An algorithm to determine regular singular Mahler systems
aa r X i v : . [ c s . S C ] F e b AN ALGORITHM TO DETERMINE REGULAR SINGULARMAHLER SYSTEMS
COLIN FAVERJON AND MARINA POULET
Abstract.
This paper is devoted to the study of the analytic properties of Mahlersystems at 0. We give an effective characterisation of Mahler systems that areregular singular at 0, that is, systems which are equivalent to constant ones. Similarcharacterisations already exist for differential and ( q -)difference systems but they donot apply in the Mahler case. This work fill in the gap by giving an algorithm whichdecides whether or not a Mahler system is regular singular at 0. Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. A characterisation of regular singularity at 0 . . . . . . . . . . . . . . 43. Index of ramification and valuation at 0 of a gauge transformation . . . . . . 54. Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . 105. The algorithm of Theorem 1 . . . . . . . . . . . . . . . . . . . . . 186. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247. Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . 27References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.
Introduction
Let K be the field of Puiseux series with algebraic coefficients i.e. the field K := [ d ∈ N ⋆ Q (cid:16)(cid:16) z /d (cid:17)(cid:17) . For an integer p ≥ φ p : K → K f ( z ) f ( z p ) . The map φ p naturally extends to matrices with entries in K . A p -Mahler system or,for short, a Mahler system is a system of the form(1) φ p ( Y ) = AY, A ∈ GL m (cid:0) Q ( z ) (cid:1) . Date : February 23, 2021.2010
Mathematics Subject Classification.
Primary: 39A06, 68W30; Secondary: 11B85.
Key words and phrases.
Mahler equations, regular singularity, algorithm.The work of the first author was supported by the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation programme under the Grant Agreement No648132. The work of the second author was performed within the framework of the LABEX MILYON(ANR-10-LABX-0070) of Universit´e de Lyon, within the program ”Investissements d’Avenir” (ANR-11-IDEX- 0007) operated by the French National Research Agency (ANR). COLIN FAVERJON AND MARINA POULET
The study of Mahler systems began with the work of Mahler in 1929 [Mah29, Mah30a,Mah30b]. Nowadays, there is an increased interest in their study because they arerelated to many areas such as automata theory or divide-and-conquer algorithm (seefor example [Cob68, MF80, Dum93, Nis97, Phi15, AF17, AF20] for a non-exhaustivebibliography). In this paper, we focus on the singularity at 0 of Mahler systems.Similarly to the case of differential or ( q -)difference systems, a singularity at 0 of aMahler system can be regular. In that case, we say that the Mahler system is regularsingular at 0. Otherwise, the system is irregular at 0. Precisely, we have the followingdefinition. Definition 1. A p -Mahler system (1) is regular singular at
0, or for short regularsingular , if there exists Ψ ∈ GL m ( K ) such that φ p (Ψ) − A Ψ is a constant matrix.The singularities of differential or ( q -)difference systems have been widely studiedand algorithms have been given. One of the main interests in studying the regularsingular systems is the good analytic properties of their solutions. Linear differentialsystems have been some of the first ones to be studied. A linear differential systemis regular singular at z = 0 if and only if all of its solutions have moderate growthat z = 0, that is, at most a polynomial growth (see for example [vdPS03, theorem5.4]). Some criteria and algorithms have been given for a linear differential system tohave a regular singularity, see for example [Bir13, Mos59, HW86, Hil87, Bar95]. Then,algorithms have been given for other systems such as difference systems and q -differencesystems (see for instance [Pra83, Bar89, BP96, BBP08]). In [BBP08], the authors givea general algorithm for recognizing the regular singularity of linear functional equationssatisfying some general properties. This algorithm applies to many systems such asdifferential systems and ( q − )difference systems. However, this general algorithm doesnot apply to the study of the regular singularity at 0 of Mahler systems because theMahler operator φ p does not preserve the valuation at 0. The aim of this paper is tofill in this gap and to present an algorithm which decides whether or not a Mahlersystem is regular singular at 0.In general, a Mahler system does not admit a fundamental matrix of solutions inGL m ( K ). To find such a matrix, one has to consider some extensions of K . Let H denote the field of Hahn power series. One can extend the operator φ p to H . In[Roq20], Roques proved that for every p -Mahler system there exists a matrix Ψ ∈ GL m ( H ) such that φ p (Ψ) − A Ψ is a constant matrix. In the mean time, for anyconstant Mahler system one can build a fundamental matrix of solutions using thefunctions log log( z ) and log a ( z ), a ∈ C (see [Roq18]). Thus, any Mahler system has afundamental matrix of solutions of the form ΨΘ, where Ψ is matrix with entries in H and Θ is a fundamental matrix of solutions of a constant system. Among them, theregular singular systems are those for which the matrix Ψ belongs to GL m ( K ). Therestriction to the subfield K of H is essential to preserve the analytic properties ofthe system. In particular, if f ∈ K m is a column vector, solution of a Mahler system,it follows from Rand´e’s Theorem [Ran92, BCR13] that the entries of f are ramifiedmeromorphic functions inside the unit disk. Definition 2.
