Computing differential Galois groups of second-order linear q -difference equations
aa r X i v : . [ m a t h . N T ] S e p COMPUTING DIFFERENTIAL GALOIS GROUPS OFSECOND-ORDER LINEAR q -DIFFERENCE EQUATIONS CARLOS E. ARRECHE AND YI ZHANG
Abstract.
We apply the differential Galois theory for difference equations developed byHardouin and Singer to compute the differential Galois group for a second-order linear q -difference equation with rational function coefficients. This Galois group encodes thepossible polynomial differential relations among the solutions of the equation. We apply ourresults to compute the differential Galois groups of several concrete q -difference equations,including for the colored Jones polynomial of a certain knot. Introduction
Consider a second-order homogeneous linear q -dilation equation y ( q x ) + a ( x ) y ( qx ) + b ( x ) y ( x ) = 0 , (1.1)whose coefficients a ( x ) , b ( x ) ∈ ¯ Q ( x ) are rational functions in x with b ( x ) = 0, and q ∈ ¯ Q isneither zero nor a root of unity. We develop algorithms that allow one to discover all thepolynomial differential equations satisfied by the solutions to (1.1), or to decide that there arenone. Our methods and results apply equally well, with small and obvious modifications, toequations (1.1) where q is not necessarily an algebraic number and the coefficients a, b ∈ C ( x )for any computable algebraically closed field C containing Q ( q ). Our strategy here is similarto the one followed in [Arr17], where analogous algorithmic results were developed in thecontext of shift difference equations. We apply the differential Galois theory for differenceequations developed in [HS08], which studies equations such as (1.1) from a purely algebraicpoint of view. This theory attaches a geometric object G to (1.1), called the differential Galoisgroup, that encodes all the difference-differential algebraic relations among the solutions to(1.1). We develop an algorithm to compute the differential Galois group G associated to(1.1) by the theory of [HS08].The differential Galois theory for difference equations of [HS08] is a generalization of the q -dilation analogue of the Galois theory for difference equations presented in [vdPS97], wherethe Galois groups that arise encode the algebraic relations among the solutions to a givenlinear difference equation. An algorithm to compute the Galois group ˜ H associated to (1.1)by the theory of [vdPS97] is developed in [Hen97]—but for technical reasons this algorithmworks only over the larger base field ¯ Q ( { x /n } n ∈ N ), rather than the field of definition ¯ Q ( x ) of(1.1). In the course of our computation of the differential Galois group G of (1.1), we alsoextend the algorithm of [Hen97] to compute the Galois group H of (1.1) over the smalleroriginal basefield ¯ Q ( x ). The work of both authors was partially supported by NSF grant CCF-1815108.
A priori one knows that the Galois group H is a linear algebraic group, and the differentialGalois group G is a linear differential algebraic group (Definition 2.8). The difference Galoisgroup H serves as a close upper bound for the difference-differential Galois group G : it isshown in [HS08] that one can consider G as a Zariski-dense subgroup of H without loss ofgenerality (see Proposition 2.12 for a precise statement). In view of this fact, our strategy tocompute G is to first apply our extension (developed in the present work) of the algorithmof [Hen97] to compute H , and then compute the additional differential-algebraic equations(if any) that define G as a subgroup of H . The computation of G in general can be muchmore difficult than that of H because there are many more linear differential algebraic groupsthan there are linear algebraic groups (more precisely, the latter are instances of the former),so identifying the correct differential Galois group from among these additional possibilitiesrequires additional work.This strategy is reminiscent of the one begun in [Dre14], and concluded in [Arr14a,Arr14b,Arr16], to compute the parameterized differential Galois group for a second-order lineardifferential equation with differential parameters, where the results of [Kov86, BD79] arefirst applied to compute the classical (non-parameterized) differential Galois group for thedifferential equation, and one then computes the additional differential-algebraic equations,with respect to the parametric derivations, that define the parameterized differential Galoisgroup inside the classical one. However, the computation of the differential Galois group G for (1.1) presents substantial new complications that do not arise in the parameterizeddifferential setting. Many of these new complications are inherent to the computation ofdifferential Galois groups of difference equations in general, and already arise in the contextof shift difference equations (see the introduction to [Arr17] for a summary), but a brand newtechnical difficulty arises for the first time in the context of q -difference equations, which wedescribe below. The same difficulties will recur, with a vengeance, in the context of Galoistheory for difference equations over elliptic curves; our hope is that the treatment developedhere will serve as a useful blueprint for that more technical setting.It is known (see [Hen97]) that the Galois group ˜ H of any q -difference equation over¯ Q ( { x /n } n ∈ N ) has a cyclic group of connected components ˜ H/ ˜ H ◦ . This fact facilitates thedevelopment of the algorithm of [Hen97]. However, the Galois group H of a q -differenceequation over ¯ Q ( x ) may admit more generally a bicyclic group of connected components,which requires the development of new techniques to identify the correct Galois group fromamong this larger set of possibilities.A theoretical consequence of the results of § σ -invariants, or the full additive group of σ -invariants. This resultwas already known when the whole differential Galois G group was already unipotent [HS08,Prop. 4.3(2)], but not when the unipotent radical is a proper subgroup of G . In othercontexts (see for example [MOS14, MOS15]) the computation of the unipotent radical hasturned out to be the main theoretical obstacle in the development of algorithms to computeGalois groups in general. We expect that this contribution to the inverse Galois problem inthe present setting will have useful ramifications in the development of future algorithms tocompute differential Galois groups for higher-order q -difference equations. IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 3 Let us now describe the contents of this work in more detail. In §
2, we summarize thedifference-differential Galois theory of [HS08], and prove some auxiliary results that will beused in the sequel. In §
3, we recall some known results, and prove some new ones, concerningdifferential relations among solutions to first-order q -dilation difference equations. In §
4, wesummarize Hendriks’ algorithm [Hen98] to compute the difference Galois group ˜ H for (1.1)over ¯ Q ( { x /n } n ∈ N ), and explain how to extend it to compute the difference Galois group H for(1.1) over ¯ Q ( x ). In §
5, we show how to compute the difference-differential Galois group G for(1.1) when H is diagonalizable in Proposition 5.2. In §
6, we show how to compute G when H is assumed to be reducible but non-diagonalizable in Proposition 6.1 and Proposition 6.2—asa consequence, we show in Corollary 6.4 that the unipotent radical of G is always of a veryspecial form. In §
7, we compute G in Proposition 7.4, Proposition 7.6, and Proposition 7.7,under the assumption that H is irreducible and imprimitive (which possibility can arise inthree different ways, as a consequence of our insistence on computing Galois groups overthe basefield ¯ Q ( x ) and not just over ¯ Q ( { x /n } n ∈ N )). In §
8, we apply results from [AS17] tocompute G in Proposition 8.1, under the assumption that H contains SL . We conclude in § q -difference equations; in particularto the one satisfied by the colored Jones polynomial of a certain knot.2. Preliminaries on differential Galois theory for difference equations
We begin with a summary of the difference-differential Galois theory presented in [HS08].Every field is assumed to be of characteristic zero, and every ring is assumed to be commu-tative unless otherwise stated.
Definition 2.1. A σδ -ring is a commutative ring R with unit, equipped with an automor-phism σ and a derivation δ such that σ ( δ ( r )) = δ ( σ ( r )) for every r ∈ R . A σδ -field isdefined analogously. We write R σ = { r ∈ R | σ ( r ) = r } ; R δ = { r ∈ R | δ ( r ) = 0 } ; and R σδ = R σ ∩ R δ , and refer to these as the subrings of σ -constants , δ -constants , and σδ -constants , respectively.A σδ - R -algebra is a σδ -ring S equipped with a ring homomorphism R → S that commuteswith both σ and δ . If R and S are fields, we also say that S is a σδ -field extension of R .The notions of σ - R -algebra, δ - R -algebra, σ -field extension, and δ -field extension are definedanalogously. If z , . . . , z n ∈ S , we write R { z , . . . , z n } δ for the smallest δ - R -subalgebra of S that contains z , . . . , z n ; as R -algebras, we have R { z , . . . , z n } δ = R [ { δ i ( z ) , . . . , δ i ( z n ) | i ∈ N } ] . If Z = ( z ij ) with 1 ≤ i, j ≤ n is a matrix, we write R = { Z } δ for R { z , . . . , z n , . . . , z n , . . . , z nn } δ . The main example of σδ -field that we will consider throughout most of this paper is k = ¯ Q ( x ), where σ denotes the ¯ Q -linear automorphism defined by σ ( x ) = qx for somefixed q ∈ ¯ Q that is neither zero nor a root of unity, and δ = x ddx . Note that in this case k σ = k δ = ¯ Q .Suppose that k is a σδ -field, and consider the matrix difference equation σ ( Y ) = AY, where A ∈ GL n ( k ) . (2.1) C.E. ARRECHE AND Y. ZHANG
Definition 2.2. A σδ -Picard-Vessiot ring (or σδ -PV ring ) over k for (2.1) is a σδ - k -algebra R such that:(i) R is a simple σδ -ring, i.e., R has no ideals, other than 0 and R , that are stable underboth σ and δ ;(ii) there exists a matrix Z ∈ GL n ( R ) such that σ ( Z ) = AZ ; and(iii) R is differentially generated as a δ - k -algebra by the entries of Z and 1 / det( Z ), i.e., R = k { Z, / det( Z ) } δ .The matrix Z is called a fundamental solution matrix for (2.1).Note that when δ = 0, this coincides with the definition of the σ -PV ring over k for (2.1)given in [vdPS97, Def. 1.5]. In the usual Galois theory of difference equations presentedin [vdPS97], the existence and uniqueness of Picard-Vessiot rings up to k - σ -isomorphism isguaranteed by the assumption that k σ is algebraically closed (see [vdPS97, § k σ is δ -closed [Kol74, Tru10]. Definition 2.3.
The ring of δ - polynomials in n variables over a δ -field C is C { Y , . . . , Y n } δ = C [ { δ i ( Y ) , . . . , δ i ( Y n ) | i ∈ N } ] , the free C -algebra on the symbols δ i ( Y j ), on which δ acts as a derivation in the obvious way.We say L ∈ C { Y , . . . , Y n } δ is a linear δ - polynomial if it belongs to the C -linear span of thesymbols δ i ( Y j ).If R is a δ - C -algebra, we say that z . . . , z n ∈ R are differentially dependent over C if thereexists a δ -polynomial 0 = P ∈ C { Y , . . . , Y n } δ such that P ( z , . . . , z n ) = 0; otherwise we saythat z , . . . , z n are δ - independent over C . When a single element z ∈ R is δ -independent(resp., δ -dependent) over C , we say that z is δ - transcendental (resp., δ - algebraic ) over C .We say the δ -field C is δ - closed if any system of δ -polynomial equations { P = 0 , . . . , P m = 0 | P i ∈ C { Y , . . . , Y n } δ for 1 ≤ i ≤ m } that has a solution in ˜ C n for some δ -field extension ˜ C ⊇ C already has a solution in C n . Theorem 2.4. (Cf. [HS08, Prop. 2.4]) If k σ = C is δ -closed, there exists a σδ -PV ring for (2.1) , and it is unique up to σδ - k -isomorphism. Moreover, R σ = k σ . The following structural result is stated in a more general context in [HS08, Lem. 6.8],with the exception of the second part of item (3), which is proved as in [vdPS97, Cor. 1.16].
Proposition 2.5.
Let R be a σδ -PV ring over k for (2.1) , where k σ is δ -closed. There existidempotents e , . . . , e t − ∈ R such that:(1) R = R ⊕ · · · ⊕ R t − , where R i = e i R ;(2) the action of σ permutes the set { R , . . . , R t − } transitively, and each R i is left in-variant by σ t ;(3) each R i is a domain, and a σ t δ -PV ring over k for σ t ( Y ) = ( σ t − ( A ) . . . σ ( A ) A ) Y . From now on, unless explicitly stated otherwise, we assume that k is a σδ -field such that k σ is δ -closed. IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 5 Definition 2.6.
The σδ -Galois group of (2.1) is the group of σδ - k -automorphisms of the σδ -PV ring R for (2.1):Gal σδ ( R/k ) = { γ ∈ Aut k -alg ( R ) | γ ◦ σ = σ ◦ γ and γ ◦ δ = δ ◦ γ } . As in the usual (non-differential) Galois theory of difference equations [vdPS97], a choiceof fundamental solution Z = ( z ij ) ∈ GL n ( R ) defines a faithful representation Gal σδ ( R/k ) ֒ → GL n ( k σ ) : γ M γ , via γ ( Z ) = γ ( z ) · · · γ ( z n )... ... γ ( z n ) · · · γ ( z nn ) = z · · · z n ... ... z n · · · z nn · M γ . A different choice of fundamental solution matrix Z ′ ∈ GL n ( R ) defines a conjugate repre-sentation of Gal σδ ( R/k ) in GL n ( k σ ). Definition 2.7.
The systems σ ( Y ) = AY and σ ( Y ) = BY for A, B ∈ GL n ( k ) are equivalent if there exists a matrix T ∈ GL n ( k ) such that σ ( T ) AT − = B . In this case, if Z is afundamental solution matrix for σ ( Y ) = AY , then T Z is a fundamental solution matrix for σ ( Y ) = BY , and therefore the σδ -PV rings of k for these systems defined by the choice offundamental solution matrices Z and T Z , and the associated representations of σδ -Galoisgroups in GL n ( k σ ), are isomorphic. Definition 2.8.
Suppose that C is a δ -closed field. A linear differential algebraic group over C is a subgroup G of GL n ( C ) defined by (finitely many) δ -polynomial equations in thematrix entries. We say that G is δ -constant if G is conjugate in GL n ( C ) to a subgroup ofGL n ( C δ ).The differential algebraic subgroups of the additive and multiplicative groups of C , whichwe denote respectively by G a ( C ) and G m ( C ), were classified in [Cas72, Prop. 11, Prop. 31and its Corollary]. Proposition 2.9. If G ≤ G a ( C ) is a differential algebraic subgroup, then there exists alinear δ -polynomial L ∈ C { Y } δ such that G = { b ∈ G a ( C ) | L ( b ) = 0 } . If G ≤ G m ( C ) is a differential algebraic subgroup, then either G = µ ℓ , the group of ℓ th roots of unity for some ℓ ∈ N , or else G m ( C δ ) ⊆ G , and there exists a linear δ -polynomial L ∈ C { Y } δ such that G = (cid:8) a ∈ G m ( C ) (cid:12)(cid:12) L (cid:0) δaa (cid:1) = 0 (cid:9) . Theorem 2.10. (Cf. [HS08, Thm. 2.6]) Suppose that k σ is δ -closed, and that R is a σδ -PVring over k for (2.1) . Then R is a reduced ring, and any choice of fundamental solutionmatrix Z ∈ GL n ( R ) identifies Gal σδ ( R/k ) with a linear differential algebraic subgroup of GL n ( k σ ) . As in [HS08, p. 337], we observe that if R is a σδ -PV ring over k for (2.1), and K is the totalring of fractions of R , then any σδ - k -automorphism of K must leave R invariant, whence thegroup Gal σδ ( K/k ) of such automorphisms coincides with Gal σδ ( R/k ). The consideration ofthe total ring of fractions of R is necessary to obtain the following Galois correspondence. C.E. ARRECHE AND Y. ZHANG
Theorem 2.11. (Cf. [HS08, Thm. 2.7]) Suppose that k σ is δ -closed, and that R is a σδ -PVring over k for (2.1) . Denote by K the total ring of fractions of R , and by F the set of σδ -rings F such that k ⊆ F ⊆ K and every non-zero divisor in F is a unit in F . Let G denote the set of linear differential algebraic subgroups G of Gal σδ ( K/k ) . There is a bijectivecorrespondence F ↔ G given by F Gal σδ ( K/F ) = { γ ∈ Gal σδ ( K/k ) | γ ( r ) = r, ∀ r ∈ F } ; and G K G = { r ∈ K | γ ( r ) = r, ∀ γ ∈ G } . In particular, an element r ∈ K is left fixed by all of Gal σδ ( K/k ) if and only if r ∈ k .The following result relates the σδ -PV rings and σδ -Galois groups of [HS08] to the σ -PVrings and σ -Galois groups considered in [vdPS97, Hen98]. Proposition 2.12. (Cf. [HS08, Prop. 2.8]) Assume k σ is δ -closed. Let R be a σδ -PV ringover k for (2.1) with fundamental solution matrix Z ∈ GL n ( R ) , and let S = k [ Z, / det( Z )] ⊂ R . Then:(i) S is a σ -PV ring over k for (2.1) ; and(ii) Gal σδ ( R/k ) is Zariski-dense in the σ -Galois group Gal σ ( S/k ) . The following result characterizes those difference equations whose σδ -Galois groups are δ -constant. Proposition 2.13. (Cf. [HS08, Prop. 2.9]) Let R be a σδ -PV ring over k for σ ( Y ) = AY ,where A ∈ GL n ( k ) and k σ is δ -closed. Then Gal σδ ( R/k ) is a δ -constant linear differentialalgebraic group if and only if there exists a matrix B ∈ gl n ( k ) such that σ ( B ) = ABA − + δ ( A ) A − . In this case, there exists a fundamental solution matrix Z ∈ GL n ( R ) that satisfies the system ( σ ( Z ) = AZ ; δ ( Z ) = BZ. Differential relations among solutions of first-order q -differenceequations In this section we recall some known results, and prove some new ones, concerning differ-ential relations among solutions of first-order q -difference difference equations. The followingresult is proved in [HS08, Prop. 3.1]. Proposition 3.1.
