algorithm is Ulmer-Weil's, published in [START_REF] Ulmer | Note on Kovacic's algorithm[END_REF], valid for the same kind of equations. Ulmer-Weil algorithm uses the symmetric powers of the differential equation. The m-th symmetric power of a linear differential equation L is another differential equation L m whose solution space is spanned by the products y 1 y 2 • • • y m of solutions y i of L. If a second-order equation L has non-zero Liouvillian solutions, either it has a solution y = 0 with y /y a rational function or L m has a rational solution for m ∈ {1, 2, 4, 6, 8, 12}. A rational solution of L m gives a solution of L algebraic of degree m. The advantage of this method is that it reduces the problem of finding Liouvillian solutions almost to the problem of finding rational solutions of linear differential equations.

Theorem 13 is proved in [START_REF] Michael | Liouvillian solutions of n-th order homogeneous linear differential equations[END_REF] as an auxiliary result for an algorithm for computing the Liouvillian solutions of a linear differential equation L of any order r over rational functions. Singer algorithm uses the symmetric powers L m up to m = I(r). There are algorithms fit to third-order equations, and [START_REF] Cormier | Résolution des équations différentielles linéaires d'ordre 4 et 5: applications à la théorie de Galois classique[END_REF] makes feasible algorithms for order 4 and 5, but the bound I prim (6) 3780 leads to a symmetric power of order greater than 10 15 .

All these algorithms are completely algebraic and implementable in symbolic computation. The aim of my thesis [START_REF] Llorente | Métodos numérico-simbólicos para calcular soluciones liouvillianas de ecuaciones diferenciales lineales[END_REF] was to give an algorithm for computing the Liouvillian solutions of an explicit differential system over the Riemann sphere of any order and size. This algorithm does not use symbolic computation alone, but a symbiosis of symbolic computation and exact numerics. These results will be published in forthcoming articles. The bound I(rn) is used as a truncation parameter of the continuous fraction expansion. If the denominator of the number to expand surpasses I(rn), then we can abort the computation and treat the number as if it was irrational.

3 Review of some results on Singer's bound

In this section, I shall review some known results concerning the value of I(n). The first known estimation of this value is based on a group-theoretical theorem of Jordan (Theorem 17), and is due to Singer (Theorem 15). I shall provide a complete proof of Singer's result, keeping a finer track of the intermediate computations in order to use them afterwards. Results of Ulmer, van der Put-Singer and Malcev-Platonov will be also presented.

Singer's and Jordan's bounds

Now I shall explain the relation between Theorem 13 and Theorem 15. Let P ∈ K[x] be the minimal polynomial of such a quotient y /y. The group Gal(∆) permutes the roots of P . The stabilizer H of y /y has index equal to the degree of P . We have that y is a common eigenvector of H. Conversely, if u = 0 is a solution of ∆ that is a common eigenvector u of H 0 < Gal(∆) with [Gal(∆) : H 0 ] = k, then u /u is algebraic over K of degree k. This correspondence explains the following group-theoretical result that defines the function I.

Theorem 15 (Singer). There exists a function I : Z >0 → Z >0 such that, for each n and any field K algebraically closed, every subgroup G of GL(n, K) with a 1-reducible subgroup of finite index admits a 1-reducible subgroup of index I(n) at most. [Sin81, prop. 2.2] Definition 16. A linear group is called m-reducible if it has an invariant subspace of dimension m. In particular, a linear group is 1-reducible if it has a common eigenvector.

The proof of Theorem 15 uses a theorem of Jordan that he proved in [Jor1877] for a weaker version of Theorem 13.

Theorem 17 (Jordan). There exists a function J : Z >0 → Z >0 such that, for each n, every finite subgroup G of GL(n, C) admits an abelian normal subgroup of index J(n) at most.

Jordan's proof does not control the growth of J. A further result of Schur gives an explicit bound

J Schur (n) = √ 8n + 1 2n 2 - √ 8n -1 2n 2
, which satisfies log J Schur (n) = O(n 2 log n) asymptotically. The proof can be found in [START_REF] Curtis | Representation theory of finite groups and associative algebras[END_REF]§36]. Blichfeldt refined the bound; in [Dor71, §30] we find

J Blich (n) = 6 (n-1)(π(n+1)+1) n!,
where π is the prime-counting function, which satisfies

log J Blich (n) = O(n 2 / log n).
This growth order was improved after the Classification of Finite Simple Groups. Weisfeiler announced in [START_REF] Weisfeiler | Post-classification version of Jordan's theorem on finite linear groups[END_REF] a bound that satisfies

log J Weis (n) = O(n log 3 n),
but unfortunately he disappeared in the Andes and his work kept unfinished and unpublished.

Collins proved in [START_REF] Michael | On Jordan's theorem for complex linear groups[END_REF] that J 0 (n) = (n + 1)! is the optimal bound for n 71. It satisfies log J 0 (n) = O(n log n) asymptotically.

Remark 18. Although Jordan's original statement is in the complex field, the only property of C he uses is, according to [START_REF] Breuillard | An exposition of Jordan's original proof of his theorem on finite subgroups of GL n (C)[END_REF]§2], that every finite-order matrix is diagonalizable. In particular, it is valid for any algebraically closed field K (of characteristic 0). Other proofs and bounds are specific to the complex field; for instance, the proof in [START_REF] Curtis | Representation theory of finite groups and associative algebras[END_REF]§36] reduces to the unitary case and the bound computes volumes. Even the proofs and bounds that use non-algebraic properties of C are valid for any algebraically closed field K thanks to the following trick that reduces to the complex field. Any given finite subgroup G of GL(n, K) is defined over the field generated by the entries of its members. Let K 0 be the algebraic closure of this field. As K 0 has a finite degree of transcendence d, it is isomorphic to Q(X 1 , X 2 , . . . , X d ), so it can be embedded in C Q(X t : t ∈ R) and thus G is defined over C. Conversely, according to [Ser77, §12.3], every complex representation of a finite group is realizable over Q, i.e., every finite subgroup of GL(n, C) is conjugated to a subgroup of GL(n, Q).

Detailed proof of Singer's theorem

I shall present a proof of Theorem 15 using the same technique as Singer did, but keeping a finer track of the bounds than he needed for J Schur . I shall start with a weaker form of Theorem 17, after recalling some basic definitions.

Definition 19. A system of imprimitivity of a subgroup G of GL(V ) is a decomposition V = V 1 ⊕ V 2 ⊕ • • • ⊕ V m such that no V i is zero and the action of G permutes the family {V 1 , V 2 , . . . , V m }. If G admits a system of imprimitivity with m > 1, G is called imprim- itive.
If G is irreducible (its only invariant subspaces are V and 0) and not imprimitive, G is called primitive.

Remark 20. This is the definition of primitive groups that Collins uses, requiring irreducibility for primitivity, which is stronger that other authors'. Nevertheless, for finite groups, by Maschke's Theorem, reducibility implies imprimitivity.

Definition 21. According to Theorem 17, the function J prim : Z >0 → Z >0 is well defined as the minimum J prim (n) such that every primitive finite subgroup G of GL(n, C) admits an abelian normal subgroup of index J prim (n) at most.

According to Remark 18, we can extend the definition of J prim to any algebraically closed field:

Theorem 22. The function J prim : Z >0 → Z >0 satisfies that, for each n and any field K algebraically closed, every primitive finite subgroup G of GL(n, K) admits an abelian normal subgroup of index J prim (n) at most.