Let p ≥ A, B ∈ GL m (cid:0) Q ( z ) (cid:1) . Let k ⊂ H be afield. The p -Mahler systems φ p ( Y ) = AY and φ p ( Y ) = BY N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS k - equivalent if there exists a matrix Ψ ∈ GL m ( k ) such that φ p (Ψ) B = A Ψ . In that case, the matrix Ψ is called the associated gauge transformation .This choice for the equivalence class ensures that if Y is such that φ p ( Y ) = AY then φ p (cid:0) Ψ − Y (cid:1) = B (cid:0) Ψ − Y (cid:1) . With this definition, the regular singular systems are theones that are K -equivalent to a constant system.A Mahler system is said to be Fuchsian at A are analytic functionsat 0 and A (0) ∈ GL m (cid:0) Q (cid:1) . To say it differently, a system is Fuchsian at φ ( Y ) = 12 (cid:18) z − z (cid:19) Y associated with the generating series of the Rudin-Shapiro sequence is not regularsingular at 0 (see Section 6). The main result of this paper reads as follow.
Theorem 1.
Let A ∈ GL m (cid:0) Q ( z ) (cid:1) and p ≥ . There exists an algorithm whichdetermines whether or not the Mahler system (1) is regular singular at . This is doneby computing the dimension of an explicit Q -vector space. If the system is regularsingular at the algorithm computes a constant matrix to which the system is equivalentand a truncation at an arbitrary order of the Puiseux development of the associatedgauge transformation. In [CDDM18], the authors built an algorithm to decide whether or not a Mahlersystem has a fundamental matrix of solutions in GL m ( K ). In that case, the system isregular singular at 0 and K -equivalent to the identity matrix. From this point of view,Theorem 1 can be seen as a generalisation of the results of [CDDM18]. Remark . It is also interesting to look at Mahler systems around other fixed pointsof φ p such as 1 and ∞ . Using the change of variable z = e u one can test the regularsingularity at 1 using the theory of q -difference linear systems. Furthermore, onecan know if a system is regular singular at ∞ by applying Theorem 1 to the system A (1 /z ). In particular, when a Mahler system is regular singular at 0, 1 and ∞ , thesecond author [Pou20] has proved a density theorem for the Galois group of the system.The paper is organised as follows. In section 2 we state Theorem 2, which refines thefirst part of Theorem 1. We define a vector space whose dimension gives a necessaryand sufficient condition for a Mahler system to be regular singular at 0. Assumingthat the system is regular singular at 0, we determine in section 3 an upper boundfor the degree of ramification of the gauge transformation Ψ and a lower bound forthe valuation of its entries. Our proof relies on the Cyclic Vector Lemma for Mahlersystems for which we provide a simple proof together with an algorithm. Section 4 isthen devoted to the proof of Theorem 2. We build an isomorphism between the Q -vector space spanned by the columns of Ψ and the vector space described in Theorem2. The algorithm of Theorem 1 is described in section 5 and a bound for its complexity COLIN FAVERJON AND MARINA POULET is given. Section 6 is devoted to the study of some examples. Eventually, in Section 7we discuss some open problems.
Notation.