Let R be a σδ - k -algebra with R σ = k σ . Suppose b , . . . , b m ∈ k and z , . . . , z m ∈ R satisfy σ ( z i ) − z i = b i ; i = 1 , . . . , m. Then z , . . . , z m are differentially dependent over k if and only if there exists a nonzero linear δ -polynomial L ( Y , . . . , Y m ) with coefficients in k σ and an element f ∈ k such that L ( b , . . . , b m ) = σ ( f ) − f. IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 7 For the remainder of this section, we restrict our attention to the σδ -field k = C ( x ), where δ ( x ) = x , C is a δ -closed field of characteristic zero, and σ is the C -linear automorphism of k defined by setting σ ( x ) = qx for some q ∈ C δ that is neither zero nor a root of unity.The following notion of q -discrete residue, defined in [CS12, Def. 2.7],will be crucial inseveral proofs in this paper. Definition 3.2.
For any non-zero β ∈ C , we call the subset[ β ] q = βq Z = { βq ℓ | ℓ ∈ Z } ⊂ C the q Z -orbit of β in C . Any f ∈ k can be decomposed into the form f = c + xp + p x s + m X i =1 n i X j =1 d i,j X ℓ =0 α i,j,ℓ ( x − β i q ℓ ) j , where p , p ∈ C [ x ]; s, m, n i , d i,j ∈ N ; c, α i,j,ℓ , β i ∈ C ; deg( p ) < s ; and the β i are non-zeroand belong to distinct q Z -orbits.The q - discrete residue of f at the q Z -orbit [ β i ] q of multiplicity j (with respect to x ) isdefined as: q -dres( f, [ β i ] q , j ) = d i,j X ℓ =0 q − ℓj α i,j,ℓ . In addition, the constant c above is the q - discrete residue of f at infinity, which we denoteby q -dres( f, ∞ ).The usefulness of the notion of discrete residue stems from the following result. Proposition 3.3. (Cf. [CS12, Prop. 2.10])Let f, g ∈ C [ x ] be non-zero, relatively prime polynomials. There exists h ∈ k such that σ ( h ) − h = f /g if and only if q - dres( f /g, ∞ ) = 0 and q - dres( f /g, [ β ] q , j ) = 0 for every j ∈ N and every = β ∈ C such that g ( β ) = 0 . The following computational lemma will be used to sharpen the conclusion of [HS08,Cor. 3.3] in the following Corollary 3.5.
Lemma 3.4.
Suppose = a ∈ C δ ( x ) , r ∈ Z ≥ , and = β ∈ C δ is a zero or pole of a . Then q - dres (cid:18) δ r (cid:18) δ ( a ) a (cid:19) , [ β ] q , r + 1 (cid:19) = ( − r · r ! · β r · q - dres (cid:18) δ ( a ) a , [ β ] q , (cid:19) . Proof.
We may assume without loss of generality that δ ( a ) a = d X ℓ =0 (cid:18) e ℓ + e ℓ q ℓ βx − βq ℓ (cid:19) (3.1)for some 0 = β ∈ C δ , d ∈ Z ≥ , and e ℓ ∈ Z for ℓ = 0 , . . . , d . Observe that in this case q -dres (cid:0) δ ( a ) a , [ β ] q , (cid:1) = P dℓ =0 βe ℓ , by Definition 3.2. We claim that δ r (cid:18) δ ( a ) a (cid:19) = d X ℓ =0 ( − r r ! q ℓ ( r +1) β r +1 e ℓ ( x − βq ℓ ) r +1 + (lower-order terms) , (3.2) C.E. ARRECHE AND Y. ZHANG which would indeed imply that q -dres (cid:18) δ r (cid:18) δ ( a ) a (cid:19) , [ β ] q , r + 1 (cid:19) = d X ℓ =0 ( − r r ! β r +1 e ℓ and conclude the proof of the Lemma. We prove (3.2) by induction. The case r = 0 is just(3.1). Assuming (3.2) for some r ≥
0, note that δ r +1 (cid:18) δ ( a ) a (cid:19) = d X ℓ =0 ( − r +1 ( r + 1)! q ℓ ( r +1) β r +1 e ℓ (cid:0) ( x − βq ℓ ) + βq ℓ (cid:1) ( x − βq ℓ ) r +2 + (lower-order terms) . This concludes the proof of the claim, and the Lemma. (cid:3)
The following result sharpens the conclusion of [HS08, Cor. 3.3].
Corollary 3.5.
Let R be a σδ - k -algebra with R σ = k σ = C . Let a , . . . , a m ∈ C δ ( x ) × and z , . . . , z m ∈ R × such that σ ( z i ) = a i z i ; i = 1 , . . . , m. Then z , . . . , z m are differentially dependent over k if and only if there exist: n , . . . , n m ∈ Z ,not all zero and with gcd( n , . . . , n m ) = 1 ; c ∈ Z ; and an element f ∈ k , such that n δ ( a ) a + · · · + n m δ ( a m ) a m = σ ( f ) − f + c. (3.3) Proof.
First suppose there exist integers n , . . . , n m , c ∈ Z as in (3.3). Since for each i =1 , . . . , m we have that σ (cid:0) δ ( z i ) z i (cid:1) = δ ( z i ) z i + δ ( a i ) a i , it follows that σ " m X i =1 n i δ (cid:18) δ ( z i ) z i (cid:19)! − δ ( f ) = m X i =1 n i δ (cid:18) δ ( z i ) z i (cid:19) + δ m X i =1 n i δ ( a i ) a i ! − σ (cid:0) δ ( f ) (cid:1) == m X i =1 n i δ (cid:18) δ ( z i ) z i (cid:19) + δ (cid:0) σ ( f ) − f + c (cid:1) − σ (cid:0) δ ( f ) (cid:1) = m X i =1 n i δ (cid:18) δ ( z i ) z i (cid:19) − δ ( f ) . Therefore, m X i =1 n i δ (cid:18) δ ( z i ) z i (cid:19) = δ ( f ) + e for some e ∈ R σ = k σ . This shows that z , . . . , z m are δ -dependent over k , after multiplyingby ( z . . . z m ) on both sides.Since σ ( δ ( z i ) z i ) = δ ( z i ) z i + δ ( a i ) a i for each i = 1 , . . . , m , Proposition 3.1 implies that the z i aredifferentially dependent over k if and only if there exists an element f ∈ k and a nonzerolinear δ -polynomial L ( Y , . . . , Y m ) = m X i =1 r i X j =0 c i,j δ j Y i , c i,j ∈ C, such that g = L (cid:18) δ ( a ) a , . . . , δ ( a m ) a m (cid:19) = σ ( f ) − f. (3.4) IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 9 Let r = max { r i | c i,r i = 0 for some i } . For each 0 = β ∈ C , it follows from (3.4),Proposition 3.3, and Lemma 3.4, that q -dres( g, [ β ] q , r + 1) = ( − r · r ! · β r · m X i =1 c i,r · q -dres (cid:18) δ ( a i ) a i , [ β ] q , (cid:19) = 0 . (3.5)On the other hand, it follows from Definition 3.2 that for each i = 1 , . . . , m we have that q -dres (cid:18) δ ( a i ) a i , [ β ] q , (cid:19) = β · e i for some e i ∈ Z . (3.6)Substituting (3.6) into (3.5), we have q -dres( g, [ β ] q , r + 1) = ( − r · r ! · β r +1 · m X i =1 c i,r · e i = 0 . (3.7)Since β = 0, the above equation is equivalent to P mi =1 c i,r · e i = 0. Since e i ∈ Z for each i =1 , . . . , m , we may take the c i,r = n i to be integers. Set c = P mi =1 n i · q -dres (cid:16) δ ( a i ) a i , ∞ (cid:17) . Sinceboth n i and q -dres (cid:16) δ ( a i ) a i , ∞ (cid:17) are integers, we see that c ∈ Z is divisible by gcd( n , . . . , n m ).Moreover, we have q -dres (cid:18) n δ ( a ) a + · · · + n m δ ( a m ) a m − c, ∞ (cid:19) = 0 . (3.8)By (3.7) and (3.8), the conclusion follows from another application of Proposition 3.3 anddividing both sides by gcd( n , . . . , n m ). (cid:3) Hendriks’ algorithm
In this section, we summarize the results of [Hen97] that we will need in our algorithm,and explain how to refine them to meet our goals. From now on, we restrict our attentionto equations of the form σ ( y ) + aσ ( y ) + by = 0 , (4.1)where a, b ∈ ¯ Q ( x ) with b = 0, and σ is the ¯ Q -linear automorphism of ¯ Q ( x ) defined by σ ( x ) = qx , where q ∈ ¯ Q is neither zero nor a root of unity. Our discussion here could begeneralized to drop the assumption that q is an algebraic number and allowing a, b ∈ C ( x ),for any computable algebraically closed field C containing Q ( q ), as we mentioned in theintroduction (see also the introduction to [Hen97]), but at the cost of overburdening thenotation.The matrix equation corresponding to (4.1) is σ ( Y ) = AY, where A := (cid:18) − b − a (cid:19) ∈ GL ( k ) . (4.2)We consider ¯ Q ( x ) as a σδ -field by setting δ = x ddx , the Euler derivation. In this section onlywe will denote k = ¯ Q ( x ), but in future sections we will recycle notation and denote by k the larger σδ -field C ( x ), where C is a δ -closure of ¯ Q , and σ is the C -linear automorphism of C ( x ) defined by σ ( x ) = qx . The algorithm of [Hen97] computes the σ -Galois group of (4.1) over the larger basefield k ∞ defined as follows. Let { q n ∈ ¯ Q | n ∈ N } denote a compatible system of n -th roots of q = q ,so that for any factorization ℓm = n we have q ℓn = q m , and consider the cyclic σ -field extension k n = ¯ Q ( x n ) of ¯ Q ( x ) such that x nn = x = x and x ℓn = x m for any factorization n = ℓm , withthe σ -field structure given by σ ( x n ) = q n x n . Then the ¯ Q -linear maps k m ֒ → k n defined by x m x ℓn are embeddings of σ -fields. Let k ∞ = lim −→ k n = S n ≥ k n . By [Hen97, Lemmas 9and 10], the σ -field k ∞ has property P : Definition 4.1.
We say a σ -field k has property P if:(1) k is a C field; and(2) if k ′ is a finite algebraic extension of k such that σ extends to an automorphism of k ′ then k ′ = k .This allows Hendriks to compute the σ -Galois group of (4.1) over k ∞ by finding a gaugetransformation T ∈ GL ( k ∞ ) that puts (4.2) in the standard form of [Hen97, Definition 8].Another special consequence of the fact that k ∞ enjoys property P (Definition 4.1) is thatthe σ -Galois group H ∞ for (4.1) over k ∞ (and in fact every difference Galois group over k ∞ )is such that its quotient H ∞ /H ◦∞ by the connected component of the identity H ◦∞ must be a(finite) cyclic group (cf. [Hen97, Thm. 6]). This facilitates the algorithm of [Hen97] by rulingout a priori the consideration of algebraic groups whose group of connected components isnot cyclic (cf [Hen97, Lem. 12]). The situation for σ -Galois groups over k is less restrictive,but we still know by [vdPS97, Prop. 12.2(1)] that the σ -Galois group H for (4.1) over k (and in fact every difference Galois group over k ) has the property that the quotient H /H ◦ is (finite) bicyclic, i.e., a product of two finite cyclic groups. Thus it is possible for us torealize additional algebraic groups H as Galois groups for (4.1) over k that do not occur inthe list [Hen97, Lem. 16 and Lem. 20] of possible σ -Galois groups over k ∞ . In particular, anyreducible algebraic subgroup of GL ( ¯ Q ) can (and does) occur as the σ -Galois group for somedifference equation (4.1), and any irreducible imprimitive algebraic subgroup of GL ( ¯ Q ) withbicyclic group of connected components can (and does) occur as a Galois group over k .The algorithm developed in [Hen97] to compute H ∞ proceeds as follows. We first decidewhether there exists a solution u ∈ k ∞ to the Riccati equation uσ ( u ) + au + b = 0 . (4.3)If such a solution u exists, then the σ -Galois group H ∞ of (4.2) over k ∞ is reducible , i.e.,conjugate to an algebraic subgroup of G m ( ¯ Q ) ⋉ G a ( ¯ Q ) ≃ (cid:26)(cid:18) α β λ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α, β, λ ∈ ¯ Q , αλ = 0 (cid:27) . Moreover, if there exist at least two distinct solutions u , u ∈ k ∞ to (4.3) then H ∞ is diagonalizable , i.e., conjugate to an algebraic subgroup of G m ( ¯ Q ) ≃ (cid:26)(cid:18) α λ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α, λ ∈ ¯ Q , αλ = 0 (cid:27) ; IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 11 and if there are at least three distinct solutions in k ∞ to (4.3) then there are infinitely many,and this occurs if and only if H ∞ is an algebraic subgroup of G m ( ¯ Q ) ≃ (cid:26)(cid:18) α α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α ∈ ¯ Q , α = 0 (cid:27) . If there is no solution u ∈ k ∞ to the Riccati equation (4.3), then H ∞ is irreducibleby [Hen97, Thm. 13]. In this case, the next step is to attempt to find T ∈ GL ( k ∞ ) and r ∈ k ∞ such that σ ( T ) (cid:18) − b − a (cid:19) T − = (cid:18) − r (cid:19) . (4.4)If a = 0 already, then we may take T = ( ) and r = b . If a = 0, we then attempt to find asolution e ∈ k ∞ to the Riccati equation eσ ( e ) + (cid:0) σ ( ba ) − σ ( a ) + σ ( b ) a (cid:1) e + σ ( b ) ba = 0 . (4.5)If there exists such a solution e ∈ k ∞ to (4.5), then it is proved in [Hen97, Thm. 18] thatthere exists a matrix T ∈ GL ( k ∞ ) such that (4.4) is satisfied with r = − aσ ( a ) + σ ( b ) + aσ ( ba ) + aσ ( e ) , (4.6)and H ∞ is imprimitive , i.e., conjugate to an algebraic subgroup of {± } ⋉ G m ( ¯ Q ) ≃ (cid:26)(cid:18) α λ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α, λ ∈ ¯ Q , αλ = 0 (cid:27) ∪ (cid:26)(cid:18) βǫ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) β, ǫ ∈ ¯ Q , βǫ = 0 (cid:27) . (4.7)Finally, if a = 0 and neither (4.3) nor (4.5) admits a solution in k ∞ , then SL ( ¯ Q ) ⊆ H ∞ ,and we compute H ∞ as in [Hen97, § H ∞ ) ⊆ G m ( ¯ Q ) ofthe determinant homomorphism.In order to produce an algorithm that computes the σ -Galois group of (4.1) over k = k ,we introduce additional notation and state some ancillary results. Let ζ n ∈ ¯ Q for n ∈ N denote a compatible system of n -th roots of unity, so that for any factorization ℓm = n wehave ζ ℓn = ζ m . Then k n is a σ -PV ring over k m for σ ( y ) = q n y with fundamental solution (1 × y = x n and cyclic σ -Galois group h τ n,m i = Gal σ ( k n /k m ) given by τ n,m ( x n ) = ζ ℓ x n .Let S ∞ denote a σ -PV ring over k ∞ for (4.2) with fundamental solution matrix Y . Then S n = k n [ Y, / det( Y )] is a σ -PV ring over k n for (4.2). Let us write H n = Gal σ ( S n /k n ) for n ∈ N ∪ {∞} . Then we see that S n is a σ -PV ring over k for the system σ ( Y n ) = − b − a
00 0 q n Y n , with fundamental solution matrix Y n = y y σ ( y ) σ ( y ) 00 0 x n . The following result is proved formally as in [Arr14b, Lem. 3.1 and Prop. 3.2] and [Arr16,Lem. 12 and Prop. 13]. Full proofs will appear in [Arr20].