As an abelian linear group is 1-reducible, we get a weaker version of Theorem 15:

Theorem 23. The function J prim : Z >0 → Z >0 satisfies that, for each n and any field K algebraically closed, every primitive finite subgroup G of GL(n, K) admits a 1-reducible subgroup of index J prim (n) at most.

This result allows us to define the following bound:

Definition 24. According to Theorem 23, the function I prim : Z >0 → Z >0 is well defined as the minimum I prim (n) such that every primitive finite subgroup G of GL(n, C) admits a 1-reducible subgroup of index I prim (n) at most.

In the same way as Theorem 22, by Remark 18, we can extend the definition of I prim to any algebraically closed field:

Theorem 25. The function I prim : Z >0 → Z >0 satisfies that, for each n and any field K algebraically closed, every primitive finite subgroup G of GL(n, K) admits a 1-reducible subgroup of index I prim (n) at most. As normality is not required in Definition 21 for its application to Definition 24, we can define an intermediate bound K prim between I prim and J prim in the following way.

Definition 27. According to Theorem 17, the function K prim : Z >0 → Z >0 is well defined as the minimum K prim (n) such that every primitive finite subgroup G of GL(n, C) admits an abelian subgroup of index K prim (n) at most.

According to Remark 18, we can extend the definition of K prim to any algebraically closed field:

Theorem 28. The function K prim : Z >0 → Z >0 satisfies that, for each n and any field K algebraically closed, every primitive finite subgroup G of GL(n, K) admits an abelian subgroup of index K prim (n) at most. Remark 29. We have the bounds I prim K prim J prim . I shall proceed with the proof of Theorem 15 by proving a chain of weaker versions starting with Theorem 25.

Proposition 30. The function I fin : Z >0 → Z >0 defined by I fin (n) = max{rI prim (s) : rs n} satisfies that, for each n and any field K algebraically closed, every finite subgroup G of GL(n, K) admits a 1-reducible subgroup of index I fin (n) at most.

Proof. Let V be a system of imprimitivity of G of maximal length. Pick V 0 ∈ V. Let V be the orbit of V 0 by the action of G. The stabilizer H of V 0 has a natural representation H → GL(V 0 ) given by the restriction; let K be the image of this representation.

Let W be a system of imprimitivity of K. Pick representatives A ⊂ G of G/H, the left cosets of H. The family W = {AW : A ∈ A, W ∈ W} yields the direct sum

W ∈W W = V ∈V V and has #W = (#V )(#W). Thus W ∪ (V \ V ) is a system of imprimitivity of G of length #V + (#V )(#W -1)
. As V has maximal length and V = ∅, necessarily #W = 1. Therefore, the only system of imprimitivity of K is W = {V 0 }.

Suppose that K has an invariant subspace W different from 0 and V 0 . As K is finite, according to Maschke's Theorem [FH91, prop. 1.5],3 there is another invariant subspace W of K complementary to W . Hence we have a system of imprimitivity {W, W } of K, in contradiction with the previous paragraph. Therefore K is irreducible.

From the two last paragraphs we conclude that K is primitive. According to Theorem 25, K admits a 1-reducible subgroup K 0 of index I prim (dim V 0 ) at most. The counterimage H 0 of K 0 is 1-reducible and satisfies

[G : H 0 ] = [G : H][H : H 0 ] = #(V )[K : K 0 ] #(V )I prim (dim V 0 ).

The vector space spanned by

V has dimension #(V ) dim V 0 n, thus [G : H 0 ] max{rI prim (s) : rs n} = I fin (n).
Proposition 31. The function I fin : Z >0 → Z >0 satisfies that, for each n and any field K algebraically closed, every subgroup G of GL(n, K) with a scalar subgroup of finite index admits a 1-reducible subgroup of index I fin (n) at most. Proposition 32. The function I norm : Z >0 → Z >0 defined by I norm (n) = max{rI fin (s) : rs n} satisfies that, for each n and any field K algebraically closed, every subgroup G of GL(n, K) with a 1-reducible normal subgroup of finite index admits a 1-reducible subgroup of index I norm (n) at most.

Proof. Let N be such a 1-reducible normal subgroup of G with [G : N ] finite. Let V = {V 1 , V 2 , .
. . , V m } be the family of maximal eigenspaces of N , following the terminology of [Sin81, prop. 2.2], whose sum is direct and hence m i=0 dim V i n. The left action of G permutes V because N G. The stabilizer K of V 1 contains N and has a natural representation K → GL(V 1 ). Let K 0 and N 0 be the respective images of K and N . As N 0 is scalar and

[K 0 : N 0 ] = [K : N ] if finite, according to Proposition 31, K 0 admits a 1-reducible subgroup H 0 of index I fin (dim V 1 ) at most. The counterimage H of H 0 is 1-reducible and satisfies [G : H] = [G : K][K : H] = #(V )[K 0 : H 0 ], where V is the orbit of V 1 , hence [G : H] mI fin (dim V 1 ). As m dim V 1 n, [G : H]
I(n) = max{rI prim (s) : rs n}. (3)
Let I prim5 be the bound of I prim taking the optimal values of I prim up to 5 (I prim (2) = 12, I prim (3) = 36, I prim (4) = 120 and I prim (5) = 55) and taking the optimal values of J prim from 6 on, where J prim is the Jordan bound restricted to primitive groups, which will be introduced in §4.1 and satisfies J prim (n) = (n + 1)! for n > 12. From I prim5 we can compute a bound

Some results of Ulmer and others

The topic of [START_REF] Michael | Liouvillian solutions of n-th order homogeneous linear differential equations[END_REF] was revisited by F. Ulmer in [START_REF] Ulmer | On Liouvillian solutions of linear differential equations[END_REF]. In his [Ulm92, thm. 3.3], also in [UC90, thm. 3.1], Ulmer gives a bound equivalent4 to

I Ulm (n) = max{r! I prim (s) : rs n}, (4)
which is a priori worse than (3) because of the factorial. Notice that (4) gives a bound I Ulm for a given bound I prim . For the bound I prim5 introduced above, the factorial in (4) makes no difference with (3), which can be proved by an argument similar to that below (3), granting that r!(s + 1)! (n + 1)! for n 15 and s 14 by counting factors, and checking the remaining finitely many cases with a computer. However, once I had refined the bound I prim , I will be able to prove that the factorial in (4) keeps the growth of I Ulm too high for fitting the reduced value J prim (n)/f n . In [START_REF] Van Der Put | Galois theory of linear differential equations[END_REF]prop. 4.18] we find a bound equivalent5 to

I vdPS (n) = max{rJ(s) : rs n}, (5)
which differs from (3) in that (5) uses the general Jordan bound J, rather than I prim .

Let us compare I vdPS , computed with the optimal J of Collins, and I 5 defined above.

In the plot of I vdPS /I 5 shown in Figure 1, we can observe the following behavior. For 2 n 5 and for 8 n 60, I vdPS (n) is several times greater than I 5 (n). For n 60, J(n) approaches (n + 1)!, which is the value taken for I prim5 , so I vdPS (n) approaches I 5 (n) and they become equal for n > 70. The anomalous behavior I vdPS (6) = I 5 (6) and I vdPS (7) < 1.43 I 5 (7) is explained by the non-optimal values of I prim taken for 6 and 7. Theorem 36 (Malcev). There exists a function µ : Z >0 → Z >0 such that, for each n > 0, Combining the theorems of Malcev and Platonov, Singer claims that Theorem 15 follows immediately, but there is still a gap, which is covered by the following preparation lemma.