We let Q denote the algebraic closure of Q in C and Q ⋆ = Q \ { } . Welet v : Q [[ z ]] Q denote the valuation at z = 0: for f ∈ Q [[ z ]], v ( f ) is the greatestinteger v such that f belongs to the ideal z v Q [[ z ]]. It extends uniquely to K . We alsoextend it to the set of matrices with entries in K where v ( U ) denotes the minimumof the valuations at 0 of the entries of a matrix U . For a matrix U ∈ M m ,m ( K ) wewrite U = X n ≥ v ( U ) U n z n/d with U n ∈ M m ,m (cid:0) Q (cid:1) , d ∈ N ⋆ . and we let U n denote the zero matrix of size m × m , when n < v ( U ). Let R = P/Q bea rational function,
P, Q ∈ Q [ z ] relatively prime. We let deg( R ) denote the maximumof the degrees of P and Q . This notation extends to matrices with entries in Q ( z ),taking the maximum of the degrees of the entries.Our bounds for the complexity of the algorithms presented here are given in termsof arithmetical operations in Q . Given f, g : N R ≥ we use the classical Landaunotation f ( n ) = O ( g ( n )) if there exists a positive real number κ such that f ( n ) ≤ κg ( n )for every integer n large enough. Given an integer n , we let M( n ) denote the complexityof the multiplication of two polynomials of degree at most n , and MM( n ) denote thecomplexity of the multiplication of two matrices of size n .For the sake of clarity, we shall denote by roman capital letters A, B, C, . . . matriceswhose coefficients are effectively known and by Greek capital letters Ψ , Θ , Λ , · · · theother matrices. While matrices are denoted by capital letters Ψ , Θ , U, . . . , the columnsof these matrices should be denoted by bold lowercase letters ψ , θ , u , . . . .2. A characterisation of regular singularity at d ≥ B ( d ) := φ d ( A ) − and we denote by B ( d ) := X n ≥ dv ( A − ) B n ( d ) z n the development of B ( d ) in Laurent power series. Set(2) µ d := ⌈− dv (cid:0) A − (cid:1) / ( p − ⌉ , and ν d := ⌈ dv ( A ) / ( p − ⌉ . Since AA − = I m , we have v ( A ) + v (cid:0) A − (cid:1) ≤ dv ( A ) / ( p − ≤ − dv (cid:0) A − (cid:1) / ( p − ν d ≤ µ d . We let M d and N d denote the block matrices M d := ( B i − pj ( d )) ν d ≤ i,j ≤ µ d and N d := ( ( B i − pj ( d )) dv ( A − ) + pν d ≤ i ≤ ν d − , ν d ≤ j ≤ µ d if ν d < µ d ∈ M ,m ( µ d − ν d +1) (cid:0) Q (cid:1) if ν d = µ d Remark . The condition ν d < µ d ensures that the matrix is well-defined because it isequivalent to(3) dv ( A − ) + pν d ≤ ν d − . N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS ν d satisfies (3), then ν d ≤ − dv ( A − ) p − − p − < µ d . Conversely, if ν d < µ d , then ν d ≤ µ d − (cid:24) − dv ( A − ) p − (cid:25) − ≤ − dv ( A − ) p − − p − , which is equivalent to (3). The last inequality comes from the fact that the number l − dv ( A − ) p − m − − dv ( A − ) p − is of the form b/ ( p −
1) with b ∈ { , . . . , p − } .Let ker( N d ) denote the (right) kernel of N d in Q m ( µ d − ν d +1) . When ν d = µ d we haveker( N d ) = Q m ( µ d − ν d +1) . For an integer n ∈ N we set M − nd · ker( N d ) = ker( N d M nd ) . We also let(4) D = { d ∈ N , ≤ d ≤ p m − | gcd( d, p ) = 1 } ⊂ N The following result gives a necessary and sufficient condition for a system to be regularsingular at 0.
Theorem 2.