Proposition 4.2.
Let ˜ H n := Gal σ ( S n /k ) and µ n denote cyclic group of n -th roots of unity.Then the intersection S ∩ k n = k m for some factorization n = ℓm , and the map ϕ : ˜ H n → H × µ n γ ( γ | S , γ | k n ) is an isomorphism onto the fiber product H × µ m µ n = { ( γ, ζ ) | γ ∈ H and ζ ∈ µ n such that γ ( x m ) = ζ ℓ x m } . (4.8)We record the following two consequences of Proposition 4.2. Corollary 4.3. S ∩ k n = k if and only if H n ≃ H , and S ∩ k n = k n if and only if H n isa normal subgroup of H of index n . Corollary 4.4.
The intersection in S ∞ given by k ∞ ∩ S = k m for some m ∈ N , and H ∞ ≃ H ℓm for every ℓ ∈ N . In particular, H ∞ is a normal subgroup of H of index m . Having computed the σ -Galois group H ∞ of (4.1) over k ∞ as in [Hen97], we can thencompute the σ -Galois group H of (4.1) over k according to the following possibilities. Theexplicit computation of H is obtained in each case as a by-product of our computation ofthe corresponding differential Galois group of (4.1) in the following sections. Proposition 4.5.
Precisely one of the following possibilities occurs.(1) There are infinitely many solutions to (4.3) in k . In this case, H is a subgroup of G m ( ¯ Q ) (included in GL ( ¯ Q ) as scalar matrices).(2) There are exactly two solutions u , u ∈ k to (4.3) . In this case, H is diagonalizable(but not contained in the group of scalar matrices).(3) There is exactly one solution u ∈ k to (4.3) . In this case, H is reducible but notdiagonalizable.(4) There are no solutions to (4.3) in k , but there are exactly two solutions u , u ∈ k \ k to (4.3) , and u = ¯ u is the Galois conjugate of u over k . In this case, H isirreducible and imprimitive.(5) There are no solutions to (4.3) in k , and either a = 0 or there is a solution e ∈ k to (4.5) . In this case, H is irreducible and imprimitive.(6) There are no solutions to (4.3) nor to (4.5) in k and a = 0 . In this case, H isirreducible and primitive, and SL ( ¯ Q ) ⊆ H .Proof. It is clear that the possibilities above are mutually exclusive. It remains to show thatthese possibilities are exhaustive, and that the σ -Galois group H is as stated in each case.Let us first show that these possibilities are exhaustive. By [Hen97, Thm. 13], there areeither zero, one, two, or infinitely many solutions to (4.3) in k ∞ . By [Hen97, Thm. 15], ifthere exists a solution u ∈ k ∞ to the Riccati equation (4.3), then there exists a solutionin k . Since the coefficients a, b ∈ k = k , for any solution u ∈ k \ k to (4.3) the Galoisconjugate ¯ u := τ , ( u ) must also satisfy (4.3). Hence, if there is exactly one solution u ∈ k ∞ to (4.3), then u ∈ k , and if there are exactly two solutions u , u ∈ k ∞ to (4.3), then either u , u ∈ k , or else u , u ∈ k \ k and u = ¯ u is the Galois conjugate of u over k . In thecase where there are infinitely many solutions to (4.3) in k ∞ , the proof of [Hen97, Thm. 15]shows that at least three of these solutions actually belong to k , in which case the proofof [Hen98, Thm. 4.2] shows that there are infinitely many solutions to (4.3) in k . This showsthat cases (1)–(4) exhaust the possibilities where there is at least one solution to (4.3) in k ∞ .Supposing now that there are no solutions to (4.3) in k ∞ and a = 0, by [Hen97, Thm. 15]we again have that if there exists at least one solution in k ∞ to (4.5), then there exists a IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 13 solution in k . This concludes the proof that the possibilities listed in Proposition 4.5 areexhaustive and mutually exclusive.The statements corresponding the form of the σ -Galois group H will be established sep-aretely in the following sections according to the possibilities listed above, depending on theexistence of solutions to (4.3) or (4.5) in k or k as discussed above. (cid:3) In view of Proposition 2.12, in order to compute the σδ -group G of (4.1), we will first applythe results of [Hen97] to compute the solutions to (4.3) and/or (4.5) in k , which accordingto the possibilities in Proposition 4.5 (and as we will show in each case in the followingsections) results in knowing whether the corresponding σ -Galois group H is: diagonalizable;reducible (but not diagonalizable); irreducible and imprimitive; or irreducible and primitive.We will then compute the additional δ -algebraic equations that define G as a subgroup of H in each case (and obtain the explicit computation of H itself along the way). In order toapply the theory of [HS08] to study (4.1), we will consider (4.1) as a difference equation overthe larger basefield C ( x ) mentioned at the beginning of this section, where we recall C is a δ -closed field extension of ( ¯ Q , δ ) such that C δ = ¯ Q (the existence of such a C is guaranteedby [Kol74, Tru10]), and the σδ -structure of C ( x ) extends that of ¯ Q ( x ): σ is the C -linearautomorphism of C ( x ) defined by σ ( x ) = qx . Remark C ( x ) to ¯ Q ( x )) . The application of the results of [Hen97] andProposition 4.5 to compute the σ -Galois group of (4.1) over C ( x ), rather than over ¯ Q ( x ),requires some justification. The point is that the number of solutions to the Riccati equations(4.3) and (4.5) in C ( x ) is the same as the number of solutions in ¯ Q ( x ). This follows froman elementary argument: suppose that a given polynomial σ -equation over ¯ Q ( x ) admits asolution pq ∈ C ( x ), where p = a n x n + · · · + a x + a and q = b m x m + · · · + b x + b . Thisis equivalent to the coefficients a i and b j satisfying b m = 0 and a system of polynomialequations defined over ¯ Q , which defines an affine algebraic variety V over ¯ Q . Since ¯ Q isalgebraically closed and C is countable, V ( C ) and V ( ¯ Q ) must have the same cardinality.Moreover, the possible defining equations for the σ -Galois groups of (4.1) over ¯ Q ( x ) andover C ( x ), whether in the reducible, irreducible and imprimitive, or irreducible and primitivecases, are all witnessed by monomial relations among (the standard form of) elements in¯ Q ( x ). Though we will see this explicitly in each situation in the following sections, it isworthwhile to emphasize now that the σ -Galois group of (4.1) over C ( x ) consists of the C -points of the σ -Galois group over ¯ Q ( x ), i.e., the former is defined as an algebraic subgroupof GL ( C ) by the same algebraic equations defining the latter as an algebraic subgroup ofGL ( ¯ Q ). 5. Diagonalizable groups
We recall the notation introduced in the previous sections: k = C ( x ), C is a δ -closureof ¯ Q with C δ = ¯ Q , σ denotes the C -linear automorphism of k defined by σ ( x ) = qx , and δ ( x ) = x . Let us first suppose that there exist at least two distinct solutions u , u ∈ ¯ Q ( x ) tothe Riccati equation (4.3) as in items (1) or (2) of Proposition 4.5. Then (4.2) is equivalentover ¯ Q ( x ) to σ ( Y ) = (cid:18) u u (cid:19) Y, in view of the following remark. Remark . Given two distinct solutions u and u to (4.3), the gauge transformation (whichis different from the one specified in the proof of [Hen98, Thm. 4.2]) T := 1 u − u · (cid:18) u − u − (cid:19) satisfies σ ( T ) AT − = (cid:0) u u (cid:1) .In this case, we compute G with the following result. Proposition 5.2.
Assume that u , u ∈ ¯ Q ( x ) are both different from , and let R be the σδ -PV ring over k corresponding to the system σ ( Y ) = (cid:18) u u (cid:19) Y. (5.1) Then G = Gal σδ ( R/k ) is the subgroup of G m ( C ) = (cid:26)(cid:18) α α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α , α ∈ C, α α = 0 (cid:27) (5.2) defined by the following conditions on α and α .(i) There exist m , m ∈ Z , not both zero, and f ∈ ¯ Q ( x ) × such that u m u m = σ ( f ) f ifand only if α m α m = 1 .(ii) There exist m , m ∈ Z , not both zero and with gcd( m , m ) = 1 ; c ∈ Z ; and f ∈ ¯ Q ( x ) such that m δ ( u ) u + m δ ( u ) u = σ ( f ) − f + c if and only if δ ( m δ ( α ) α + m δ ( α ) α ) = 0 .Moreover, c = 0 if and only if δ ( α m α m ) = 0 .(iii) If neither of the conditions above is satisfied, then G = H = G m ( C ) .Proof. We begin by observing that, if we can find f ∈ k witnessing the relations in items(i) or (ii), then we may take f ∈ ¯ Q ( x ), since u i ∈ ¯ Q ( x ) (cf. [Har08, Lem. 2.4, Lem. 2.5] andRemark 4.6). Note that by Theorem 2.4, R σ = C . Let y , y ∈ R be non-zero elements suchthat σ ( y i ) = u i y i . Then (cid:0) y y (cid:1) is a fundamental solution matrix for (5.1), so y , y ∈ R × and the embedding of G into (5.2) is given by γ ( y i ) = α γ,i y i for i = 1 , γ ∈ G .The proof of item (i) is standard: given m , m ∈ Z we have that α m γ, α m γ, = 1 for every γ ∈ G if and only if γ ( y m y m ) = y m y m for every γ ∈ G . By Theorem 2.11, this isequivalent to y m y m = f ∈ k , which in turn is equivalent to σ ( f ) f = u m u m .Setting δ ( y i ) y i =: g i ∈ R for i = 1 ,
2, we see that σ ( g i ) − g i = δ ( u i ) u i and γ ( g i ) = g i + δ ( α γ,i ) α γ,i . (5.3)By Corollary 3.5, y and y are differentially dependent over k if and only if there exist m , m ∈ Z , not both zero and with gcd( m , m ) = 1, c ∈ Z , and f ∈ k such that m δ ( u ) u + m δ ( u ) u = σ ( f ) − f + c. (5.4) IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 15 Hence, if there do not exist such m , m , c ∈ Z and f ∈ k , y and y are δ -independent over k , which implies that G = G m ( C ) by [HS08, Prop. 6.26]. This proves item (iii).Let us establish item (ii). It follows from (5.3) that for any m , m ∈ Z we have σ ( m g + m g ) − ( m g + m g ) = m δ ( u ) u + m δ ( u ) u ; and γ ( m g + m g ) = ( m g + m g ) + m δ ( α γ, ) α γ, + m δ ( α γ, ) α γ, . Suppose there exists f ∈ k satisfying (5.4) with c = 0 and gcd( m , m ) = 1. Then m g + m g − f ∈ k σ , which implies that m g + m g ∈ k and therefore δ ( α m γ, α m γ, ) = 0for every γ ∈ G . On the other hand, if δ ( α m γ, α m γ, ) = 0 for every γ ∈ G with at least one m i = 0, then the same relation holds after replacing m i with m i gcd( m ,m ) and we see that m g + m g = f ∈ k satisfies (5.4) with c = 0.More generally, suppose there exist f ∈ k and c ∈ Z satisfying (5.4) with gcd( m , m ) = 1.Then we see that m δ ( g ) + m δ ( g ) − δ ( f ) ∈ k σ , and therefore m δ ( g ) + m δ ( g ) ∈ k , whichimplies that δ (cid:18) m δ ( α γ, ) α γ, + m δ ( α γ, ) α γ, (cid:19) = 0 for every γ ∈ G. (5.5)On the other hand, assuming (5.5) with at least one m i = 0, then the same relation holdsafter replacing m i with m i gcd( m ,m ) , and we have that m δ ( g )+ m δ ( g ) = g ∈ k , and therefore m δ (cid:18) δ ( u ) u (cid:19) + m δ (cid:18) δ ( u ) u (cid:19) = σ ( g ) − g. By Proposition 3.3, for each β ∈ Q × we have that0 = q -dres (cid:18) δ (cid:18) δ ( u m u m ) u m u m (cid:19) , [ β ] q , (cid:19) = − β · q -dres (cid:18) δ ( u m u m ) u m u m , [ β ] q , (cid:19) , where the second equality follows from Lemma 3.4. Hence, letting c := q -dres (cid:18) δ ( u m u m ) u m u m , ∞ (cid:19) = m · q -dres (cid:18) δ ( u ) u , ∞ (cid:19) + m · q -dres (cid:18) δ ( u ) u , ∞ (cid:19) , (5.6)we have that c ∈ Z and every q -discrete residue of m δ ( u ) u + m δ ( u ) u − c is 0. By anotherapplication of Proposition 3.3, there exists f ∈ k satisfying (5.4) with c as in (5.6). (cid:3) Remark . To compute the difference-differential Galois group G for (4.1) when there existat least two distinct solutions u , u ∈ ¯ Q ( x ) to the Riccati equation (4.3), we apply Propo-sition 5.2 as follows. First, compute the q -discrete residues r i ([ β ] q ) := q -dres (cid:16) δ ( u i ) u i , [ β ] q , (cid:17) at each q Z -orbit [ β ] q for β ∈ ¯ Q × as in Definition 3.2 (note these will be zero for any β thatis neither a zero nor a pole of u or u ). Then decide whether there exist relatively prime m , m ∈ Z such that m r ([ β ] q ) + m r ([ β ] q ) = 0 for every q Z -orbit [ β ] q simultaneously (ingeneral this will be an overdetermined linear system over ¯ Q , so the task is to decide whetherthere exists a non-zero solution in ¯ Q and then whether such a solution can be taken tobe in Z ). For any such pair (0 , = ( m , m ) ∈ Z , taking c ∈ Z as in (5.6) the proof of Proposition 5.2 shows that there exists f ∈ ¯ Q ( x ) satisfying (5.4); it is not necessary todetermine what the certificate f actually is.The Z -module M generated by all pairs ( m , m ) ∈ Z as in Proposition 5.2(ii) is freeof rank r ≤
2. It follows from the proof of Corollary 3.5 that Z /M is torsion-free, andtherefore also free of rank 2 − r , since if ( dm , dm ) ∈ M then ( m , m ) ∈ M also for any d ∈ Z . Thus, if the rank of M is r = 2 then M = Z . It follows that the defining equationsfor G arising from Proposition 5.2(ii) are given by either: a single pair ( m , m ), unique upto multiplication by ± c in (5.6) is 0; or else the two relations corresponding to (1 , , c i := q -dres( δ ( u i ) u i ) = 0 for both i = 1 ,