Some results of Malcev and Platonov

1. µ(n) µ(n 1 )µ(n 2 ) • • • µ(n k ) for any partition n = n 1 + n 2 + • • • + n k , 2.
Lemma 38. Every subgroup G of GL(n, K) with a 1-reducible subgroup of finite index admits a non-null invariant subspace V such that the restriction of G to V admits a solvable/triangularizable normal subgroup of finite index.

As the main application of Theorem 15 is Theorem 13, the case where G is a differential Galois group is an important one. In this case, Lemma 38 follows from Theorem 10 by taking V the subspace of Liouvillian solutions. In the general case, Lemma 38 holds for any field K. The proof follows by taking V the linear span of the orbit {Kv 1 , Kv 2 , . . . , Kv k } of an invariant line of a 1-reducible finite-index subgroup of G and the stabilizer of the family

{Kv 1 + Kv 2 + • • • + Kv i } k i=1 .
The alternate proof of Theorem 15 using Lemma 38, Theorem 37 and Theorem 35 yields a bound given by I(n) = µ(n)J(n), for the functions µ of Theorem 36 and J of Theorem 17. The following result allows us to compare µ(n)J(n) with J(n).

Theorem 39 (Dornhoff). Every finite solvable subgroup G of GL(n, K) admits an abelian normal subgroup H G of index [G : H] 2 (4n-3)/3 • 3 (10n-3)/9 , and this bound is achieved for K = C and n = 3

• 4 k with k 0. [Dor70]
The example for n = 3 is the so-called Hessian group of [Bli17, p. 109], which is defined over Q. The examples for the rest of the geometric progression are based on this group and also defined over Q, so the examples are valid for any algebraically closed field K of characteristic 0. Therefore we can bound µ(n) 2 (4n-3)/3 • 3 (10n-3)/9 for n = 3 • 4 k and k 0. We can construct a lower bound µ 0 of µ using these values and Theorem 36.1, though it depends on successive Euclidean divisions on n, and its computational implementation shows a growth like α n with α = 2 4/3 • 3 10/9 ≈ 8.54, behaving empirically like Ω(β n ) for any 0 < β < α. Theoretically speaking, we can grant µ

(n) = Ω(6 n ) from µ(n + 3) µ(n)µ(3) = 6 3 µ(n).

Some results of Collins

Jordan's bound refined by Collins

In [START_REF] Michael | On Jordan's theorem for complex linear groups[END_REF], M. J. Collins gives the optimal function J of Theorem 17, depending on his work [START_REF]Bounds for finite primitive complex linear groups[END_REF], where he computes the optimal function J prim of the restriction of Theorem 17 to finite primitive groups. The value of J prim is given in [Col08, thm. A], but with a typo in J prim (8). It is J prim (n) = (n + 1)! with the exceptions listed in the left-hand-side The optimal value of J is given in [START_REF] Michael | On Jordan's theorem for complex linear groups[END_REF], though not explicitly. It is J(n) = (n + 1)! with the following exceptions: J(n) = J prim (n) for 2 n 6, the exceptions of the right table of Figure 2, J(2r) = 60 r r! for 10 r 35, and J(2r + 1) = 60 r r! for 10 r 30. The exceptional domain, the range 2 n 62 and n ∈ {64, 66, 68, 70}, can be divided into three stretches. The initial stretch is the range 2 n 6, where the group H n reaching the bound J(n) is primitive. According to Remark 42, the abelian normal subgroup is the center Z(H n ), of index J prim (n). Such a group is unique up to isoclinism, but Collins takes the example of [Col07, prop. C] for the sake of construction of further examples, since these groups are the building blocks of the rest of the examples attaining J(n) in the exceptional domain. The middle stretch is the range 7 n 19, where the groups reaching the bound J(n) are given by [Col07, thm. D]. The groups achieving J(n) in this stretch are direct products of wreath products i (H k i wr S r i ), with H k wr S 1 = H k . The abelian normal subgroup is the direct product i r i Z(H k i ), with index i (r i ! J prim (k i ) r i ). The last stretch, the range 20 n 62 and n ∈ {64, 66, 68, 70}, yields a bound J(n) = 60 r r! with r = n ÷ 2 the Euclidean quotient, according to [Col07, thm. B.(i)], reached by groups described in [Col07, thm. B.(a)], whence one can construct the wreath product H 2 wr S r as an easy example.

The primitive groups attaining the bound J prim (n) are described in [Col08, thm. A]. This group is generically S n+1 , with exceptions for 2 n 9 and n = 12. Let me start recalling two results about finite primitive groups mentioned in [START_REF]Bounds for finite primitive complex linear groups[END_REF].

Lemma 41. Every abelian normal subgroup of a finite primitive subgroup

G of GL(n, K) is central in G. [Col08, lem. 1]
Remark 42. Lemma 41 allows us to assume that the abelian normal subgroup involved in Jordan's bound is the center, as observed in Remark on [Col08, p. 762], so

J prim (n) = max{[G : Z(G)] : G < GL(n, C) finite primitive}.
Lemma 43 (Schur). The center of an irreducible lineal group consists of scalar matrices/automophisms. [FH91, lem. 1.7.2] Corollary 44. The center of an irreducible lineal group is the subgroup of its scalar matrices/automophisms.

Framework for Collins's proof

The core of [START_REF]Bounds for finite primitive complex linear groups[END_REF] consists of the proof I shall sketch below and refine in §5 and §6 for the goal of this article. For this proof I recall some results of Collins.

Definition 45. A group G is quasisimple if it is perfect (G = G) and G/Z(G) is simple (G has no intermediate normal subgroup). A subgroup H of G is subnormal if there is a chain of subgroups H = H 0 < H 1 < • • • < H m = G such that each subgroup is normal in the next.
If G is a finite group whose order is divisible by the prime p, then the p-core of G is its largest normal p-subgroup O p (G). A component of a finite group is a quasisimple subnormal subgroup. A quasicomponent of a finite group is a non-cyclic p-core for a prime p. Remark 49. Collins uses the dot notation both for group extensions and products. In Lemma 46 and Lemma 48 he refers to a product, and these products happen to be central because one factor is the center.

A key result of structure is the following one.

Theorem 50. Let G be a non-abelian primitive linear group with quasicomponents P 1 , . . . , P r and components E 1 , . . . , E s . As each P i /Z(P i ) is an elementary abelian group of order an even prime-power, let p 2n i i be this order. Let Aut c denote the subgroup of automorphisms that fix the center, and Out c the corresponding quotient. Assume that the family of components splits into t isomorphism classes of lengths l 1 , . . . , l t . Then, G admits a normal subgroup N such that the following structure holds.

1. F * (G)/Z(G) is the direct product of the P i /Z(P i ) and the E j /Z(E j ).

N/F

* (G) is isomorphic to a subgroup of the direct product r i=1 Sp(2n i , p i ) × s j=1 Out c (E j ). 3. G/N is isomorphic to a subgroup of the direct product of symmetric groups S l 1 × • • • × S lt .
[Col08, thm. 5]

According to this structure result, the contribution of each quasicomponent P i to the index [G : Z(G)] is bounded by p 2n i i • # Sp(2n i , p i ), and the contribution of each isomorphism class of components is bounded by

[E j : Z(E j )] • # Out c (E j ) • l i ! = # Aut c (E j ) • l i !.
As the elements of Aut c (E j ) are in bijection with those of Aut(E j /Z(E j )), the bound for the contribution is # Aut(E j /Z(E j )) • l i !, which depends only on the simple group E j /Z(E j ) and the multiplicity l i . The value of [G : Z(G)] is bounded by the product of the contributions of the quasicomponents and the components.