The system (1) is regular singular at if and only if there exists aninteger d ∈ D such that the dimension of the Q -vector space X d := \ n ∈ Z M nd · ker( N d ) is greater than or equal to m . In that case, it is equal to m and the system (1) is Q (( z /d )) -equivalent to a constant system. Furthermore, there is an algorithm tocompute this integer d .Remark . We show in Lemma 16 that X d = \ − c d ≤ n ≤ c d M nd · ker( N d )where c d := m ( µ d − ν d + 1).3. Index of ramification and valuation at of a gauge transformation Assume that the system is regular singular at 0. The entries of the associated gaugetransform belong to z v Q [[ z /d ]] for some integers v ∈ Z and d ≥
1. The aim of thissection is to provide an upper bound for the ramification index d and a lower boundfor the valuation v .3.1. The Cyclic Vector Lemma.
For the sake of completeness, we develop here aproof of a result called the Cyclic Vector Lemma. Any Mahler system is associatedwith an homogeneous Mahler equation, that is an equation of the form q y + q φ p ( y ) + q φ p ( y ) + · · · + q m − φ m − p ( y ) − φ mp ( y ) = 0 , with q , . . . , q m − ∈ Q ( z ). This result is known as the Cylic Vector Lemma. We providea proof of this result here, together with an algorithm to realize it. COLIN FAVERJON AND MARINA POULET
Theorem 3 (Cyclic Vector Lemma) . Every Mahler system (1) is Q ( z ) -equivalent toa companion matrix system, i.e., there exist P ∈ GL m (cid:0) Q ( z ) (cid:1) and q , . . . , q m − ∈ Q ( z ) such that φ p ( P ) AP − = A comp where (5) A comp = · · · ... . . . . . . ...... . . . . . . · · · · · · q · · · · · · · · · q m − . Proof.
We adapt the proof of Birkhoff given in [Bir30, § P , we build its rows r , . . . , r m .These rows must be linearly independent and must satisfy φ p ( r i ) A = r i +1 for 1 ≤ i ≤ m − . Therefore, we are looking for a vector r ∈ Q ( z ) m such that the vectors r := r , r i +1 := φ p ( r i ) A , 1 ≤ i ≤ m − Q ( z ) m . For this purpose, we choosea z ∈ Q ⋆ not a root of unity such that A ( z ) , . . . , A (cid:16) z p m − (cid:17) ∈ GL m (cid:0) Q (cid:1) (such a z exists because the matrix A has finitely many singularities). Now, by interpolation,since z , z p , . . . , z p m − are all different, we may choose r ∈ Q ( z ) m such that(6) r ( z ) = e r ( z p ) = e A ( z ) − ... r ( z p m − ) = e m A ( z ) − . . . A (cid:16) z p m − (cid:17) − where e , . . . , e m is the canonical basis of Q ( z ) m . Set r := r and define recursively r i +1 := φ p ( r i ) A , 1 ≤ i ≤ m −
1. By construction, r i ( z ) = e i . The matrix P whose rows are r , . . . , r m satisfies P ( z ) = I m . Thus, P ∈ GL m (cid:0) Q ( z ) (cid:1) . Set ( q , . . . , q m − ) := φ p ( r m ) AP − . Then A comp = φ p ( P ) AP − is a companion matrix of the form (5). (cid:3) Recall that, from the Lagrange Theorem, the roots of a polynomial p := p + p z + p z + · · · + p h z h with p , . . . , p h − ∈ C , p h ∈ C ⋆ have a module strictly less than 1 plus the max of | p k || p h | , 0 ≤ k ≤ h −
1. We thus set k p k := 1 + max (cid:26) | p k || p h | , ≤ k ≤ h − (cid:27) . The following algorithm takes a Mahler system as input and returns a matrix P suchthat φ p ( P ) AP − is a companion matrix together with the last row of φ p ( P ) AP − . N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS Algorithm 1:
Find a cyclic vector associated with the Mahler system (1)
Input: A , p . Output:
The last row of a companion matrix Q ( z )-equivalent to A and itsassociated gauge transform.Set m the size of A .Set f the numerator of det( A ) and z := k f k . for ≤ i, j ≤ m do Let g ∈ Q [ z ] stand for the denominator of the entry ( i, j ) of A . if k g k ≥ z then z := k g k . endend Let r ∈ Q [ z ] m satisfy the interpolation (6).Set P the matrix with rows r := r , r i +1 := φ p ( r i ) A , 1 ≤ i ≤ m − return P and φ p ( r m ) AP − . Proposition 4.