2, in which case we obtain an additional relation given by δ ( α d α εd ) = 0 with d i := lcm( | c | , | c | ) c i and ε = ( c c ∈ Z < ; − c c ∈ Z > . (5.7)Having computed all possible relations arising from Proposition 5.2(ii), let us now showhow to find the possible relations arising from Proposition 5.2(i), and thus determine alldefining equations for G ⊆ G m ( C ) . We still denote by M ⊆ Z the Z -submodule generatedby pairs ( m , m ) as in Proposition 5.2. If M = { (0 , } then G = G m ( C ) as in Proposi-tion 5.2(iii), so from now on we assume M is not trivial. We saw above that either M = Z or else M = Z · ( m , m ) with gcd( m , m ) = 1.Suppose M = Z · ( m , m ). If the value of c given in (5.6) is not 0, then G is defined bythe single equation δ (cid:16) δ ( α m α m ) α m α m (cid:17) = 0 as in Proposition 5.2(ii). On the other hand, if this c = 0, then we must decide whether there exist: a primitive n -th root of unity ζ n , integers r, s such that 0 ≤ r < s and gcd( r, s ) = 1, and g ∈ ¯ Q ( x ) × such that u m u m = ζ n q rs σ ( g ) g . Ifso, then G is defined by the single equation ( α m α m ) ℓ = 1 as in Proposition 5.2(i), where ℓ := lcm( n, s ), the least common multiple of n and s ; otherwise, α m α m has infinite orderin G m ( C δ ) for every (cid:0) α α (cid:1) ∈ G , and G is defined by the single equation δ ( α m α m ) = 0 asin Proposition 5.2(ii) only.If M = Z , let again c i := q -dres( δ ( u i ) u i ). If exactly one c i is 0, say c = 0 = c , then wemust decide whether there exist: a primitive n -th root of unity ζ n , integers r, s such that0 ≤ r < s and gcd( r, s ) = 1, and g ∈ ¯ Q ( x ) × such that u = ζ n q rs σ ( g ) g . If so, then G is definedby the equations: α ℓ = 1 as in Proposition 5.2(i), with ℓ := lcm( n, s ), and δ (cid:16) δ ( α ) α (cid:17) = 0as in Proposition 5.2(ii); otherwise, G is defined instead by δ ( α ) = 0 and δ (cid:16) δ ( α ) α (cid:17) = 0.The case where c = 0 = c is analogous. If c , c = 0, then we must decide whether thereexist: a primitive n -th root of unity ζ n , integers r, s such that 0 ≤ r < s and gcd( r, s ) = 1,and g ∈ ¯ Q ( x ) × such that u d u εd = ζ n q rs σ ( g ) g , with d , d , ε defined as in (5.7). If so, then G is defined by the equations δ (cid:16) δ ( α i ) α i (cid:17) = 0 for i = 1 , α d α εd ) ℓ = 1 as in Proposition 5.2(i), where ℓ := lcm( n, s ); otherwise, G is defined by δ (cid:16) δ ( α i ) α i (cid:17) = 0 for i = 1 , IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 17 The case where M = Z and c i := q -dres( δ ( u i ) u i ) = 0 for both i = 1 , m , m ∈ Z , a primitive n -th root of unity ζ n , integers r, s such that 0 ≤ r < s and gcd( r, s ) = 1, and g ∈ ¯ Q ( x ) × such that u m u m = ζ n q rs σ ( g ) g .If so, then G is defined by δ ( α i ) = 0 for i = 1 , α m α m ) ℓ = 1 as in Proposition 5.2(i), where ℓ := lcm( n, s ); otherwise, G is defined by δ ( α i ) = 0 for i = 1 , u := u m u m = ζ n q rs σ ( g ) g as above for a given single pair( m , m ) is addressed in [Hen97, § reduced form ˜ u = hx n pq where: h ∈ ¯ Q × ; n ∈ Z ; p, q ∈ ¯ Q [ x ]are monic such that gcd( p, σ m ( q )) = 1 for every m ∈ Z ; if h = ζ q t for some root of unity ζ and t ∈ Q , then 0 ≤ t <
1; and such that there exists g ∈ ¯ Q ( x ) × with u = ˜ u σ ( g ) g for some g ∈ ¯ Q ( x ) × . Thus we only need to check whether ˜ u = ζ n q rs .In the case where M = Z and c i := q -dres( δ ( u i ) u i ) = 0 for both i = 1 ,
2, one can show thatthe standard form ˜ u i = h i with h i ∈ ¯ Q × , and one needs to decide whether h and h aremultiplicatively independent modulo q Z . We do not know how to produce a priori boundson the possible coefficients ( m , m ) such that h m h m ∈ q Z in general, so in this case onlywe offer no improvements on the algorithm in [Hen97, § m , m ) ∈ Z , although this requires the ability to computethe q -discrete residues of δ ( u i ) u i (cf. [vdPS97, § Reducible non-diagonalizable groups
We recall the notation introduced in the previous sections: k = C ( x ), where C is a δ -closure of ¯ Q , σ denotes the C -linear automorphism of k defined by σ ( x ) = qx , and δ ( x ) = x .We now proceed to define the additional notation that we will use throughout this section.We will assume that there exists exactly one solution u ∈ ¯ Q ( x ) to the Riccati equation (4.3),so that the σ -Galois group H for (4.1) is reducible but not diagonalizable as in Proposi-tion 4.5(3), and the difference operator implicit in (4.1) factors as σ + aσ + b = ( σ − bu ) ◦ ( σ − u ) , as we saw in §
4. This means that there is a C -basis of solutions { y , y } in any σδ -PV ring R for (4.1) such that y , y = 0 satisfy σ ( y ) = uy and σ ( y ) − uy = y , where y = 0 satisfies σ ( y ) = bu y . A fundamental solution matrix for (4.2) is given by (cid:18) y y σ ( y ) σ ( y ) (cid:19) = (cid:18) y y uy uy + y (cid:19) . (6.1)If we now let A = ( − b − a ), T = (cid:0) − u − u (cid:1) , and v = bu = − σ ( u ) − a (since u satisfies (4.3)),we have that σ ( T ) AT − = (cid:18) − σ ( u ) 1 − σ ( u ) 1 (cid:19) (cid:18) − b − a (cid:19) (cid:18) − u − u (cid:19) = (cid:18) u − u + v v (cid:19) =: B. Therefore, the systems (4.2) and σ ( Z ) = BZ are equivalent (in the sense of Definition 2.7),and a fundamental solution matrix for the latter system is given by T Y = (cid:18) − u − u (cid:19) (cid:18) y y uy uy + y (cid:19) = (cid:18) y y + y y (cid:19) = Z. For any γ ∈ H , the σ -Galois group for (4.1), we have that γ (cid:18) y y + y y (cid:19) = (cid:18) y y + y y (cid:19) (cid:18) α γ ξ γ λ γ (cid:19) = (cid:18) α γ y ξ γ y + λ γ y + λ γ y λ γ y (cid:19) , (6.2)and therefore the action of H on the solutions is defined by γ ( y ) = α γ y ; γ ( y ) = λ γ y ; and γ ( y ) = λ γ y + ξ γ y . (6.3)It will be convenient to define the auxiliary elements w = y uy and z = y y , (6.4)on which σ acts via σ ( w ) = buσ ( u ) w ; σ ( z ) = z + w, (6.5)and H acts via γ ( w ) = λ γ α γ w ; γ ( z ) = λ γ α γ z + ξ γ α γ . (6.6)We observe that the σ -PV ring S = k [ y , y + y , y , ( y y ) − ] = k [ y , w, z, ( y w ) − ]and the σδ -PV ring R = k { y , y + y , y , ( y y ) − } δ = k { y , w, z, ( y w ) − } δ . Our computation of the σδ -Galois group G for (4.1) in this section will be accomplishedby studying the action of G on y , w , and z . We begin by defining the unipotent radicals R u ( H ) = H ∩ (cid:26)(cid:18) ξ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ξ ∈ C (cid:27) and R u ( G ) = G ∩ (cid:26)(cid:18) ξ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ξ ∈ C (cid:27) , (6.7)and observe that R u ( H ) (resp., R u ( G )) is an algebraic (resp., differential algebraic) subgroupof G a ( C ), the additive group of C . By [Hen97, Thm. 13(2)], R u ( H ) = G a ( C ) if and only ifthere exists exactly one solution u ∈ k to (4.3). We observe that R u ( G ) = { γ ∈ G | γ ( y i ) = y i for i = 0 , } . The reductive quotient
G/R u ( G ) ≃ (cid:26)(cid:18) α γ λ γ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) γ ∈ G (cid:27) is the σδ -Galois group corresponding to the matrix equation σ ( Y ) = (cid:18) u v (cid:19) Y, (6.8) IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 19 which we compute with Proposition 5.2 and Remark 5.3.In the following result, we compute the defining equations for the σδ -Galois group G for(4.1) in a special case. Recall that u ∈ ¯ Q ( x ) denotes the unique solution to the Riccatiequation (4.3), H denotes the σ -Galois group for (4.1), and w is as in (6.4). Proposition 6.1.
Suppose there is exactly one solution u ∈ k to (4.3) and H is commutative.Then H is a subgroup of G m ( C ) × G a ( C ) = (cid:26)(cid:18) α ξ α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α, ξ ∈ C, α = 0 (cid:27) (6.9) with R u ( H ) = G a ( C ) . Moreover, there exists w ∈ ¯ Q ( x ) satisfying (6.5) , and G is thesubgroup of (6.9) defined by the following conditions on α and ξ .(i) There exist m ∈ N and f ∈ ¯ Q ( x ) × such that u m = σ ( f ) f if and only if α m = 1 .(ii) There exist c ∈ Z and f ∈ ¯ Q ( x ) such that δ ( u ) u = σ ( f ) − f + c if and only if δ (cid:16) δ ( α ) α (cid:17) = 0 . Moreover, c = 0 if and only if δ ( α ) = 0 .(iii) There exist: c ∈ ¯ Q ; f ∈ ¯ Q ( x ) ; and a linear δ -polynomial L ∈ ¯ Q { Y } δ such that L ( δ ( u ) u ) − w = σ ( f ) − f + c if and only if δ (cid:0) ξα (cid:1) = L (cid:16) δ (cid:16) δ ( α ) α (cid:17)(cid:17) . Moreover, c = 0 ifand only if ξ = α L ( δ ( α ) α ) .(iv) If none of the conditions above is satisfied, then G = H = G m ( C ) × G a ( C ) .Proof. First recall that when there is exactly one solution u ∈ k to (4.3) the σ -Galois group H of (4.1) is reducible but not diagonalizable by [Hen97, Thm. 13], and therefore H is anon-diagonalizable subgroup of G m ( C ) ⋉ G a ( C ) = (cid:26)(cid:18) α ξ λ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α, ξ, λ ∈ C, αλ = 0 (cid:27) . In particular, R u ( H ) = G a ( C ) and a straightforward computation shows that H is commu-tative if and only if it is actually a subgroup of (6.9). We recall the notation introduced atthe beginning of this section: v = bu , { y , y } is a C -basis of solutions for (4.1) such that σ ( y ) = uy and σ ( y ) − uy = y , where y = 0 satisfies σ ( y ) = vy . The embedding H ֒ → GL ( C ) : γ M γ is as in (6.2), and the action of H on the solutions is given in(6.3). The auxiliary elements w and z are defined as in (6.4); they are acted upon by σ asin (6.5) and by H as in (6.6). The relation γ ( w ) = λ γ α γ w for each γ ∈ H from (6.6), togetherwith Theorem 2.11, imply that w ∈ k . Since σ ( w ) = buσ ( u ) w from (6.5) and b, u ∈ ¯ Q ( x ),if w ∈ k we may actually take w ∈ ¯ Q ( x ) by [Har08, Lem. 2.5] (cf. Remark 4.6). Thus, ifwe can find f ∈ k witnessing the relations in items (i) or (ii), then we may take f ∈ ¯ Q ( x ),as already discussed in the proof of Proposition 5.2, and similarly if we can find f ∈ k and c ∈ C witnessing the relation in item (iii), then we may take f ∈ ¯ Q ( x ) and c ∈ ¯ Q .Items (i) and (ii) were already established in Proposition 5.2. Let us prove item (iii).Setting δ ( y ) y =: g ∈ R we have that σ ( g ) − g = δ ( u ) u and γ ( g ) = g + δ ( α γ ) α γ (6.10) for γ ∈ H (cf. the proof of Proposition 5.2). On the other hand, the actions of σ and γ ∈ H on the element z ∈ R defined in (6.4) in this case is given by σ ( z ) − z = w and γ ( z ) = z + ξ γ α γ . (6.11)Consider the relation stipulated in item (iii): L (cid:18) δ ( u ) u (cid:19) − w = σ ( f ) − f + c, (6.12)where L ∈ ¯ Q { Y } δ is a linear differential polynomial, f ∈ ¯ Q ( x ), and c ∈ ¯ Q . It follows from(6.10) and (6.11) that, for any linear differential polynomial L ∈ C { Y } δ and γ ∈ G , we havethat σ ( L ( g ) − z ) − ( L ( g ) − z ) = L (cid:18) δ ( u ) u (cid:19) − w ; and γ ( L ( g ) − z ) = ( L ( g ) − z ) + L (cid:18) δ ( α γ ) α γ (cid:19) − ξ γ α γ . Suppose there exist f ∈ ¯ Q ( x ) and a linear differential polynomial L ∈ ¯ Q { Y } δ satisfying(6.12) with c = 0. Then L ( g ) − z − f ∈ k σ , which implies that L ( g ) − z ∈ k , and therefore L (cid:16) δ ( α γ ) α γ (cid:17) = ξ γ α γ for every γ ∈ G by Theorem 2.11. On the other hand, if L ∈ ¯ Q { Y } δ is a lineardifferential polynomial such that L (cid:16) δ ( α γ ) α γ (cid:17) = ξ γ α γ for every γ ∈ G , then L ( g ) − z = f ∈ k satisfies (6.12) with c = 0.More generally, suppose there exist f ∈ ¯ Q ( x ), c ∈ ¯ Q , and a linear differential polynomial L ∈ ¯ Q { Y } δ satisfying (6.12). Then we see that L ( δ ( g )) − δ ( z ) − δ ( f ) ∈ k σ , and therefore L ( δ ( g )) − δ ( z ) ∈ k , which implies that L (cid:18) δ (cid:18) δ ( α γ ) α γ (cid:19)(cid:19) = δ (cid:18) ξ γ α γ (cid:19) (6.13)for every γ ∈ G . On the other hand, if L ∈ ¯ Q { Y } δ is a linear differential polynomial suchthat (6.13) holds for every γ ∈ G , then L ( δ ( g )) − δ ( g ) =: h ∈ k , and therefore the element L ( g ) − z ∈ R is differentially dependent over k . It then follows from [HS08, Prop. 3.10(2.a)]that there exist f ∈ k and c ∈ C δ = ¯ Q satisfying (6.12). This concludes the proof ofitem (iii).By [HS08, Cor. 3.2], g and z are differentially dependent over k if and only if there existlinear differential polynomials L , L ∈ ¯ Q { Y } δ , not both zero, and ˜ f ∈ ¯ Q ( x ), such that L (cid:18) δ ( u ) u (cid:19) − L ( w ) = σ ( ˜ f ) − ˜ f . (6.14)Hence if there do not exist such L i and ˜ f , the elements g, z ∈ R are differentially independentover k , which implies that G = G m ( C ) ⋉ G a ( C ) by [HS08, Prop. 6.26]. Thus, assume theredo exist L , L ∈ ¯ Q { Y } δ , not both zero, and ˜ f satisfying (6.14). If L = 0, then L = 0and it follows from (6.14) and (6.10) that g is differentially dependent over k , and thereforeso is y . By Corollary 3.5, this implies that there exist f ∈ ¯ Q ( x ) and c ∈ Z such that IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 21 δ ( u ) u = σ ( f ) − f + c , as in item (ii). To prove item (iv), let us show that if there exist lineardifferential polynomials L , L ∈ ¯ Q { Y } δ with L = 0 and ˜ f ∈ ¯ Q ( x ) satisfying (6.14), thenwe can construct a linear δ -polynomial L ∈ ¯ Q { Y } δ and c ∈ ¯ Q such that L (cid:18) δ ( u ) u (cid:19) − w = σ ( f ) − f + c for some f ∈ ¯ Q ( x ) , as in item (iii). Let ord( L i ) = m i and L i = P m i j =0 c i,j δ j ( Y ) for i = 1 ,
2; if L = 0, we set m = 0, and we adopt the convention that c i,j := 0 for every j > m i . By Proposition 3.3,the existence of ˜ f ∈ k as in (6.14) implies that0 = q -dres (cid:18) L (cid:18) δ ( u ) u (cid:19) − L ( w ) , [ β ] q , j (cid:19) (6.15)for every q Z -orbit [ β ] q with β ∈ ¯ Q × and every j ∈ N . Let r ∈ N be the largest order suchthat q -dres( w, [ β ] q , r ) = 0 for some q Z -orbit [ β ] q . Then it follows from (6.15) and Lemma 3.4 that, for each q Z -orbit [ β ] q with β ∈ ¯ Q × , the q -discrete residues c ,m + r − ( − m + r − ( m + r − β m + r − q -dres (cid:0) δ ( u ) u , [ β ] q , (cid:1) = q -dres (cid:16) L (cid:0) δ ( u ) u (cid:1) , [ β ] q , m + r (cid:17) and c ,m ( − m ( m + r )!( r − β r − q -dres( w, [ β ] q , r ) = q -dres( L ( w ) , [ β ] q , m + r )are equal. Since β = 0, the above equality is equivalent to c ,m + r − c ,m ( − r − ( r − β r − q -dres (cid:0) δ ( u ) u , [ β ] q , (cid:1) = q -dres( w, [ β ] q , r ) (6.16)Set c r − = c ,m r − c ,m . Then (6.16) is equivalent to q -dres (cid:18) c r − δ r − (cid:18) δ ( u ) u (cid:19) − w, [ β ] q , r (cid:19) = 0for every q Z -orbit [ β ] q with β ∈ ¯ Q × simultaneously.We continue by taking the next highest r ′ ≤ r − q -dres( w, [ β ] q , r ′ ) = 0 for some[ β ] q , and proceed as above to find the coefficient c r ′ − ∈ ¯ Q of L such that q -dres (cid:18) c r − δ r − (cid:18) δ ( u ) u (cid:19) + c r ′ − δ r ′ − (cid:18) δ ( u ) u (cid:19) − w, [ β ] q , r ′ (cid:19) = 0 . Eventually we will have constructed a linear δ -polynomial L ∈ ¯ Q { Y } δ such that q -dres (cid:18) L (cid:18) δ ( u ) u (cid:19) − w, [ β ] q , j (cid:19) = 0for every q Z -orbit [ β ] q with β ∈ ¯ Q × and every j ∈ N . Set c = q -dres (cid:16) L (cid:16) δ ( u ) u (cid:17) − w, ∞ (cid:17) ∈ ¯ Q .Then it follows from Proposition 3.3 that L ( δ ( u ) u ) − w − c = σ ( f ) − f for some f ∈ k . By [Har08, Lem. 2.4] (cf. Remark 4.6), we may take f ∈ ¯ Q ( x ), so we are indeed in case (iii),as we wanted to show. (cid:3) The main ideas for the proof of the following result were communicated to the first authorin [Sin15], during the development of the algorithm in [Arr17].