According to [Col08, §2, post thm. 5], as G is quasiprimitive and E 1 (G) a normal subgroup, the restriction of the natural representation to E 1 (G) is homogeneous, so all its irreducible constituents are equivalent and, in particular, have the same degree. We can thus assume that E 1 (G) is irreducible of a degree m that divides the degree n of G. According to [Gor80, thms. 3.7.1 and 3.7.2 ], the irreducible representations of E 1 (G) are (equivalent to) tensor products of irreducible representations of the components and quasicomponents. If we want the product representation to be faithful, all the factors must be faithful too. As we look for lower bounds for the degree n, such a bound is the product of the minimal degree of an irreducible faithful representation of a component or quasicomponent. For a quasicomponent P with [P : Z(P )] = p 2n , such a minimal degree is p n . For a component E, we look for the minimal degree of a projective representation of the simple group E/Z(E). Such lower bounds can be found in [START_REF] Hoffman | Cross characteristic projective representations for some classical groups[END_REF] for the families of simple groups and in the Atlas [START_REF] Conway | Maximal subgroups and ordinary characters for simple groups[END_REF] for the sporadic simple groups.

What Collins does in [START_REF]Bounds for finite primitive complex linear groups[END_REF]§4] is to compute the cases of maximal contribution for a fixed degree m, obtaining that a maximally contributing component is A m+1 , with a table of exceptions for m ∈ {2, 3, 4, 5, 6, 7, 8, 9, 12} shown in Figure 3, and that a maximally contributing quasicomponent for m = p n contributes less than a maximally contributing component of this degree, with the exception of m = 9.

Sketch of Collins's proof

In order to make more understandable the reasoning of Collins, I introduce two useful concepts.

Definition 51. I call a contribution record a record with two mandatory fields and two optional ones, all of them positive integers. The first mandatory field is the degree r, which represents the degree of the representation of the component or quasicomponent. The other mandatory field is the contribution c, which represents # Aut(E/Z(E)) for a component E and p 2n • # Sp(2n, p) for a quasicomponent P with [P : Z(P )] = p 2n . The first optional field is the multiplicity l, which represents the length of its isomorphism class. By default, we assume l = 1. For quasicomponents l = 1. The total contribution is c l l!. The other optional field is the reduction factor f , which represents a number by which we can divide the contribution. By default, we assume f = 1. The reduction factor will make sense in §6. The reduced total contribution is c l l!/f . Definition 52. I call a contribution setting a family of contribution records, maybe repeated, together with its degree as a positive integer field. The total multiplicity of an element is the product of how many times it appears in the family by its multiplicity field. The total contribution of a setting is the product of the total contributions of its elements. We restrict the definition to those degrees that are multiple of the product of the degrees of the elements counted with their total multiplicity.

In this language, what Collins wants to do is maximizing the total contribution of a contribution setting for a given degree n, especially proving that the component A n+1 attains such a maximum for G = S n+1 , and also tabulating the exceptions. He proceeds by replacing contribution records with other more contributing ones. For this goal, he first discards as not maximizing the contribution records with multiplicity l > 1. In [START_REF]Bounds for finite primitive complex linear groups[END_REF]lem. 11] he proves that A r l +1 contributes more than any record of degree r and multiplicity l > 1, with the exception of r = 2 and l ∈ {2, 3}. For these exceptions, he proves in [START_REF]Bounds for finite primitive complex linear groups[END_REF]lem. 12] that a maximally contributing record of degree 2 l and multiplicity 1 contributes more than any record of degree 2 and multiplicity l. Then, he proves in [START_REF]Bounds for finite primitive complex linear groups[END_REF]lem. 12] that the records of a maximally contributing setting do not have repeated degrees. In [Col08, lem. 13], he proves that a maximally contributing setting has a single record (of multiplicity l = 1) and that the degree of the setting is the degree of the record. Finally, he notes that the maximally-contributing settings are afforded by primitive groups for degree n 6. This proof will be adapted in §6, using the reduction factor of the contribution records, for K prim . First, I shall study in §5 the maximal contributions of a single contributor using reduction factors in the generic case.

Study of a single contributor

The aim of this section is to go through the Classification of Finite Simple Groups in order to refine Collins's computations of [START_REF]Bounds for finite primitive complex linear groups[END_REF]§4] for using the reduction factors in the generic case. He gives a table of the families of finite simple groups with an upper bound N (q, n) of the total contribution and a lower bound D(q, n) of the minimal degree of a projective representation. His aim was to compute the exceptions to N (q, n) < (D(q, n) + 1)!, and in our case the inequation would be N (q, n) < (D(q, n) + 1)!/f for the reduction factor computed below.

For the record corresponding to an alternating group A n , we have a degree r = n -1, a contribution c = n!, a multiplicity l = 1 and as reduction factor the maximal order f of an abelian subgroup of S n . According to [START_REF] Burns | Maximal order abelian subgroups of symmetric groups[END_REF],

f = 3 k if n = 3k, f = 4 • 3 k-1 if n = 3k + 1, excluding n = 1, and f = 2 • 3 k if n = 3k + 2. This yields f = 4 3 3 r/3 if r ∈ 3Z, f = 2 3 3 √ 3 2 3 r/3 if r ∈ 3Z + 1, and f = 3 √ 3 • 3 r/3 if r ∈ 3Z + 2. So, 4 3 3 r/3 f 3 √ 3 • 3 r/3 regardless the residue of r modulo 3.
So, in our case, the inequation would be

N (q, n) < (D(q, n) + 1)! 3 (D(q,n)+1)/3 , (6)
with D(q, n) > 12, as the lower-degree cases are considered by him apart. For granting that a group that does not satisfy (6) has all its representation either of degree 12 at most, and thus considered apart, or of degree greater than 12 but safe, I shall consider the inequation

N (q, n) < 14! 3 14/3 . (7)
For this, I shall consider the function f 0 : N → R given by f 0 (x) = x!/3 x/3 . Then, (6) can be read as N (q, n) < f 0 (D(q, n) + 1). Bounding log f , we get

log f 0 (x) = log(x!) - log 3 3 x x log x -x + 1 - log 3 3 x,
so, defining the function f 1 : R + → R given by

f 1 (x) = x log x -1 - log 3 3 + 1,
we get log f 0 (x) f 1 (x). The function f 1 is convex, as f 1 (x) = 1/x is positive, and has slope f 1 (x 0 ) = 1 at x 0 = 3 √ 3 exp(1). The affine function f 2 (x) = x -3 falls below f 1 at x 0 , so it falls below it in all R + by convexity. Hence, log f 0 (x) f 2 (x) = x -3 and thus f 0 (x) exp(x -3). Therefore, (6) can be substituted with the stronger inequation

N (q, n) < exp D(q, n) -2 . ( 8
)
Furthermore, as the expressions of N and D are complicated to manipulate, I shall consider rough approximations Ñ N and D D, in such a way that

Ñ (q, n) < exp D(q, n) -2 (9)
is stronger than (8).

Recall that some families depend on two parameters q and n, where q = p m , while others depend only on q and there are even exceptional cases, including the sporadic simple groups. So, first I shall consider the uniparametric families, including the sections for n fixed of the biparametric families. Then, I shall consider the biparametric families, taking in account the sections for n fixed, and finally I shall consider the exceptional cases. family Ñ (q) D(q) restriction L 2 (q) q 4 q -1 q even L 2 (q) q 4 (q -1)/2 q odd G 2 (q) q 15 q(q 2 -1) q 5 F 4 (q) q 53 q 4 (q 6 -1) q odd F 4 (q) q 53 q 7 (q 3 -1) q 4 even E 6 (q) q 79 q 9 (q 2 -1) E 7 (q) q 134 q 15 (q 2 -1) E 8 (q) q 249 q 27 (q 2 -1) 3 D 4 (q) q 28 q 3 (q 2 -1) 2 E 6 (q) q 79 q 9 (q 2 -1) quasicomp. q 5 q q prime Figure 4: Table of uniparametric families in q, derived form [Col08, §4].