Algorithm 1 has complexity O (MM( m )M( p m (deg( A ) + m ))) . Proof.
The computation of det( A ) has complexity O (MM( m )M(deg( A ))). Given apolynomial P , the complexity of computing k P k is O (deg( P )). Thus, the computationof z has complexity O ( m deg( A ) + MM( m )M(deg( A ))) . To obtain r we need to compute the matrices (cid:16) A ( z ) A ( z p ) A (cid:16) z p (cid:17) . . . A (cid:16) z p k (cid:17)(cid:17) − , ≤ k ≤ m − . This has complexity O (cid:0) m deg( A ) + m MM( m ) (cid:1) . Then, we must interpolate at m points, for each of the m entries of r , which has complexity O ( m M( m ) log( m )). Since r has degree m − P has complexity O ( m M( p m − (deg( A ) + m ))).Then we need to compute the inverse of P which takes O (MM( m )M( p m − (deg( A ) + m ))) . Thus, the computation of φ p ( r m ) AP − has complexity O (MM( m )M( p m (deg( A )+ m ))).Eventually, Algorithm 1 has complexity O (MM( m )M( p m (deg( A ) + m ))) . (cid:3) On the possible ramification indexes of a gauge transformation.
Recallthat, from (4), D is the set of d ∈ { , . . . , p m − } such that p and d are relativelyprime. The aim of this subsection is to prove the following result. Proposition 5.
Assume that the system (1) is regular singular at with an associatedgauge transformation Ψ ∈ GL m ( K ) . Then, the matrix Ψ belongs to GL m (cid:0) Q (( z /d )) (cid:1) for some d ∈ D . Furthermore, Algorithm 2 below computes such an integer d . COLIN FAVERJON AND MARINA POULET
Let(7) Λ := φ p (Ψ) − A Ψ ∈ GL m ( Q )denote the constant matrix which is K -equivalent to A . Without loss of generality, weassume that Λ has a Jordan normal form that isΛ := J s ( λ ) J s ( λ ) . . . J s r ( λ r ) where λ , . . . , λ r are algebraic numbers and J s i ( λ i ) is the Jordan block of size s i asso-ciated to the eigenvalue λ i . Let ψ , , . . . , ψ ,s , ψ , , . . . , ψ ,s , . . . , ψ r, , . . . , ψ r,s r denote the columns of Ψ indexed according to the Jordan normal form of Λ. From (7),for any i, j ∈ N such that 1 ≤ i ≤ r , 2 ≤ j ≤ s i one has(8) λ i φ p ( ψ i, ) = A ψ i, , and λ i φ p ( ψ i,j ) + φ p ( ψ i,j − ) = A ψ i,j . Before proving Proposition 5 we need two lemmas about the solutions of homogeneousand inhomogeneous linear Mahler equations.
Lemma 6.
For any system (1) , there exists an integer d ∈ D such that for any λ ∈ Q ⋆ and f ∈ K m satisfying (9) λφ p ( f ) = A f we have f ∈ Q (cid:0)(cid:0) z /d (cid:1)(cid:1) m .Proof. By Theorem 3 there exists P ∈ GL m ( Q ( z )) such that A comp = φ p ( P ) AP − isa companion matrix, that is,(10) A comp = · · · · · · · · · q · · · · · · · · · q m − where q , . . . , q m − ∈ Q ( z ) . We let f , λ be as in the lemma. We have λφ p ( P f ) = A comp P f . Substituting P f to f , we might assume that A = A comp . Let f , . . . , f m ∈ K be theentries of f . Since A is a companion matrix, for every i ∈ { , . . . , m − } , f i +1 = λφ p ( f i ) . Thus, in order to prove that f ∈ Q (cid:0)(cid:0) z /d (cid:1)(cid:1) m , it is enough to prove that f ∈ Q (( z /d )).It follows from (9) and (10) that f is a solution of the Mahler equation(11) q y + λq φ p ( y ) + · · · + λ m − q m − φ m − p ( y ) − λ m φ mp ( y ) = 0 . From [CDDM18, Prop. 2.19], there exists an integer d ∈ D such that any solution y ∈ K of (11) belongs to Q (cid:0)(cid:0) z /d (cid:1)(cid:1) . Furthermore, as shown in [CDDM18], this integer d depends only on the valuation of the rational functions which are the coefficients of N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS λ . Precisely, let H ∈ R denote the lower hull of the set { ( p i , v ( q i ) } ∪ { ( p m , } . The integer d can be taken to be the least common multiple of the denominators ofthe slopes of H which are prime together with p . We fix such an integer d . Therefore, f ∈ Q (( z /d )) and d does not depend on λ . (cid:3) Lemma 7.