Proposition 6.2.
Suppose there exists exactly one solution u ∈ ¯ Q ( x ) to (4.3) and H is notcommutative. Then R u ( G ) = R u ( H ) = G a ( C ) .Proof. We recall the notation introduced at the beginning of this section: u ∈ ¯ Q ( x ) is theunique solution in k to the Riccati equation (4.3), v = bu , { y , y } is a C -basis of solutionsfor (4.1) such that σ ( y ) = uy and σ ( y ) − uy = y , where y = 0 satisfies σ ( y ) = vy .The embedding G ֒ → GL ( C ) : γ M γ is as in (6.2), and the action of G on the solutionsis given in (6.3). The auxiliary elements w and z are defined as in (6.4); they are acted uponby σ as in (6.5) and by G as in (6.6).By Proposition 2.9, either R u ( G ) = G a ( C ), or else R u ( G ) = (cid:26)(cid:18) ξ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) β ∈ C, L ( ξ ) = 0 (cid:27) . (6.17)for some nonzero, monic linear δ -polynomial L ∈ C { Y } δ . Since R u ( G ) is normal in G , thisimplies that (cf. [HS08, Lem. 3.6]) M γ (cid:18) β (cid:19) M − γ = (cid:18) α γ ξ γ λ γ (cid:19) (cid:18) ξ (cid:19) (cid:18) α − γ − α − γ λ − γ ξ γ λ − γ (cid:19) = (cid:18) α γ λ − γ ξ (cid:19) ∈ R u ( G )for each γ ∈ G and (cid:0) ξ (cid:1) ∈ R u ( G ). If L is as in (6.17), then L ( ξ ) = 0 ⇒ L ( α γ λ − γ ξ ) = 0.By [HS08, Lem. 3.7], this implies that if ord( L ) = 0, then δ ( α γ λ − γ ) = 0 for every γ ∈ G .But since L 6 = 0, ord( L ) = 0 if and only if R u ( G ) = { } , which is impossible, for then wewould have that G ≃ G/R u ( G ) ≃ (cid:26)(cid:18) α γ λ γ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) γ ∈ G (cid:27) is commutative, and since G is Zariski-dense in H by Proposition 2.12, this would force H to be commutative also, contradicting our hypotheses.We proceed by contradiction: assuming R u ( G ) = G a ( C ), we will show that R u ( H ) = { } ,contradicting our hypotheses. We have shown above that if R u ( G ) = { } then there existsa monic linear δ -polynomial L ∈ C { Y } δ with ord( L ) ≥ R u ( G ) is as in (6.17)and δ (cid:16) α γ λ γ (cid:17) = 0 for every γ ∈ G . It follows from (6.6) and Theorem 2.11 that the group { λ γ α − γ | γ ∈ G } ⊆ G m ( ¯ Q ) is the σδ -Galois group for the system σ ( W ) = (cid:18) buσ ( u ) (cid:19) W, which by Proposition 2.13 must be integrable over k in the sense of [AS17, Def. 3.3]. It isshown in [AS17, Prop. 3.6] that this system must then be integrable over ¯ Q ( x ), and therefore IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 23 by [SS19, Thm. 2], there exist t ∈ ¯ Q ( x ) × and c, d ∈ ¯ Q with c = 0 such that ˜ w := t − w satisfies σ ( ˜ w ) = c ˜ w ; δ ( ˜ w ) = d ˜ w. It is convenient to point out now that c = q r for any r ∈ Z , because otherwise we wouldhave ˜ w = ex r for some e ∈ C , which would imply that w ∈ k , contradicting our hypothesisthat H is not commutative (cf. the proof of Proposition 6.1: H is commutative if and onlyif α γ = λ γ for every γ ∈ H if and only if w ∈ k by (6.6) and Theorem 2.11). We will needto use the fact that c / ∈ q Z at the end of the proof.We claim that ˜ w − L ( w ) =: f L ∈ k, and moreover f L = ˜ L ( t ) (6.18)for some linear differential polynomial 0 = ˜ L ∈ C { Y } δ . In fact, this is true for any non-zerolinear differential polynomial in C { Y } δ , not just for the specific L ∈ C { Y } δ in (6.17). Itsuffices to show that ˜ w − δ n ( w ) belongs to the C -linear span D of { δ j ( t ) | j ∈ Z ≥ } for every n ∈ N . We prove this by induction: the case n = 0 is clear, since ˜ w − w = t ∈ D . Assumingthat ˜ w − δ n ( w ) = f n ∈ D , we see that˜ w − δ n +1 ( w ) = ˜ w − δ ( δ n ( w )) = ˜ w − δ ( f n ˜ w ) = δ ( f n ) + df n ∈ D as well. Moreover, this computation also shows that L , ˜ L ∈ C { Y } δ have the same order andthe same leading coefficient.By (6.5) and (6.6), the element L ( z ) ∈ R satisfies σ ( L ( z )) − L ( z ) = L ( w ) , and γ ( L ( z )) = λ γ α γ L ( z ) + L (cid:18) ξ γ α γ (cid:19) for every γ ∈ G , since δ ( λ γ α − γ ) = 0 for γ ∈ G . Hence γ ( L ( z )) = L ( z ) for every γ ∈ R u ( G ),and therefore by Theorem 2.11 we have that L ( z ) ∈ k h y , y i δ =: F , the total ring offractions of the σδ -PV ring k { y , y , ( y y ) − } δ for (6.8); we emphasize that the latter ringis not necessarily a domain, so F is not necessarily a field.For γ ∈ G/R u ( G ) ≃ Gal σδ ( F/k ) =: ¯ G given by (cid:16) α γ λ γ (cid:17) ∈ G m ( C ) , let τ γ := γ ( ˜ w − L ( z )) − ˜ w − L ( z ) , (6.19)where we note that since L ( z ) ∈ F is fixed by R u ( G ), the action of the reductive quotient ¯ G on L ( z ) is well-defined. We claim that { τ γ | γ ∈ ¯ G } is a 1-cocycle of ¯ G with values in the ¯ G -module M := C · ˜ w − (see [Lan02, VI.10]). Since M is the solution space for σ ( W ) = c − W in F , it is clear that M is stabilized by ¯ G . Moreover, it follows from (6.18) that σ ( τ γ ) = (( γ − ◦ σ )( ˜ w − L ( z )) = ( γ − (cid:0) c − ˜ w − L ( z ) + c − ˜ w − L ( w ) (cid:1) = c − τ γ , since c − ˜ w − L ( w ) = c − f L ∈ k and therefore γ ( c − f L ) = c − f L for every γ ∈ ¯ G . Hence τ γ ∈ M for each γ ∈ ¯ G . To verify the cocycle condition, note that for γ, θ ∈ ¯ G we have that τ γθ = γθ ( ˜ w − L ( z )) − ˜ w − L ( z ) = γ (cid:0) θ ( ˜ w − L ( z )) − ˜ w − L ( z ) (cid:1) + (cid:0) γ ( ˜ w − L ( z )) − ˜ w − L ( z ) (cid:1) = γ ( τ θ ) + τ γ . Since G is not commutative (for otherwise H would be commutative, as discussed aboveand contrary to our hypotheses), there exists γ ∈ ¯ G such that α γ = λ γ , and therefore m γ ( m ) − m is a ¯ G -automorphism of M for such a γ ∈ ¯ G , since ¯ G is commutative.By Sah’s Lemma [Lan02, Lem. VI.10.2], the cohomology group H ( ¯ G, M ) = { } , and inparticular { τ γ | γ ∈ ¯ G } is a 1-coboundary, i.e., there exists e ˜ w − ∈ C · ˜ w − = M such that τ γ = γ ( e ˜ w − ) − e ˜ w − . It follows from the definition of τ γ in (6.19) that γ ( ˜ w − L ( z ) − e ˜ w − ) = ˜ w − L ( z ) − e ˜ w − for every γ ∈ ¯ G , which implies that ˜ w − L ( z ) − e ˜ w − =: g ∈ k by Theorem 2.11. Hence f L ˜ w = L ( w ) = σ ( L ( z )) − L ( z ) = σ ( g ˜ w ) − g ˜ w = ( cσ ( g ) − g ) ˜ w, and therefore, since c ∈ ¯ Q × , ˜ L ( c − t ) = c − f L = σ ( g ) − c − g, where 0 = ˜ L ∈ C { Y } δ is the linear differential polynomial defined implicitly in (6.18). Since c / ∈ q Z , it follows from [HS08, Prop. 6.4(2)] that there exists h ∈ k such that c − t = σ ( h ) − c − h. But then h ˜ w satisfies σ ( h ˜ w ) − h ˜ w = ( cσ ( h ) − h ) ˜ w = t ˜ w = w, and therefore σ ( z − h ˜ w ) − ( z − h ˜ w ) = 0 by (6.5), which implies that z − h ˜ w ∈ C andtherefore z ∈ k [ w ] is fixed by R u ( H ). But γ ( z ) = z + ξ γ for every γ ∈ R u ( H ), and therefore R u ( H ) = { } , which contradicts our hypotheses and concludes the proof. (cid:3) Remark . To compute the difference-differential Galois group G for (4.1) when there existsexactly one solution u ∈ ¯ Q ( x ) to (4.3), we apply Propositions 5.2, 6.1, and 6.2 as follows.First, compute the defining equations for the reductive quotient¯ G := G/R u ( G ) = (cid:26)(cid:18) α γ λ γ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) γ ∈ G (cid:27) , which is the σδ -Galois group for the system (6.8), as in Proposition 5.2 and Remark 5.3,with u = u and u = v . In particular, this requires computing the q -discrete residues q -dres (cid:16) δ ( u ) u , [ β ] q , (cid:17) for each q Z -orbit [ β ] q with β ∈ ¯ Q × . Note that this will produce all thedefining equations for G relating α and λ only, and it remains to compute the remainingdefining equations for G , if there are any.If uv − = σ ( w ) w for any w ∈ ¯ Q ( x ) as in Proposition 5.2(i), then R u ( G ) = G a ( C ) by Propo-sition 6.2, and therefore there are no more defining equations for G . Otherwise, computesuch a w ∈ ¯ Q ( x ), as well as its q -discrete residues q -dres( w, [ β ] q , j ) for every q Z -orbit [ β ] q and j ∈ N (only finitely many of these are non-zero). In this case, G ⊆ ¯ G × G a ( C ) = (cid:26)(cid:18) α ξ α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) α α (cid:19) ∈ ¯ G, ξ ∈ C (cid:27) , (6.20)and this containment is proper if and only if there exist f ∈ ¯ Q ( x ), a linear differentialpolynomial L ∈ ¯ Q { Y } δ , and c ∈ ¯ Q as in Proposition 6.1(iii). IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 25 Let us first compute the defining equations of G in (6.20) when q -dres( w, [ β ] q , j ) = 0 forevery q Z -orbit [ β ] q and j ∈ N , in which case q -dres( w, ∞ ) =: c = 0 and − w = σ ( f ) − f − c for some f ∈ ¯ Q ( x ) by Proposition 3.3, as in Proposition 6.1(iii). In this case, G is containedin the subgroup of (6.20) defined by δ (cid:0) ξα (cid:1) = 0, and R u ( G ) ⊆ G a ( C δ ). If δ ( u ) u = σ ( ˜ f ) − ˜ f forsome ˜ f ∈ ¯ Q ( x ) as in Proposition 5.2(ii), so that δ ( α γ ) = 0 for every γ ∈ G , then G is thesubgroup of (6.20) defined by δ ( ξ ) = 0, and R u ( G ) = G a ( C δ ). If there exist ˜ f ∈ ¯ Q ( x ) and0 = ˜ c ∈ Z as in Proposition 5.2(ii), so that δ (cid:16) δ ( α γ ) α γ (cid:17) = 0 for every γ ∈ G but there exists γ ∈ G such that δ ( α γ ) = 0, then G is the subgroup of (6.20) defined by ˜ cξ = cδ ( α ), and R u ( G ) = { } . If there are no ˜ f ∈ ¯ Q ( x ) and ˜ c ∈ Z such that δ ( u ) u = σ ( ˜ f ) − ˜ f + ˜ c , then G isprecisely the subgroup of (6.20) defined by δ (cid:0) ξα (cid:1) = 0, and R u ( G ) = G a ( C δ ).Assuming now that some q -discrete residue q -dres( w, [ β ] q , j ) = 0, let r ∈ N be as large aspossible such that q -dres( w, [ β ] q , r ) = 0 for some q Z -orbit [ β ] q . Write the linear differentialpolynomial L = r − X i =0 c i δ i ( Y ) ∈ ¯ Q { Y } δ with undetermined coefficients, and decide whether the system of linear equations over ¯ Q defined by setting q -dres (cid:18) L (cid:18) δ ( u ) u (cid:19) − w, [ β ] q , j (cid:19) = 0 (6.21)for every q Z -orbit [ β ] q and 1 ≤ j ≤ r admits a solution. If there is no solution, then againwe have that R u ( G ) = G a ( C ) and G is precisely the group in (6.20). If there is a solution,then it it is unique and c r − = 0. In this case, setting c := c · q -dres (cid:18) δ ( u ) u , ∞ (cid:19) − q -dres ( w, ∞ ) , (6.22)there exists f ∈ ¯ Q ( x ) as in Proposition 6.1(iii) by Proposition 3.3, and G is the subgroupof (6.20) defined by the corresponding relation stipulated in Proposition 6.1, depending onwhether the c ∈ ¯ Q defined in (6.22) is zero or not. If c = 0 then R u ( G ) = 0, and if c = 0then R u ( G ) = G a ( C δ ).Since R u ( G ) = { } whenever there is not exactly one solution u ∈ ¯ Q ( x ) to (4.3) (i.e.,either there is no solution or there is more than one solution to (4.3) in ¯ Q ( x )), we deducethe following result from Remark 6.3, which generalizes [HS08, Prop. 4.3(2)]. Corollary 6.4. If G is the σδ -Galois group of (4.1) , then the unipotent radical R u ( G ) iseither { } , G a ( C δ ) , or G a ( C ) . Irreducible and imprimitive groups