Uniparametric families

The families in the single parameter q will be listed in Figure 4, where I include the quasicomponents for n = 1. For all these families in q, we shall proceed proving (9) by induction. For L 2 (q) and q even, (9) is false for q 8, but it is true for q = 16. Nevertheless, for these exceptions, (7) is satisfied. For L 2 (q) and q odd, (9) is false for q 31, but it is true for q = 37. Again, all the exceptions satisfy (7). For quasicomponents, (9) is false for q 13, but it is true for q = 17, and all the exceptions satisfy (7). The rest of the families satisfy (9) for the initial value of q in the family, which is q = 2 with the exceptions noted in the restriction column of the table.

Following with the inductive argument, we shall grant the implication

Ñ (q) < exp D(q) -2 ⇒ Ñ (q + 1) < exp D(q + 1) -2 . ( 10
)
According to the technique of ratios that Collins suggests, (10) follows from

Ñ (q + 1) Ñ (q) < exp D(q + 1) -2 exp D(q) -2 = exp D(q + 1) -D(q) . (11)
As Ñ (q) = q α for all the families considered, the left-hand side is

(q + 1) α q α = 1 + 1 q α ,
which reaches its maximum at the initial value of q for the induction. For the righthand side, notice that D(q) has the form λq a (q b -1), with λ and b positive, except for quasicomponents. I need a lemma that will be also useful later.

Lemma 53. For a function g(x) = λx a (x b -1) with x 2, the difference d k = g(k + 1)g(k) is an increasing sequence, provided 2a + b > 1.

Proof. As g (x) = λ(a + b)(a + b -1)x a+b-2 -λa(a -1)x a-2 , we have g (x) > 0 if and only if (a + b)(a + b -1)x b > a(a -1). We have (a + b)(a + b -1) > a(a -1) if and only if 2a + b -1 > 0. If the condition 2a + b > 1 holds, then g (x) > 0 and thus g is an increasing function. By the Lagrange Mean-Value Theorem, for each k 2, there exists

c k ∈ (k, k + 1) such that d k = g(k + 1) -g(k) = g (c k ).
As g is an increasing function, In all the families considered, except L 2 (q) and the quasicomponents, the condition 2a + b > 1 holds. For L 2 (q), g(x) = λ(x -1), so d k = λ. For quasicomponents, g(x) = x and d k = 1. Therefore, the right-hand side of (11) reaches its minimum at the initial value of q for the induction. Hence, it suffices to check (11) for the initial value q 0 of the induction and then (9) is proved for all the values q q 0 .

d k = g (c k ) is an increasing sequence. family Ñ (l) D(l) restr. 2 B 2 (2 2l+1 ) 2 11l+6 2 3l l 2 2 G 2 (3 2l+1 ) 3 3l+2 3 4l+1 2 F 4 (2 2l+1 ) 2 53l+27 2 11l+4
For L 2 (q) and q even, it holds for q 0 = 16, and for q odd it does for q 0 = 37. For G 2 (q) it holds for q 0 = 5. For F 4 (q) and q odd, it holds for q 0 = 3, and for q even it does for q 0 = 4. For quasicomponents, it holds for q 0 = 17. For the rest of the families, it holds for q 0 = 2. Hence, we have proved that no group of these families is a new obstruction for refining Collins's results.

Apart, we add the families in the single parameter l in Figure 5. In a similar way, we shall proceed proving (9) by induction for these families in l. For 2 B 2 (2 2l+1 ), (9) is true for l = 2, and for 2 G 2 (3 2l+1 ) and 2 F 4 (2 2l+1 ) it holds for l = 1. For the induction step, we shall prove

Ñ (l + 1) Ñ (l) < exp D(l + 1) -D(l) (12)
as we did for the families in q. The left-hand side is 2 11 , 3 3 and 2 57 respectively. For the right-hand side, D is again a convex function in l, so the difference d l = D(l + 1) -D(l) is an increasing sequence. As (12) holds for the three families for their initial values of l, we have proved (9) by induction. Therefore, no group of the uniparametric families is a new obstruction for refining Collins's results.

Biparametric families

The families in the two parameters (q, n) will be listed in Figure 6, where I have added the quasicomponents because they form a biparametric family that can be addressed in the same way. For the aim of proving (9) for these families by induction in both parameters, I shall first prove the induction steps in q

Ñ (q, n) < exp D(q, n) -2 ⇒ Ñ (q + 1, n) < exp D(q + 1, n) -2 , ( 13
)
and in n

Ñ (q, n) < exp D(q, n) -2 ⇒ Ñ (q, n + 1) < exp D(q, n + 1) -2 , ( 14
)
by means of the same technique of ratios that Collins suggested. First, for proving (13), the ratio inequation is

Ñ (q + 1, n) Ñ (q, n) < exp D(q + 1, n) -D(q, n) . ( 15
)
For the same reason as in the uniparametric families, the left-hand side attains its maximum at the initial value of q for the induction. As D(q, n) = g(q) for certain function q 2an+a+b q 4n+5 . The expression D(q, n + 1) -D(q, n) can be bounded from below by q n-2 , and thus the right-hand side by exp(q n-2 ). So, we have the stronger inequation q 4n+5 < exp(q n-2 ) or q < exp q n-2 4n + 5 . (18)

As h(x) = exp( 1 37 x 6 ) is a convex function and the identity function is below h at x = 2, with h (2) > 1, then the identity function is below h for x 2. Thus, we have proved (18) for n = 8. As q n-2 4n + 5

∞ n=8
is an increasing sequence for any q 2, we have proved (18) for n 8, and thus ( 16) and ( 14) with the same restriction.

In order to prove (9), it suffices to prove it for n 8, which yields finitely many uniparametric families whose checking can be reduced to the starting point of the induction. If we repeat the check of (15) for n up to 8, we find no new exception, only the old ones: (4, 2) is the starting point for S 2n (q) and q even, (4, 3) is for U n (q) and n odd, and (19, 2) is for quasicomponents. The check of (9) yields the following exceptional starting points: {(4, 3), (3, 4), (3, 5)} for L n (q), {(7, 2), (5, 3), (4, 4)} for S 2n (q) and q odd, {(6, 2), (3, 3), (3, 4)} for S 2n (q) and q even, {(3, 4)} for O - 2n (q), {(5, 4), (3, 5), (3, 6), (3, 7)} for U n (q) and n even, {(8, 3), (5, 4), (3, 5), (3, 6), (3, 7)} for U n (q) and n odd, and {(5, 2), (4, 3), (3, 4), (3, 5), (3, 6)} for quasicomponents. For each staring point (q 0 , n 0 ), we check if Ñ (q, n 0 ) < f 0 (14) or Ñ (q, n 0 ) < f 0 D(q, n 0 ) + 1 for q < q 0 . The only negative results gotten are (3, 3) for S 2n (q) and q odd, (3, 4) and (5, 3) for U n (q) and n odd, and (2, 4) for quasicomponents. The exception (3, 4) cannot happen in a family restricted to n odd. The exception (5, 3) for U n (q) satisfies (6). So, we have the only exceptions S 6 (3) and the quasicomponents for q = 2 and n = 4.