In this section we will denote k = C ( x ), where C is a δ -closure of ¯ Q , σ denotes the C -linear automorphism of k defined by σ ( x ) = qx , and δ ( x ) = 1. It will be convenient touse similar notation as that of Section 4: fix once and for all q ∈ ¯ Q such that q = q , andlet k := C ( x ) be the σδ -field extension of k defined by setting x = x , σ ( x ) = q x , and δ ( x ) = x . Let us now suppose that there are no solutions in ¯ Q ( x ) to the first Riccati equation (4.3).According to Proposition 4.5, under these conditions the σ -Galois group H for (4.1) over k should be irreducible, and H should be imprimitive if and only if one of the followingpossibilities holds:(1) there exist two solutions u , u ∈ ¯ Q ( x ) \ ¯ Q ( x ) to the first Riccati equation (4.3) suchthat u = ¯ u is the Galois conjugate of u over ¯ Q ( x ); or(2) either a = 0 or else there exists a solution e ∈ ¯ Q ( x ) to the second Riccati equation(4.5); or(3) a = 0 and there exist two solutions e , e ∈ ¯ Q ( x ) \ ¯ Q ( x ) to the second Riccati equation(4.5) such that e = ¯ e is the Galois conjugate of e over ¯ Q ( x ).Note that (2) and (3) above are mutually exclusive and together exhaust the possibility thatthe more compact Proposition 4.5(5) holds. We will address each of the possibilites (1),(2), and (3) above in turn, in Sections 7.1, 7.2, and 7.3, respectively, and establish in eachcase that H is indeed irreducible and imprimitive in each of these scenarios, as stated inProposition 4.5.By [vdPS97, Prop. 12.2(1)], in any case the group of connected components H/H ◦ mustbe bicyclic. The irreducible and imprimitive algebraic subgroups of GL ( ¯ Q ) with bicyclicgroup of connected components are listed in the following result, which we prove usingthe classification of the algebraic subgroups of GL ( C ) developed in [NvdPT08]. In theclassification below we denote {± } × {± } by {± } and G m ( C ) × G m ( C ) by G m ( C ) . Lemma 7.1. If H is an irreducible and imprimitive algebraic subgroup of GL ( C ) such that H/H ◦ is bicyclic, then H is the subgroup of {± } ⋉ G m ( C ) = (cid:26)(cid:18) α α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α , α ∈ C × (cid:27) ∪ (cid:26)(cid:18) λ λ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) λ , λ ∈ C × (cid:27) (7.1) defined by precisely one of the following sets of conditions on α , α , λ , and λ .(1) H = D − m for some m ∈ N , defined as the subgroup of (7.1) such that ( α α ) m = 1 and ( λ λ ) m = − ; or(2) H = D + m for some m ∈ N , defined as the subgroup of (7.1) such that ( α α ) m = 1 and ( λ λ ) m = 1 ; or(3) H = {± } ⋉G m ( C ) , defined as the subgroup of (7.1) such that α = α and λ = λ ;or(4) H = {± } ⋉ G m ( C ) as in (7.1) , with no other conditions on α , α , λ , and λ .Proof. The algebraic subgroups H ⊆ GL ( ¯ Q ) are classified in [NvdPT08] according to theirprojective image ¯ H ⊆ PGL ( ¯ Q ). Since H is irreducible and imprimitive with bicyclic groupof connected components, it is an infinite non-commutative subgroup of (7.1), and thereforeits projective image is either ¯ H = D n , the dihedral group of order 2 n for some n ≥
2, or else¯ H = ¯ D ∞ , the projective image of D ∞ = (cid:26)(cid:18) α α − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α ∈ ¯ Q × (cid:27) ∪ (cid:26)(cid:18) − λλ − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) λ ∈ ¯ Q × (cid:27) . If ¯ H = D n then D n must be commutative, since the algebraic quotient map H → ¯ H factorsthrough H/H ◦ , which we are assuming is abelian, and therefore n = 2 (corresponding to IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 27 D n ≃ K , the Klein four-group) in this case. By [NvdPT08, Thm. 4], the minimal subgroups(see [NvdPT08, §
2] for the definition) of GL ( ¯ Q ) having projective image D are D ,ℓ forsome ℓ ∈ Z ≥ , where D ,ℓ := (cid:28) ζ ℓ +1 (cid:18) i − i (cid:19) , (cid:18) ii (cid:19)(cid:29) and ζ ℓ +1 denotes a primitve (2 ℓ +1 )-th root of unity. Therefore the only infinite subgroups H ⊆ GL ( ¯ Q ) having projective image D are given by ¯ Q × · D ,ℓ , which are all equal to π − ( D ), where π : GL ( ¯ Q ) → PGL ( ¯ Q ) is the projection map. Finally, note that ¯ Q × · D ,ℓ for any ℓ ∈ Z ≥ is precisely the subgroup of (7.1) defined by the conditions in item (3): α = α and λ = λ .If ¯ H = ¯ D ∞ , then either H = {± } ⋉ G m ( ¯ Q ) in (7.1) as in item (4), or else H = µ n · D ∞ ,ℓ for some ℓ ∈ Z ≥ and some n ∈ N , where µ n denotes the group of n -th roots of unity, and D ∞ ,ℓ := (cid:28)(cid:26)(cid:18) α α − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α ∈ ¯ Q × (cid:27) , (cid:18) ζ ℓ +1 ζ ℓ +1 (cid:19)(cid:29) , where again ζ ℓ +1 is a primitive (2 ℓ +1 )-th root of unity, since by [NvdPT08, Thm. 4] the D ∞ ,ℓ are all the minimal subgroups of GL ( ¯ Q ) with projective image ¯ D ∞ . All of these groups havethe property that H/H ◦ is bicyclic. It remains to show that for any n ∈ N and ℓ ∈ Z ≥ thegroup µ n · D ∞ ,ℓ is one of the groups described by the conditions in either item (1) or item (2).Let us write ∆ n,ℓ := (cid:8)(cid:0) α α (cid:1) (cid:12)(cid:12) α , α ∈ ¯ Q × (cid:9) ∩ ( µ n · D ∞ ,ℓ ), the group of all diagonal matricescontained in µ n · D ∞ ,ℓ , and ∇ n,ℓ := (cid:8)(cid:0) λ λ (cid:1) (cid:12)(cid:12) λ , λ ∈ ¯ Q × (cid:9) ∩ ( µ n · D ∞ ,ℓ ) for the complementarycoset of ∆ n,ℓ in µ n · D ∞ ,ℓ consisting of all the antidiagonal matrices contained in µ n · D ∞ ,ℓ .Then we see that ∇ n,ℓ = ζ ℓ +1 · ∆ n,ℓ · ( ), and ∆ n,ℓ = h ζ n , ζ ℓ i · (cid:8)(cid:0) α α − (cid:1) (cid:12)(cid:12) α ∈ ¯ Q × (cid:9) .Therefore,∆ n,ℓ = (cid:26)(cid:18) α α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α , α ∈ ¯ Q × , ( α α ) m = 1 (cid:27) , where m := lcm( n, ℓ ) if ℓ ≥ n if ℓ = 0 and 2 | n ; n if ℓ = 0 and 2 ∤ n ;because ( h ζ n , ζ ℓ i ) = h ζ m i with m defined as above. Since ∇ n,ℓ = (cid:26)(cid:18) α ζ ℓ +1 α ζ ℓ +1 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α , α ∈ ¯ Q × , ( α α ) m = 1 (cid:27) , we have that µ n · D ∞ ,ℓ is the group described in item (2) if and only if 2 ℓ | m (which occursprecisely when either ℓ = 0 or else ℓ ≥ ℓ +1 | n ), and µ n · D ∞ ,ℓ is the group described initem (1) otherwise, since for ℓ ≥ ℓ − | m , and therefore ( ζ ℓ +1 α α ) m =( ζ ℓ α α ) m = −
1, precisely when 2 ℓ ∤ m , 2 ℓ − | m (with ℓ ≥ α α ) m = 1. (cid:3) Remark . Given an irreducible and imprimitive algebraic subgroup H ⊆ GL ( ¯ Q ) such thatthe group of connected components H/H ◦ is bicyclic, we can uniquely identify it among thepossibilities listed in Lemma 7.1 by the knowledge of two auxiliary groups:∆( H ) := (cid:26)(cid:18) α α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α , α ∈ ¯ Q × (cid:27) ∩ H and det( H ) = { det( h ) | h ∈ H } ⊆ G m ( ¯ Q ) , respectively the subgroup of diagonal matrices in H and the image of H under the de-terminant map. Indeed, ∆( H ) = G m ( ¯ Q ) if and only if H = {± } ⋉ Q m ( ¯ Q ) as inLemma 7.1(4); ∆( H ) = (cid:8)(cid:0) α α (cid:1) (cid:12)(cid:12) α , α ∈ ¯ Q × , α = α (cid:9) if and only if H = {± } ⋉G m ( ¯ Q )is as in Lemma 7.1(3); and ∆( H ) = (cid:8)(cid:0) α α (cid:1) (cid:12)(cid:12) α , α ∈ ¯ Q × , ( α α ) m (cid:9) for some m ∈ N if and only if H is one of the groups D − m or D + m described respectively in items (1) or(2) of Lemma 7.1. To decide between these cases, note that det( H ) = h α α , − λ λ i hasdet( H ) m = h ( − m ( λ λ ) m i ; hence, if m is even, then H = D − m if and only if det( H ) = µ m and H = D + m if and only if det( H ) = µ m ; and if m is odd, then H = D − m if and only ifdet( H ) = µ m and H = D + m if and only if det( H ) = µ m .7.1. Irreducible and imprimitive (1): diagonalizable over the quadratic extension.
Supposing there are no solutions to (4.3) in ¯ Q ( x ), but there are two solutions u, ¯ u ∈ ¯ Q ( x )to (4.3), Galois-conjugate over ¯ Q ( x ), the system (4.2) σ ( Y ) = (cid:18) − b − a (cid:19) Y with fundamental solution matrix Y = (cid:18) y y σ ( y ) σ ( y ) (cid:19) ; and σ ( Z ) = (cid:18) u
00 ¯ u (cid:19) Z with fundamental solution matrix Z = (cid:18) z z (cid:19) (7.2)are equivalent over ¯ Q ( x ) via the gauge transformation Z = T Y , where (cf. Remark 5.1) T := (cid:18) ¯ uu − ¯ u − u − ¯ uuu − ¯ u − u − ¯ u (cid:19) ∈ GL ( ¯ Q ( x )) . (7.3)Let us write S = k [ Y, det( Y ) − ] = k [ z , z , ( z z ) − ] for the σ -PV ring for (4.2) (or equiva-lently for (7.2)) over k . Then S = k [ Y, det( Y ) − ] ⊂ S is a σ -PV ring for (4.2) over k . Letus also write H i = Gal σ ( S i /k i ) for i = 1 ,
2, and ˜ H = Gal σ ( S /k ). Since (7.2) is a diagonalsystem, the group H is diagonalizable. By Proposition 4.2,˜ H ≃ H × µ m µ , where m ∈ { , } is determined by the intersection S ∩ k = k m inside S , and H is anindex- m subgroup of H . We claim that any ˜ τ ∈ ˜ H such that ˜ τ ( x ) = − x has the propertythat τ := ˜ τ | S ∈ H is given by an anti-diagonal matrix. From this it will follow that H has index exactly 2 in H , and H = H ∪ H · τ is irreducible and imprimitive, as claimedin Proposition 4.5(4).To see this, let M τ ∈ GL ( C ) such that τ ( Y ) = Y M τ . Then for the gauge transformation T given in (7.3) we see that˜ τ ( Z ) = ˜ τ ( T Y ) = ¯
T Y M τ = ( ) ZM τ . On the other hand, we see that σ (˜ τ ( z )) = ˜ τ ( σ ( z )) = ˜ τ ( uz ) = ¯ u ˜ τ ( z ), and therefore˜ τ ( z ) = λ z for some λ ∈ C × . A similar computation shows that ˜ τ ( z ) = λ z for some λ ∈ C × . From this it follows that˜ τ ( Z ) = (cid:18) ˜ τ ( z ) 00 ˜ τ ( z ) (cid:19) = (cid:18) λ z λ z (cid:19) = (cid:18) (cid:19) Z (cid:18) λ λ (cid:19) . Hence M τ = (cid:0) λ λ (cid:1) , as we wanted to show. IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 29 Remark . Having established that the σ -Galois group H for (4.2) over k is indeedirreducible and imprimitive as claimed in Proposition 4.5(4), we can compute this H fromamong the possibilities listed in Lemma 7.1 as explained in Remark 7.2, by determining thesubgroup ∆( H ) of diagonal matrices in H , and the group det( H ) ⊆ G m ( C ).Since det( H ) is the σ -Galois group for the system σ ( y ) = by , we see that det( H ) = µ m if and only if m ∈ N is the smallest positive integer such that b m = σ ( f ) f for some f ∈ ¯ Q ( x ),and if there is no such m then det( H ) = G m ( C ).Since ∆( H ) = H is the σ -Galois group for (7.2) over k , we can compute the definingequations for ∆( H ) ⊆ (cid:26)(cid:18) α α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α , α ∈ C × (cid:27) as follows:(1) ( α α ) m = 1 if and only if ( u ¯ u ) m = σ ( f ) f for some f ∈ ¯ Q ( x ) × (and in this case H is D − m or D + m );(2) α = α if and only if (cid:0) u ¯ u (cid:1) = σ ( f ) f for some f ∈ ¯ Q ( x ) (and in this case H = {± } ⋉ G m ( C )) ;(3) if none of these possibilities holds, then ∆( H ) = G m ( C ) (and in this case H = {± } ⋉ G m ( C ) ).The computation of the σδ -Galois group G for (4.2) over k , assuming that the corre-sponding σ -Galois group H has already been computed as in Remark 7.3, will by achievedanalogously in the following result, by studying the σδ -Galois group G for (4.2) over k . Proposition 7.4.
Suppose there are no solutions to (4.3) in ¯ Q ( x ) , and let u, ¯ u ∈ ¯ Q ( x ) satisfy (4.3) . Then G is the subgroup of {± } ⋉ G m ( C ) = (cid:26)(cid:18) α α (cid:19) , (cid:18) λ λ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α α = 0 , λ λ = 0 (cid:27) (7.4) defined by the following conditions on α , α , λ , and λ .(1) If H = D − m as in Lemma 7.1(1) or H = D + m as in Lemma 7.1(2), then G = H .(2) If H = {± } ⋉ G m ( C ) as in Lemma 7.1(3), then:(a) there exist = c ∈ Z and g ∈ ¯ Q ( x ) such that δ ( b ) b = σ ( g ) − g + c if and only if G is the subgroup of H defined by δ (cid:16) δ ( α ) α + δ ( α ) α (cid:17) = 0 = δ (cid:16) δ ( λ ) λ + δ ( λ ) λ (cid:17) ;(b) otherwise, G = H .(3) If H = {± } ⋉ G m ( C ) as in Lemma 7.1(4), then:(a) there exist c ∈ Z and g ∈ ¯ Q ( x ) such that δ ( b ) b = σ ( g ) − g + c if and onlyif δ (cid:16) δ ( α ) α + δ ( α ) α (cid:17) = 0 = δ (cid:16) δ ( λ ) λ + δ ( λ ) λ (cid:17) ; moreover, c = 0 if and only if δ ( α α ) = 0 = δ ( λ λ ) ;(b) otherwise, G = H .Proof. Since they systems (4.2) and (7.2) are equivalent over k , and the latter system isdiagonal, we can compute G with Proposition 5.2 and Remark 5.3, but with a small caveat.Namely, after replacing δ with δ := 2 δ , we see that k as a σδ -field behaves just as k : σ ( x ) = q x and δ ( x ) = x . Thus we may compute the δ -algebraic group G ⊆ G m ( C ) over k using the procedure described in Remark 5.3 exactly as stated there, and then simplyreplace every instance of δ in the defining equations for G with δ a posteriori. But sincethe system (7.2) has such a special form, not every possibility listed in Proposition 5.2 mayoccur.We saw in Remark 5.3 that G is a proper subgroup of G m ( C ) if and only if there exist: m , m ∈ Z , not both zero and with gcd( m , m ) = 1; c ∈ Z ; and g ∈ ¯ Q ( x ), such that m δ ( u ) u + m δ (¯ u )¯ u = σ ( g ) − g + c ⇐⇒ m δ ( u ) u + m δ (¯ u )¯ u = σ (¯ g ) − ¯ g + c. (7.5)Let us consider the submodule M ⊆ Z generated by relatively prime pairs ( m , m ) suchthat there exist g ∈ ¯ Q ( x ) and c ∈ Z satisfying the above conditions. Then, as we saw inRemark 5.3, either M = { } is trivial; or M = Z · ( m , m ) is infinite cyclic; or M = Z .Moreover, M = { } is trivial if and only if the σδ -Galois group G for (7.2) is all of G m ( C ) .In this case we must have G = H = {± } ⋉ G m ( C ) , because G is Zariski-dense in H by Proposition 2.12, and therefore G contains at least one anti-diagonal matrix, whence itcontains all anti-diagonal matrices.From now on we assume that M is not trivial. It follows from (7.5) that at least one of(1 ,
1) or (1 , −
1) belongs to M . In any case it is useful to observe that q -dres (cid:18) δ ( u ) u , ∞ (cid:19) = d = q -dres (cid:18) δ (¯ u )¯ u , ∞ (cid:19) , where d ∈ Z is the common degree of u and ¯ u considered as rational functions in x .Therefore, (1 , − ∈ M if and only if δ ( α α ) = 0 for every (cid:0) α α (cid:1) ∈ G .We claim that actually (1 , − ∈ M if and only if H = {± } ⋉G m ( C ) as in Lemma 7.1(3).As explained in Remark 7.3, H = {± } ⋉ G m ( C ) if and only if there exists f ∈ ¯ Q ( x ) × such that (cid:0) u ¯ u (cid:1) = σ ( f ) f , which in turn implies that δ ( u ) u − δ (¯ u )¯ u = σ (cid:18) δ ( f ) f (cid:19) − δ ( f ) f . Thus if H = {± } ⋉ G m ( C ) then (1 , − ∈ M . To establish the opposite implication, let usstudy the reduced form of u : there exists v ∈ ¯ Q ( x ) such that u σ ( v ) v = ex n p p , where e ∈ ¯ Q × is such that if e ∈ q Z then e = 1, n ∈ Z is arbitrary, and p , p ∈ ¯ Q [ x ] are monic such thatgcd( x , p ) = gcd( x , p ) = gcd( p , σ m ( p )) = 1 for every m ∈ Z . We say that ex n p p is the reduced form of u . We then see that the reduced form of ¯ u is ( − n ex n p ¯ p . Although it neednot be the case that the reduced form of u ¯ u is exactly( − n p ¯ p p ¯ p , (because it is possible for gcd( p , σ m (¯ p )) = 1 for some m ∈ Z ), we see that in any case thereduced form of u ¯ u is similarly given by ( − n ˜ p ˜ p IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 31 for some ˜ p , ˜ p ∈ ¯ Q [ x ] monic and such that gcd( x , ˜ p ) = gcd( x , ˜ p ) = gcd(˜ p , σ m (˜ p )) = 1for every m ∈ Z . But then we see that if, say, ˜ p = 1, then there exists β ∈ ¯ Q × such that˜ p ( β ) = 0, and we have that q -dres (cid:18) δ (˜ p )˜ p , [ β ] q , (cid:19) = 0 = q -dres (cid:18) δ (˜ p )˜ p , [ β ] q , (cid:19) , and similarly if we assume instead that ˜ p = 1. Therefore, if either ˜ p = 1 or ˜ p = 1, it isimpossible to have (1 , − ∈ M . Or in other words, if (1 , − ∈ M then u ¯ u = ( − n σ ( ˜ f )˜ f forsome ˜ f ∈ ¯ Q ( x ) × . But in this case we then see that n must be odd, for otherwise we wouldhave that α = α for every (cid:0) α α (cid:1) ∈ G , and since G is Zariski-dense in H the samerelation would be satisfied by every diagonal matrix in H , but this does not occur for anyof the possibilities for H listed in Lemma 7.1. This conlcudes the proof that (1 , − ∈ M ifand only if H = {±} ⋉ G m ( C ) as in Lemma 7.1(3).In case we do have (1 , − ∈ M , we must decide whether M = Z · (1 , −
1) or M = Z .We have that M = Z · (1 , −
1) if and only if G = H , which implies that G = H . Onthe other hand, we have M = Z if and only if (1 , ∈ M also, i.e., (7.5) is satisfied with m = 1 = m . But then after adding those two equations together we see that there exists f ∈ ¯ Q ( x ) (not just in ¯ Q ( x )), such that δ ( b ) b = δ ( u ) u + δ (¯ u )¯ u + σ (cid:18) δ ( w ) w (cid:19) − δ ( w ) w = σ ( f ) − f + 2 c. Indeed, writing u − ¯ u = x w with w ∈ ¯ Q ( x ) × and f := ( g + ¯ g ) + δ ( w ) w ∈ ¯ Q ( x ), where g ∈ ¯ Q ( x ) and c ∈ Z are as in (7.5), the above equation results from comparing determinantsin σ ( T ) AT − = ( u
00 ¯ u ) with T as in (7.3). Furthermore, in this case we must have c = 0,for otherwise we would have that G ⊆ GL ( C δ ) is differentially constant, which by [AS17,Thm. 3.7(ii)] would imply that G is commutative. But this is impossible, since G isZariski-dense in H by Proposition 2.12, so H would have to be commutative also, yieldinga contradiction. Thus, G is a proper subgroup of H if and only if G = (cid:26)(cid:18) α α (cid:19) , (cid:18) α − α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α ∈ C × with δ (cid:18) δ ( α ) α (cid:19) = 0 (cid:27) . Since for any (cid:0) ± λλ (cid:1) ∈ G we have that λ ( ) ∈ G , we see that δ (cid:16) δ ( λ ) λ (cid:17) = 0 also,concluding the proof of item (2).It remains to show that the statements in items (1) and (3) are correct when M = Z · (1 , H = D − m or H = D + m , then G = H and therefore G = H . This establishes item (1).Finally, supposing H = {± } ⋉ G m ( C ) and M = Z · (1 , δ ( b ) b = σ ( f ) − f + 2 c , if and only ifdet( G ) ⊆ n α ∈ C × (cid:12)(cid:12)(cid:12) δ (cid:16) δ ( α ) α (cid:17)o , with equality if and only if c = 0, and moreover c = 0 ifand only if det( G ) = { α ∈ C × | δ ( α ) = 0 } . Since G has index 2 in G , det( G ) has index atmost 2 in det( G ); but since det( G ) is divisible in either case, we see that det( G ) = det( G ),concluding the proof of item (3). (cid:3) Irreducible and imprimitive (2): rational system of imprimitivity.
Supposingthere are no solutions to (4.3) in ¯ Q ( x ), and either a = 0 or there exists a solution e ∈ ¯ Q ( x )to (4.5), we proceed as follows. The non-existence of solutions to (4.3) in k implies thereare no solutions in k ∞ either, which in turn implies that the σ -Galois group H ∞ for (4.2)over k ∞ is irreducible, and since H ∞ ⊆ H , the σ -Galois group for (4.2) over k , we thenhave that H must be irreducible also.The system (4.2) in this case is equivalent to σ ( Y ) = (cid:18) − r (cid:19) Y, (7.6)for some r ∈ ¯ Q ( x ) as we saw in §
4, which implies that H ∞ is imprimitive. Since H ∞ hasfinite index in H , the classification of algebraic subgroups of GL ( C ) from [NvdPT08] thenimplies that H must also be imprimitive, and therefore H must be one of the irreducibleimprimitive subgroups of GL ( C ) with bicyclic group of connected components listed inLemma 7.1. Remark . In this case, we can compute the σ -Galois group H for (7.6) over k fromamong the possibilities listed in Lemma 7.1 with the aid of Remark 7.2 by computing thediagonal subgroup ∆( H ) and the image of the determinant det( H ) as follows. As before,det( H ) = µ m , the group of m -th roots of unity, if and only if m is the smallest positiveinteger such that b m = σ ( f ) f for some f ∈ ¯ Q ( x ) × ; if there is no such m , then det( H ) = G m ( C ).On the other hand, ∆( H ) is precisely the σ -Galois group for the system σ ( Z ) = (cid:18) − r − σ ( r ) (cid:19) Z (7.7)over k , which we can compute as in Proposition 5.2 and Remark 5.3 by considering k as a σ -field. We see that ∆( H ) ⊆ (cid:26)(cid:18) α α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α , α ∈ C × (cid:27) is the subgroup defined by the following conditions on α and α :(1) ( α α ) m = 1 if and only if m is the smallest positive integer such that ( rσ ( r )) m = σ ( f ) f for some f ∈ ¯ Q ( x ) × ;(2) otherwise ∆( H ) = G m ( C ) .The omission of the possibility that H = {± } ⋉ G m ( C ) as in Lemma 7.1(3) is deliberate.This is impossible under the present assumptions because α = α for every (cid:0) α α (cid:1) ∈ ∆( H )if and only if (cid:16) σ ( r ) r (cid:17) = σ ( f ) f for some f ∈ ¯ Q ( x ) × . But if we let v ∈ ¯ Q ( x ) × such that r σ ( v ) v = ex n p p is reduced , with e ∈ ¯ Q × such that e ∈ q Z if and only if e = 1, n ∈ Z , and p , p ∈ ¯ Q [ x ] monic such that gcd( x, p ) = gcd( x, p ) = gcd( p , σ m ( p )) for every m ∈ Z ,we would then have that the reduced form of σ ( r ) is exactly σ ( r ) σ ( v ) σ ( v ) = eq n x n σ ( p ) σ ( p ) , andtherefore σ ( r ) r σ ( v ) v = q n σ ( p ) p p σ ( p ) . IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 33 This element is not necessarily reduced with respect to σ , but the reduced form of σ ( r ) r withrespect to σ is given by q ε ˜ p ˜ p , where ε = 0 if n is even and ε = 1 if n is odd, and ˜ p , ˜ p ∈ ¯ Q [ x ] are again monic suchthat gcd( x, ˜ p ) = gcd( x, ˜ p ) = gcd(˜ p , σ m (˜ p )) = 1 for every m ∈ Z . We then have thatthe reduced form of σ ( r ) r with respect to σ is (cid:16) ˜ p ˜ p (cid:17) , and therefore (cid:16) σ ( r ) r (cid:17) = σ ( f ) f for some f ∈ ¯ Q ( x ) × if and only if ˜ p = 1 = ˜ p , but this would imply that σ ( r ) r = σ ( ˜ f )˜ f for some˜ f ∈ ¯ Q ( x ) × already, which in turn would imply that α = α for every (cid:0) α α (cid:1) ∈ ∆( H ),which is not possible according to the classification of Lemma 7.1.In fact, we may pursue this further to conclude that it is also impossible to have δ ( σ ( r )) σ ( r ) − δ ( r ) r = σ ( g ) − g + c (7.8)for some g ∈ ¯ Q ( x ) and c ∈ Z . This is because if, say, ˜ p = 1, then there would exist β ∈ ¯ Q × such that ˜ p ( β ) = 0, and then we would have that q -dres (cid:18) δ (˜ p )˜ p , [ β ] q , (cid:19) = 0 = q -dres (cid:18) δ (˜ p )˜ p , [ β ] q , (cid:19) , and similarly with the roles of ˜ p and ˜ p exchanged. But since δ ( σ ( r )) σ ( r ) − δ ( r ) r = δ (˜ p )˜ p − δ (˜ p )˜ p modulo ( σ − Q ( x )), we see that (7.8) is impossible unless ˜ p = 1 = ˜ p , which we alreadyruled out above.Having computed the σ -Galois group H for (4.2) over k as above, we can now computethe σδ -Galois group G for (4.2) over k with the following result. Proposition 7.6.
Suppose there are no solutions to (4.3) in ¯ Q ( x ) , and either a = 0 orthere exists a solution to (4.5) in ¯ Q ( x ) . Then H = {± } ⋉ G m ( C ) as in Lemma 7.1(3),and G is the subgroup of {± } ⋉ G m ( C ) = (cid:26)(cid:18) α α (cid:19) , (cid:18) λ λ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α α = 0 , λ λ = 0 (cid:27) (7.9) defined by the following conditions on α , α , λ , and λ .(1) If H = D − m as in Lemma 7.1(1) or H = D + m as in Lemma 7.1(2), then G = H .(2) If H = {± } ⋉ G m ( C ) as in Lemma 7.1(4), then:(a) there exist c ∈ Z and g ∈ ¯ Q ( x ) such that δ ( b ) b = σ ( g ) − g + c if and onlyif δ (cid:16) δ ( α ) α + δ ( α ) α (cid:17) = 0 = δ (cid:16) δ ( λ ) λ + δ ( λ ) λ (cid:17) ; moreover, c = 0 if and only if δ ( α α ) = 0 = δ ( λ λ ) ;(b) otherwise, G = H . Proof.
The fact that H = {± } ⋉ G m ( C ) as in Lemma 7.1(3) under these conditionswas already established in Remark 7.5. Let us denote by ∆( G ) the subgroup of diagonalmatrices in G , which coincides with the σ δ -Galois group for (7.7) over k . We may compute∆( G ) using the results of Proposition 5.2 and Remark 5.3. We again denote by M ⊆ Z the submodule generated by ( m , m ) ∈ Z , not both zero and with gcd( m , m ) = 1, suchthat there exist c ∈ Z and g ∈ ¯ Q ( x ) such that m δ ( σ ( r )) σ ( r ) + m δ ( r ) r = σ ( g ) − g + c, (7.10)which is equivalent to m δ ( σ ( r )) σ ( r ) + m δ ( r ) r = σ (cid:18) σ ( g ) − m δ ( r ) r (cid:19) − (cid:18) σ ( g ) − m δ ( r ) r (cid:19) + c. As we saw in Remark 5.3, either M = { } is trivial; or M = Z · ( m , m ); or M = Z . Butit follows from the above computation that if M is not trivial, then at least one of (1 ,
1) or(1 , − M . But we saw in Remark 7.5 that we cannot have (1 , − ∈ M , sincethe relation 7.8 is impossible. The only possibilities that remain are therefore M = { } or M = Z · (1 , M = { } , then ∆( G ) = G m ( C ) , and therefore G = H = {± } ⋉ G m ( C ) . Let usnow suppose that M = Z · (1 , H = D − m as in Lemma 7.1(1) or H = D + m as inLemma 7.1(2), then ∆( G ) = ∆( H ), which implies that G = H , as claimed in item (1). Itremains to establish item (2) under the assumption that M = Z · (1 , G )) ⊆ n α ∈ C × (cid:12)(cid:12)(cid:12) δ (cid:16) δ ( α ) α (cid:17)o with equality if and only if c = 0 in (7.10), andmoreover this c = 0 if and only if det(∆( G )) = { α ∈ C × | δ ( α ) = 0 } . Since ∆( G ) has finiteindex in G and det(∆( G )) is divisible in either case, we obtain that det( G ) = det(∆( G )),which concludes the proof of item (2). (cid:3) Irreducible and imprimitive (3): quadratic system of imprimitivity.