There are three subfamilies which are parameterized only by n: O 2n+1 (3), O + 2n (2) and O + 2n (3), all of them for n 4. The first family has Ñ (n) = 3 2n 2 +n and D(n) = 3 2n-3 . The second family has Ñ (n) = 6 • 2 2n 2 +n and D(n) = 2 2n-4 . The third family has Ñ (n) = 6 • 3 2n 2 +n and D(n) = 3 2n-4 . For n = 4, (9) is satisfied by the first and third families, while the second family satisfies Ñ (4) < f 0 ( D(4) + 1), so the induction start holds. Particularizing (16), we have the inequations 3 4n+3 < exp(2 3 3 2n-3 ), 2 4n+3 < exp(2 2n-4 3) and 3 4n+3 < exp(2 3 3 2n-4 ),

which hold for n = 4. Taking logarithms in (19), we get (4n + 3) log(3) < 2 3 3 2n-3 , (4n + 3) log(2) < 2 2n-4 3 and (4n + 3) log(3) < 2 3 3 2n-4 . (20) The right-hand sides have the form h i (n) with h i convex functions, while the left-hand sides are affine functions g i (n). We have proved that the line g i is below the respective h i at n = 4. As h i (4) is greater than the slope of g i , then we have g i (n) < h i (n) for any n 4, so proving (20) and thus (19) and ( 16). This completes the induction, so these families are also safe.

Sporadic groups and other exceptions

The sporadic groups complete the finite simple groups together with those individual groups detached from their families. The first kind of groups is well known and their values of N and D can be found in the Atlas and the corresponding GAP libraries [?]. The second kind of groups consists of those listed in Figure 7. Checking satisfied by all these groups except those with D 12. Anyway, these exceptions satisfy N < f 0 (14), so they are studied apart. Among the sporadic groups, only M 22 , J 2 and Suz fail to satisfy (6). The three groups happen to have D 12, and N < f 0 (14) for M 22 and J 2 . The group Suz requires closer attention. The group 6 . Suz has a faithful representation of minimal degree 12. The rest of the faithful representations of m . Suz have degree 66 at least. As N < f 0 (67), the only representation to worry about is that of degree 12, and it is considered apart.

Finally, I shall consider the alternating groups A n in n, which can yield proper covers as non-alternating components. For n 7, the alternating group A n has Out(A n ) C 2 , so the contribution of any component E with E/Z(E)

A n is c = n!. For n 6, the minimal degree of a faithful representation of A n is r = n -1. For n 8, the Schur multiplier of A n is C 2 , so the only components E with E/Z(E) A n are A n and its Schur cover A n = 2.A n . For n 10, the minimal degree of a faithful representation of the latter is 2 (n-2)/2 at least, according to [Sch1911, §44]6 . If A n has a faithful representation of degree r, then n -1 < 2 (n-2)/2 r, so A r+1 will be better contributing than A n for this degree. Hence, we can discard A n when maximizing the contributions.

Reduction to a single contributor

In §5, I refined Collins's arguments in order to prove that the maximal contributors for degree r > 12 are still given by the symmetric groups S r+1 even using the reduction factor corresponding to the maximal order of their abelian subgroups. In this section, I shall refine other Collins's arguments in order to prove that the maximal contributions are achieved by single contributors, even with the considered reduced contributions.

As we want to maximize contributions, we can assume that all the contribution records correspond to the maximal contributors of Collins. In the lesser side of the inequations, we use unreduced records. In the greater side of the inequations, we will use reduced records for the alternating components, so it will be useful to bound f 3 (r+1)/3 for proving stronger inequations.

Refining Lemma 11 of Collins

For a contribution record (r, N, l), [START_REF]Bounds for finite primitive complex linear groups[END_REF]lem. 11] proves that N l • l! < (r l + 1)! for l > 1 with two exceptions, which bounds the total contribution of the said record with the total

Refining Lemma 12 of Collins

The total contribution of a given degree in a setting is maximized when all the components or quasicomponents of this degree are the same and of maximal contribution for the degree. In the case of components, they are grouped in a single record. By [Col08, lem. 12], the total contribution of all the degrees is maximized when there is a single component or quasicomponent of each degree. The proof proceeds by substituting the contribution of a degree r with multiplicity l by another contribution of degree r l , generically a symmetric group. According to Proposition 54, this substitution is still maximizing, with exceptions, even if we consider reduced contributions in the greater side of the inequations. So, we need only to consider the exceptions of Proposition 54. The exceptions for r = 2 were treated by Collins, and he substitutes two or three icosahedral components7 by a single maximizing one of degree 4 or 8 respectively. The exception r = 3 and l = 2 is new, but the same approach is valid: two Valentiner components8 can be substituted by a single maximizing quasicomponent of degree 9. Hence, we have proved the following analogue of [Col08, lem. 12].

Proposition 55. A maximally-contributing setting has a single component or quasicomponent record of each contributing degree, even using the reduced contributions for the alternating components.

Refining Lemma 13 of Collins

Collins proves in [START_REF]Bounds for finite primitive complex linear groups[END_REF]lem. 13] that the total contribution of a setting is maximized when there is a single degree with a single component or quasicomponent. We already knew that a maximal-contribution setting has a single component or quasicomponent for each contributed degree. For this, he proves it by replacing two components or quasicomponents of degrees p < q with a single one of degree pq. If their contributions are N p , N q and N pq , he grants N p N q < N pq . I shall strengthen his result by taking the reduced contributions of the alternating components.

Proposition 56. A maximally-contributing setting has a single component or quasicomponent record, even using the reduced contributions for the alternating components.

Proof. Proceeding as Collins, if p < q 12, the problem reduces to a finite check. If p 12 < q, we have to prove N p (q + 1)! < (pq + 1)! 3 (pq+1)/3 , which is equivalent to

N 3 p <
(pq + 1)! 3 3 pq+1 (q + 1)! 3 .

(

) 23
The numerator of the right-hand side has 3pq factors, while the denominator has (pq + 1) + 3q, and the factors of the numerator can be chosen bounding the factors of the denominator, so it suffices to prove N 3 p < 2 q(2p-3)-1 for q = 13, which reduces to a finite check. This check succeeds for 3 p 12, but fails for p = 2 because it was oversimplified. However, N 3 2 < 2 q-1 holds for q 22 and we can check that the inequation (23) holds for p = 2 and 13 q 21, so the claim is proved in this case.

Finally, if 12 < p < q, we have to prove (p + 1)! • (q + 1)! < (pq + 1)! 3 (pq+1)/3 , which is equivalent to (p + 1)! 3 (q + 1)! 3 3 pq+1 < (pq + 1)! 3 .

The left-hand side has 3p + 3q + (pq + 1) factors, while the right-hand side has 3pq ones, which can be chosen bounding the left-hand-side ones. For 12 < p < q, the number of right-hand-side factors exceeds the left-hand-side ones, so the inequation holds.

In [START_REF]Bounds for finite primitive complex linear groups[END_REF]lem. 13], Collins also proves that the degree of the only component or quasicomponent is the degree of the maximizing contribution setting. The key is that, if the degree r of the component or quasicomponent were a strict divisor of the degree d of the maximizing setting, we could add a component or quasicomponent of degree d/r, getting a setting with greater contribution. His lemma ends with the remark that these maximizing settings are afforded by primitive groups for degree n 6.