Suppos-ing there are no solutions to (4.3) in ¯ Q ( x ), a = 0, and there are no solutions to (4.5) in¯ Q ( x ), let us now assume that there is a solution e ∈ ¯ Q ( x ) to (4.5), and therefore the Galoisconjugate ¯ e of e over ¯ Q ( x ) also satisfies (4.5), since this Riccati equation with respect to σ is defined over ¯ Q ( x ). Here again we have that the non-existence of solutions to (4.3)in ¯ Q ( x ) implies that there are no solutions to (4.3) in all of k ∞ , which implies that H ∞ is irreducible as explained in §
4. Since H ∞ ⊆ H ⊆ H (which again denote the σ -Galoisgroups for (4.2) over k ∞ , k , and k , respectively), we then have that H and H must alsobe irreducible. Moreover the existence of the solution e ∈ k to (4.5) implies that H ∞ mustbe imprimitive, and since H ∞ has finite index in H and in H , the classification of thealgebraic subgroups of GL ( C ) of [NvdPT08] implies that H and H must be imprimitivealso. By [vdPS97, Prop. 12.2(1)], both H and H must have bicyclic groups of connectedcomponents, and thus they must both be included in the list of irreducible imprimitive sub-groups given in Lemma 7.1. By Corollary 4.3, H ⊆ H has index either 1 or 2. We willshow that H = H , which implies that the index of H in H is exactly 2. A straightforwardcomputation shows that the only groups listed in Lemma 7.1 admitting another such group IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 35 as an index-2 subgroup are H = D + m with m even, with H then given by one of the groups D − m/ or D + m/ .To see that H = H in this case, recall from [Hen97, Thm. 18] that e ∈ ¯ Q ( x ) satisfies(4.5) if and only if d := e + ba has the property that dy + σ ( y ) =: z d satisfies σ ( z d ) + rz d = 0with r := − aσ ( a ) + σ ( b ) + aσ ( d ) if and only if y satisfies (4.1). We see that this is equivalentto z ¯ d := ¯ dy + σ ( y ) satisfying σ ( z ¯ d ) + ¯ rz ¯ d = 0, where ¯ d and ¯ r denote the Galois conjugatesof d, r ∈ ¯ Q ( x ) over ¯ Q ( x ). Since e = ¯ e , we also have d = ¯ d and r = ¯ r . At this point, wecould compute H directly as in Remark 7.5, where in particular the subgroup of diagonalmatrices ∆( H ) in H corresponds to the σ -Galois group for the system σ ( Y (2) ) = A (2) Y (2) (7.11)over k , where A (2) := σ ( A ) A . A computation shows that setting T := (cid:18) d d (cid:19) ∈ GL ( ¯ Q ( x ))we have that σ ( T ) A (2) T − = (cid:18) − r − ¯ r (cid:19) , and therefore (7.11) is equivalent over k to the system σ ( Z (2) ) = (cid:18) − r − ¯ r (cid:19) Z (2) (7.12)via the gauge transformation Z (2) = T Y (2) .If, contrary to our contention, we did have that H = H , then the σ -Galois group H (2)1 for the system (7.11) over k would coincide with ∆( H ), and in particular we would have H (2)1 = ∆( H ) = ∆( H ) being diagonal. We will show that this is not the case. For this,consider the system σ ( W ) = (cid:18) A (2) q (cid:19) W, with fundamental solution matrix W = (cid:18) Y (2) x (cid:19) , (7.13)where Y (2) in turn denotes a (2 ×
2) fundamental solution matrix for (7.11) over k . Let˜ H (2) denote the σ -Galois group for the system (7.13) over k . Let ˜ τ ∈ ˜ H (2) such that˜ τ ( x ) = − x , and let τ := ˜ τ | S ∈ H (2)1 denote the restriction of ˜ τ to the σ -PV ringcorresponding to the system (7.11): S (2)1 := k [ Y (2) , det( Y (2) ) − ]. Let M τ ∈ GL ( C ) denotethe matrix correspondiong to τ ∈ H (2)1 , so that τ ( Y (2) ) = Y (2) M τ . Since the system (7.12)is diagonal, we have that T Y (2) = Z (2) = (cid:18) z z (cid:19) , where σ ( z ) = − rz and σ ( z ) = − ¯ rz . But then we see that˜ τ ( Z (2) ) = ˜ τ ( T Y (2) ) = ¯
T Y (2) M τ = (cid:18) (cid:19) Z (2) M τ . On the other hand, σ (˜ τ ( z )) = ˜ τ ( σ ( z )) = ˜ τ ( − rz ) = − ¯ r ˜ τ ( z ), and therefore ˜ τ ( z ) = λ z for some λ ∈ C × . Similarly we see that ˜ τ ( z ) = λ z for some λ ∈ C × , and therefore˜ τ ( Z (2) ) = (cid:18) (cid:19) Z (2) (cid:18) λ λ (cid:19) . This shows that M τ = (cid:0) λ λ (cid:1) , as we wanted to show. Proposition 7.7.
Suppose there are no solutions to (4.3) in ¯ Q ( x ) , a = 0 , and there are nosolutions to (4.5) in ¯ Q ( x ) but there exists a solution to (4.5) in ¯ Q ( x ) . Then G = H = D + m for the smallest even positive integer m ∈ N such that b m = σ ( f ) f for some f ∈ ¯ Q ( x ) × .Proof. The remarks above show that under these assumptions the σ -Galois group H for(4.2) over k has index exactly 2 in the σ -Galois group H for (4.2) over k . Thus H = D + m as in Lemma 7.1(2) for some even positive integer m ∈ N , and H is then one of D − m/ or D + m/ . In either case, it follows from Proposition ?? , applied over k instead of k , that H = G is also the σδ -Galois group G for (4.2) over k . Since the index of G in G , the σδ -Galois group for (4.2) over k , is also 2 = [ H : H ], in then follows that G = H = D + m in this case, as claimed. (cid:3) Irreducible and primitive groups
Let us denote again k = C ( x ), where C is a δ -closure of ¯ Q , σ denotes the C -linearautomorphism of k defined by σ ( x ) = qx , and δ ( x ) = 1. We write H for the σ -Galois groupand G for the σδ -Galois group for σ ( Y ) = (cid:18) − b − a (cid:19) Y (8.1)over k , where a, b ∈ ¯ Q ( x ) and b = 0. In this section we consider the case where a = 0 andthere are no solutions in ¯ Q ( x ) to (4.3) nor to (4.5), which is equivalent to the condition thatSL ( C ) ⊆ H by the results of [Hen97] summarized in §
4. In this case, H is reductive and theconnected component of the identity H ◦ is either SL ( C ) or GL ( C ), and in either case thederived subgroup H ◦ , der = SL ( C ). Therefore by [AS17, Thm .5.2] SL ( C ) ⊆ G , and hence G ⊆ GL ( C ) is determined by the image the determinant map det( G ) ⊆ G m ( C ), which isthe σδ -Galois group for σ ( y ) = by over k . The proof of the following result is immediate. Proposition 8.1.
Suppose there are no solutions to (4.3) in ¯ Q ( x ) , a = 0 , and there are nosolutions to (4.5) in ¯ Q ( x ) . Then det( G ) ⊆ G m ( C ) is determined as follows.(1) There exist a smallest positive integer m ∈ N and f ∈ ¯ Q ( x ) × such that b m = σ ( f ) f ifand only if det( G ) = µ m , the group of m -th roots of unity.(2) There exist = c ∈ Z and f ∈ ¯ Q ( x ) such that δ ( b ) b = σ ( f ) − f + c if and only if det( G ) = n α ∈ C × (cid:12)(cid:12)(cid:12) δ (cid:16) δ ( α ) α (cid:17) = 0 o .(3) There exists f ∈ ¯ Q ( x ) such that δ ( b ) b = σ ( f ) − f if and only if det( G ) = { α ∈ C × | δ ( α ) = 0 } .(4) Otherwise, det( G ) = G m ( C ) . IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 37 Examples
In this section we compute the σδ -Galois group G associated to some concrete second-order linear difference equations over ¯ Q ( x ) with respect to the q -dilation operator σ ( x ) = qx ,where q ∈ C × is not a root of unity. We will first apply the algorithm of [Hen97] to computethe σ -Galois group H associated to the equation, and then apply the procedures developedin this paper to compute G .9.1. Example.
Let us consider (4.1) with b = q x ( x − ( q x + 6 qx + 6) x + 6 x + 6 ; and a = − q x (2 q ) x + 4( q + q ) x + (7 q − q + 7) x − q + 1) x + 12 x + 6 x + 6 . Applying the procedure in [Hen97, § u ∈ ¯ Q ( x ) to the first Riccatiequation (4.3) in this case, given by u := x ( x − . After computing buσ ( u ) = ( x − ( q x + 6 qx + 6)( qx − ( x + 6 x + 6) , we see that w := x + 6 x + 6( x − ∈ ¯ Q ( x ) × satisfies σ ( w ) = buσ ( u ) w , and therefore we are in the setting of Proposition 6.1. After verifyingthat δ ( u ) u = 5 + 2 x − = σ ( f ) − f + c for any f ∈ ¯ Q ( x ) and c ∈ Z , we proceed to attempt to find a linear differential operator L ∈ ¯ Q [ δ ] of smallest possibleorder such that there exist f ∈ ¯ Q ( x ) and c ∈ ¯ Q satisfying L (cid:18) δ ( u ) u (cid:19) − w = σ ( f ) − f + c. Since w = x + 6 x + 6( x − = 5( x − + 8 x − q -discrete residues: q -dres( w, [1] q ,
2) = 5; q -dres( w, [1] q ,
1) = 8; and q -dres( w, ∞ ) = 1 , we see that if there exists such an L ∈ ¯ Q [ δ ] then its order must be exactly 1. Writing L = e δ + e , we find that L (cid:18) δ ( u ) u (cid:19) − w = − e − x − + − e + 2 e − x − e − , which has the desired form σ ( f ) − f + c for some f ∈ ¯ Q ( x ) and c ∈ ¯ Q if and only if − e − − e + 2 e − ⇐⇒ e = − and e = . The corresponding value of c = 5 e − = 0. With this, we conclude that the σδ -Galoisgroup for (4.1) over k for this choice of coefficients a, b ∈ ¯ Q ( x ) is G = (cid:26)(cid:18) α ξ α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α, ξ ∈ C, α = 0 , δ (cid:18) ξα (cid:19) = − δ (cid:18) δ ( α ) α (cid:19) + 32 δ (cid:18) δ ( α ) α (cid:19)(cid:27) . Example.
Let us consider (4.1) with a = − ( q + q / ) x and b = q / ( x − x ) . Since the valuations at x = 0 of the coefficients are v ( a ) = 1 and v ( b ) = 1, we are in the casewhere v ( b ) ≤ v ( a ) and v ( b ) is odd, and therefore there are no solutions to (4.3) in ¯ Q ( x )(cf. [Hen97, § u = x + x / ∈ ¯ Q ( x / ) and ¯ u = x − x / ∈ ¯ Q ( x / ) both satisfy(4.3). Since δ ( b ) b = 2 + 1 x − = σ ( f ) − f + c for any f ∈ ¯ Q ( x ) and c ∈ Z , we deduce that det( G ) = G m ( C ). Since there is no f ∈ ¯ Q ( x / ) × such that (cid:16) u ¯ u (cid:17) = x (cid:18) x / + 1 x / − (cid:19) = σ ( f ) f , we conclude that the σδ -Galois group G for (4.1) over k for this choice of coefficients a, b ∈ ¯ Q ( x ) is G = {± } ⋉ G m ( C ) = (cid:26)(cid:18) α α (cid:19) , (cid:18) λ λ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α , α , λ , λ ∈ C × (cid:27) . Example.
Let us consider (4.1) with a = 0 and b = − q / x . This example was discussedin [AS17, § σ -Galois group H was solvable but not abelian; in fact it was proved there using ad-hoc methods that H = (cid:26)(cid:18) α α (cid:19) , (cid:18) α − α (cid:19) , (cid:18) λλ (cid:19) , (cid:18) λ − λ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α, λ ∈ C × (cid:27) . We can now prove this systematically, as well as find the corresponding σδ -Galois group G ,using the results of § x = 0 of the coefficients are v ( a ) = ∞ and v ( b ) = 1, we are in the case where v ( b ) ≤ v ( a ) and v ( b ) is odd, and therefore thereare no solutions to (4.3) in ¯ Q ( x ) (cf. [Hen97, § u = x / ∈ ¯ Q ( x / ) and ¯ u = − u = − x / ∈ ¯ Q ( x / ) both satisfy (4.3). Since δ ( b ) b = 1 , we see thatdet( G ) = n α ∈ C × (cid:12)(cid:12)(cid:12) δ (cid:16) δ ( α ) α (cid:17) = 0 o . We also verify that (cid:0) u ¯ u (cid:1) = ( − = 1 . This concludesthe computation that G = (cid:26)(cid:18) α α (cid:19) , (cid:18) α − α (cid:19) , (cid:18) λλ (cid:19) , (cid:18) λ − λ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α, λ ∈ C × , δ (cid:18) δ ( α ) α (cid:19) = 0 = δ (cid:18) δ ( λ ) λ (cid:19)(cid:27) . IFFERENTIAL GALOIS GROUPS OF q -DIFFERENCE EQUATIONS 39 Example.
In [KZ18] the authors develop algorithms for desingularization of q -difference–differential operators. In [KZ18, Example 5.2], those results were applied in the study of thedifference equations satisfied by the colored Jones polynomials of several knots. In spite ofthe name, a colored Jones polynomial is not actually a polynomial in general, but rather con-sist of an infinite sequence of rational functions in Q ( q ), where q is a formal indeterminate.We refer to [KZ18, §
5] and the references therein for additional details.This second-order difference equation is satisfied by the colored Jones polynomial (afternormalization) of the knot K twist − ; we emphasize that the name “polynomial” may be mis-leading: in general, the colored Jones polynomial of a knot actually consists of an infinitesequence of rational functions in Q ( q ). Let us consider (4.1) with a = ( qx − qx + 1) (cid:0) q x − q x − q x − qx − qx + 1 (cid:1) q x (cid:0) qx − (cid:1) and b = q x − qx − y ( q x ) + a ( x ) y ( qx ) + b ( x ) y ( x ) = 0 . (9.1)The corresponding second-order linear difference equation with this choice of coefficients a, b ∈ Q ( q )( x ), where q is a formal indeterminate, is satisfied by the colored jones polynomialfor K twist − (see [KZ18, §
5, Fig. 1]).To compute the σδ -Galois group for this equation over C ( x ), where C is a δ -closure of the δ -constant field Q ( q ), proceeds as follows. Using the QHypergeometricSolution commandincluded in the Maple package QDifferenceEquations, we have verified that the Riccati equa-tions (4.3) and (4.5) do not admit any solutions in C ( x / ). Therefore SL ( C ) ⊆ G ⊆ H ,where H denotes the σ -Galois group, as discussed in § §
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Department of Mathematical Sciences, The University of Texas at Dallas, Texas, USA
E-mail address : [email protected] Department of Applied Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou, China
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