Conclusions

In §6, I have refined 3 lemmas of Collins in order to grant that the maximizing groups for n > 16, which are the symmetric groups S n+1 , are also maximizing if their order is reduced by the maximal-order of an abelian subgroup. Recall that, as S n+1 is centerless, according to Lemma 41, the only normal abelian subgroup of S n+1 is the trivial subgroup. This reduction factor is, according to [START_REF] Burns | Maximal order abelian subgroups of symmetric groups[END_REF], f n = 4 • 3 (n-3)/3 if n ∈ 3Z, f n = 2 • 3 (n-1)/3 if n ∈ 3Z + 1, and f n = 3 (n+1)/3 if n ∈ 3Z + 2. This gives a bound K prim (n) = (n + 1)!/f n , which allows us to get a finer Singer bound I(n) for Liouvillian integration.

For n 16, we can still reduce by the respective reduction factor, but this requires a detailed study of the primitive linear groups of these degrees, which will be done in subsequent articles. Nevertheless, for 13 n 16, there are only two exceptions that fail to satisfy (6). These exceptions are the components with inner group S 6 (3) and the quasicomponents for q = 2 and n = 4. Notice that the case of S 6 (3) has no impact above degree 14 since N < f 0 (16). When in further work we prove that these exceptions satisfy also the generic bound K prim (n), we would have proved that K prim (n) has the generic value for 13 n 16.

With the bounds proved up to now, we can compute I now (n) according to (3) taking the optimal values I prim (n) for n 5, the values I prim (n) = J prim (n) of Collins for 6 n 16, and the just proven values I prim (n) = (n + 1)!/f n for n > 16. Let me compare I now and I 5 defined below (3). In the plot of I 5 /I now shown in Figure 8, we can observe the following behavior. For 2 n 16, I now (n) is equal to I 5 (n) because both used the same estimate I prim5 of I prim . For n 19, the graph is almost a straight line taking the values of the reduction factor f n . There is a transition period in between, which takes in account the influence of the imprimitive cases. The values of the graph for n 19 can be taken as the result of this article. The exceptional values for n 16 and their influence area up to n = 18 will be sharpened in subsequent articles.

We can also compute I now Ulm (n) according to (4) taking the same I prim as for I now . Let me compare I now Ulm and I now . The plot of I now Ulm /I now is also shown in Figure 8, and we can observe that it is close to the graph of I 5 /I now . Moreover, I now Ulm (n) n! afforded by the monomial case r = n and s = 1, while I 5 (n) = (n + 1)!, so the reduction factor I 5 /I now Ulm is linear at most, while I 5 /I now is exponential. This shows how the factorial in (4) is an actual overestimation.

Remark 26 .

 26 The bound I prim J prim , given by Theorem 23, is too rough. According to [Cor01, Chap. 4], we have the values I prim (2) = 12, I prim (3) = 36, I prim (4) = 120 and I prim (5) = 55, but I prim (6) 3780.

  Proof. Let N be such a scalar subgroup of G with [G : N ] finite. Thus G 0 = (K * G) ∩ SL(n, K) is finite. According to Proposition 30, G 0 admits a 1-reducible subgroup H 0 of index I fin (n) at most. The group H = (K * H 0 ) ∩ G is 1-reducible, and a subgroup of G with [G : H] [G 0 : H 0 ] I fin (n).

  I 5 according to (3), getting the following values: I 5 (2) = 12, I 5 (3) = 36, I 5 (4) = I 5 (5) = 120, I 5 (6) = I 5 (7) = 7! 6 4 , I 5 (8) = I 5 (9) = I 5 (10) = 10! 96, I 5 (11) = 12!, I 5 (12) = I 5 (13) = 13! 72 and I 5 (14) = 15!. If s 14 and n 15 in (3), rI prim5 (s) 15! r (n + 1)!. If s 14, rI prim5 (s) = r(s + 1)! (n + 1)!. This proves that I 5 (n) = (n + 1)! for n 14.

Figure 1 :

 1 Figure 1: Plot of I vdPS /I 5 in logarithmic scale.

Remark 40 .

 40 Collins's statements are for C, but the trick in Remark 18 allows us to extend both the results and the bounds to any algebraically closed field. The generic bound (n + 1)! is achieved by the symmetric group S n+1 embedded in GL(n, C). The permuted elements are e 1 , e 2 , . . . , e n of the standard basis and -e 1e 2 -• • • -e n . If φ : S n+1 → GL(n, C) is this embedding, then σ → (-1) σ φ(σ) is another embedding. The abelian normal subgroup is the trivial one. According to [Col07, thm. A] and [Col07, thm. B.(ii)], this generic bound holds for the range n 71 and for n ∈ {63, 65, 67, 69}.

Lemma 46 .

 46 A quasicomponent P = O p (G) of a finite primitive subgroup G of GL(n, C) is a central product Z(P ) • E, with E an extraspecial p-group, which can be chosen of exponent p for p odd. [Col08, lem. 2] Definition 47. For a finite primitive linear group G, Collins defines the subgroup E 1 (G) of G as the central product of its components and quasicomponents. Lemma 48. The generalized Fitting subgroup F * (G) of a finite primitive linear group G is the central product Z(G) • E 1 (G). [Col08, §2, ante prop. 4]

Figure 5 :

 5 Figure 5: Table of uniparametric families in l, derived form [Col08, §4].

Figure 8 :

 8 Figure 8: Plot of I 5 /I now (upper graph) and I now Ulm /I now (lower graph) in logarithmic scale.

  max{rI fin (s) : rs n} = I norm (n).Proposition 33. The function I norm : Z >0 → Z >0 satisfies that, for each n and any field K algebraically closed, every subgroup G of GL(n, K) with a 1-reducible subgroup of finite index admits a 1-reducible subgroup of index I norm (n) at most. Proof. Let H be such a 1-reducible subgroup of G with [G : H] finite. The left action of G permutes G/H, the left cosets of H, and gives a natural representation G → Sym(G/H). Its kernel K is contained in H and is thus 1-reducible. As K G and [G : K] # Sym(G/H) = [G : H]!, according to Proposition 32, G admits a 1-reducible subgroup H 0 of index I norm (n) at most.

	Now we can finally prove a more constructive version of Theorem 15:
	Remark 34. Therefore, Theorem 15 is proved with

  Singer pointed out in [Sin81, p. 681] the following results of Malcev and Platonov as an alternate proof of Theorem 15, his [Sin81, prop. 2.2].

	Theorem 35 (Malcev). Every solvable subgroup G of GL(n, K) admits a triangularizable
	normal subgroup H G of index [G : H]	µ(n), for the function µ of Theorem 36.
	[Weh73, thm 3.6, p. 45]

  every completely reducible solvable subgroup G of GL(n, K) admits an abelian normal subgroup of index µ(n) at most. Theorem 37 (Platonov). Every solvable-by-finite subgroup G of GL(n, K) admits a solvable normal subgroup H G of index [G : H] J(n), for the function J of Theorem 17.[START_REF] Bertram | Infinite linear groups[END_REF] corol. 10.11, p. 142]

	[Weh73, thm 3.5, p. 44]

  table of Figure 2.

			n J(n)
			7 J prim (4) J prim (3)
	n J prim (n)	8 J prim (4) 2 2!
	2	60	9 J prim (6) J prim (3)
	3	360	10 J prim (6) J prim (4)
	4	5! 6 3	11 J prim (4) 2 2! J prim (3)
	5	5! 6 3	12 J prim (4) 3 3!
	6	7! 6 4	13 J prim (4) 3 3!
	7	8! 6 2	14 J prim (2) 7 7!
	8	10! 96	15 J prim (4) 3 3! J prim (3)
	9	6 7 15	16 J prim (4) 4 4!
	12	13! 72	17 J prim (4) 4 4!
			18 J prim (2) 9 9!
			19 J prim (4) 4 4! J prim (3)

Figure 2: The left-hand-side table shows the exceptions to J prim (n) = (n + 1)!, corrected from [Col08, thm. A]. The right-hand-side table shows the non-systematic exceptions of J(n) = (n + 1)!, elaborated from [Col07, thm. D].

  Table of maximal contributions for degrees m when it is not given by a component A m+1 , corrected from [Col08, p. 772], with the Atlas notation [CCN85].

	m max. contrib. component
	2	120 2.A 5
	3	720 3.A 6
	4	2 • 5! 6 3 2.S 4 (3)
	5	2 • 5! 6 3 S 4 (3)
	6	7! 6 4 6 1 .U 4 (3)
	7	8! 6 2 S 6 (2)
	8	10! 96 2.O + 8 (2)
	9	6 7 15 quasicomp.
	12	13! 72 6.Suz
	Figure 3:

If f = 0 but f 1 = 0, we reorder the variables, so we proceed without loss of generality.

This proposition is claimed for K = C, but its second proof is valid for any field whose characteristic does not divide the order of the group.

This bound is reconstructed from Ulmer's ones by considering his bounds for imprimitive and primitive groups together and also reducible groups.

This bound appears almost in the shown form, but in a proposition stated in a more geometric language.

Collins omitted the floor function when citing Schur's bound, but this floor function is necessary as the examples of n = 11 shows, since A 11 has an irreducible faithful representation of degree 16.

The icosahedral group is the maximally contributing in degree 2.

The Valentiner group is the maximally contributing in degree 3.

Acknowledgments

I want to thank Jorge Mozo-Fernández, who was my PhD advisor, for his guidance in the preparation of this article.

family Ñ (q, n) D(q, n) restriction L n (q) q n 2 q n-1 -1 n 3, not valid for (3, 4) S 2n (q) q 2n 2 +n+1 (q n -1)/2 n 2, q odd S 2n (q) q 2n 2 +n+1 q 2n-1 /8 n 2, q even O 2n+1 (q) q 2n 2 +n+1 q 2n-2 -1 n 3, q 5 odd O 2n+1 (q) q 2n 2 +n+1 q 2n-1 /8 q = 3, n 4 O + 2n (q) q 2n 2 -n+1 q n-3 (q n -1) -2 n 4, q 3 O + 2n (q) q 2n 2 -n+1 q n-1 (q n-2 -1) n 4, q 3 O - 2n (q) q 2n 2 +n+1 q n-1 (q n-2 -1) -1 n 4 U n (q)

q n 2 +2n-1 (q n -1)/(2q) n 3 even U n (q)

q n 2 +2n-1 (q n-1 -1)/2 n 3 odd quasicomp. q 2n 2 +3n

q n q prime, n 2 g with g(x) = λx a (x b -1), except when g(x) = 1 8 x 2n-1 or g(x) = x n , the difference d k = g(k+1)-g(k) is an increasing sequence, provided 2a+b > 1, according to Lemma 53. This condition holds for all the families and all the admissible values of n. The case when g(x) = 1 8 x 2n-1 or g(x) = x n satisfies also that d k is an increasing sequence. Hence, it suffices to prove (15) for the initial value of q for the induction.

Checking (15) for the initial point (q 0 , n 0 ), we prove that, if (9) holds for (q 1 , n 0 ) with q 1 q 0 , then (9) holds for any (q, n 0 ) with q q 1 . Such initial point is (2, 3) for L n (q), (3, 2) for S 2n (q) and q odd, (2, 2) for S 2n (q) and q even, (4, 3) for O 2n+1 (q) and q 4, (4, 4) for O + 2n (q) and q 4, (2, 4) for O - 2n (q), (2, 4) for U n (q) and n even, (2, 3) for U n (q) and n odd, and (2, 1) for the quasicomponents. The results of the check are positive for all the cases except for S 2n (q) and q even, for U n (q) and n odd, and for quasicomponents. These exceptional families have the starting point for the induction step in q in (4, 2), (4, 3) and (19, 2) respectively.

In a second place, for proving (14), the ratio inequation is

As Ñ (q, n) has the form q an 2 +bn+c , the left-hand side of the ratio inequation is q 2an+a+b . We can get rid of n by taking ratios a second time, getting the inequation

As a ∈ {1, 2}, the left-hand side of (17) can be bounded as q 2a q 4 . The expression D(q, n + 2) -2 D(q, n + 1) + D(q, n) can be bounded from below by q n-2 , and thus the right-hand side by exp(q n-2 ). So, we have the stronger inequation q 4 < exp(q n-2 ) or q < exp( 1 4 q n-2 ). As h(x) = exp( 1 4 x 2 ) is a convex function and the identity function is below h at the point with h (x) = 1, then the identity function is below h in the whole domain. Thus, we have proved q < exp( 1 4 q 2 ) exp( 1 4 q n-2 ) for n 4, hence (17) is proved for n 4 and any q, and thus the induction step in n for (16).

In order to complete the induction proof of (16), we shall prove it for certain n = n 0 and general q. As a 2 and b 3, the left-hand side of ( 16) can be bounded as contribution of an alternating group A r l +1 . The analogue for reduced contributions is the following.

Proposition 54. The total contribution of a record (r, N, l), with l 2, is less than the reduced contribution of an alternating group A r l +1 , except for (r, l) ∈ {(2, 2), (2, 3), (3, 2)}.

Proof. The inequation to prove is

which can be proved by induction in l 2. First, for r = 2, the cases l ∈ {2, 3} are exceptions also for [START_REF]Bounds for finite primitive complex linear groups[END_REF]lem. 11], so a fortiori for our claim. Thus, our starting point for r = 2 is l = 4, which satisfies the claim. Then, for r = 3, it appears a new exception for l = 2, which is also an exception for the sharp value (3 2 + 1)! 4 • 3 2 = 100800 of the right-hand side. The starting point for r = 3 is l = 3, which satisfies the claim. For 4 r 12, we can check that the claim is satisfied for l = 2. For r > 12, N = (r + 1)!, so the inequation to prove for l = 2 is 2(r + 1)! 2 3 (r 2 +1)/3 < (r 2 + 1)!, or its cube

The left-hand side has 3 + 6r + (r 2 + 1) factors that can be bounded by factors among the 3r 2 ones of the right-hand side. It suffices to prove that 3 + 6r + (r 2 + 1) < 3r 2 for r > 12, but the parabola 2r 2 -6r -4 is positive in this range, so the bound holds. Now, assuming (21) holds, we have to prove N l+1 • (l + 1)! < (r l+1 + 1)! 3 (r l+1 +1)/3 . Following the technique of Collins, N l+1 • (l + 1)! < N (l + 1) (r l + 1)! 3 (r l +1)/3 , so the claim holds if N (l + 1) (r l + 1)! 3 (r l +1)/3 (r l+1 + 1)! 3 (r l+1 +1)/3 , which is equivalent to

Leaving N 3 apart, the left-hand side has 3 + 3r l + r l+1 -r l factors that can be bounded by factors among the 3r l+1 ones of the right-hand side. The right-hand side has an excess of 2r l (r -1) -3 factors, which is a positive number in our ranges of l and r, so it suffices to check N 3 2 2r l (r-1)-3 in our ranges. For 3 r 12 and l = 2, it holds, and so for l 2, hence the induction is complete in this case. For r = 2 and l = 4, it holds too, and so for l 4, completing the induction also in this case. For r > 12, N = (r + 1)!, so we are comparing 3r + 3 + 3r l + r l+1 -r l factors with 3r l+1 ones in the inequation (22). The cubic 2r 3 -2r 2 -3r -3 is positive for r > 12, so the induction is completed in the last case.