Consider a system (1) and let d be the integer obtained in the Lemma 6for this system. If g ∈ Q (( z /d )) m , λ ∈ Q ⋆ and f ∈ K m satisfy (12) φ p ( f ) λ + g = A f then f ∈ Q (( z /d )) m .Proof. By Theorem 3, there exists P ∈ GL m ( Q ( z )) such that A comp = φ p ( P ) AP − isa companion matrix of the form (10). We let f , g , λ be as in the lemma. We have φ p ( P f ) λ + φ p ( P ) g = A comp P f so up to replace P f with f and φ p ( P ) g with g , we might assume that A = A comp .Since A is a companion matrix, for every i ∈ { , . . . , m − } , f i +1 = g i + λφ p ( f i ) . Thus, it is enough to prove that f ∈ Q (( z /d )). It follows from (10) and (12) that(13) q f + λq φ p ( f ) + · · · + λ m − q m − φ m − p ( f ) − λ m φ mp ( f ) = g , where g ∈ Q (cid:0)(cid:0) z /d (cid:1)(cid:1) is a Q ( z )-linear combination of the φ jp ( g i ), i ∈ { , . . . , m } , j ∈ { , . . . , m − } . Assume that f does not belong to Q (cid:0)(cid:0) z /d (cid:1)(cid:1) and let h be the sumof all the monomials in the Puiseux series development of f whose power cannot bewritten as k/d , with k ∈ Z . Since gcd( d, p ) = 1, none of the monomials of the powerseries φ ℓp ( h ), ℓ ∈ N , belong to Q (cid:0)(cid:0) z /d (cid:1)(cid:1) . On the other hand, since f − h ∈ Q (cid:0)(cid:0) z /d (cid:1)(cid:1) , φ ℓp ( f − h ) ∈ Q (cid:0)(cid:0) z /d (cid:1)(cid:1) for every ℓ ∈ N . It thus follows from (13) that h is a solutionof (11), the Mahler homogeneous equation associated with (13). From Lemma 6, h ∈ Q (cid:0)(cid:0) z /d (cid:1)(cid:1) , a contradiction. (cid:3) The following algorithm computes an integer d ∈ D satisfying the properties ofLemma 6. Algorithm 2:
The integer d Input: A , p . Output:
An integer d satisfying the properties of Lemma 6Compute the last row ( q , . . . , q m − ) of the matrix A comp with Algorithm 1.Compute the lower hull H of the set { ( p i , v ( q i ) } ∩ { ( p m , } .Compute S the set of denominators of the slopes of H which are primetogether with p . return lcm( S ).We can now prove Proposition 5.0 COLIN FAVERJON AND MARINA POULET
Proof of Proposition 5.
Let d denote the integer given by Lemma 6. Fix an integer i ∈ { , . . . , r } . We prove by induction on j ∈ { , . . . , s i } that ψ i,j ∈ Q (cid:0)(cid:0) z /d (cid:1)(cid:1) m forevery j ∈ { , . . . , s i } . From (8), the vector ψ i, satisfies λ i φ p ( ψ i, ) = A ψ i, . Thus, from Lemma 6, ψ i, ∈ Q (cid:0)(cid:0) z /d (cid:1)(cid:1) m . Fix an integer j ∈ { , . . . , s i } and assumethat ψ i,j − ∈ Q (cid:0)(cid:0) z /d (cid:1)(cid:1) m . From (8), ψ i,j satisfies λ i φ p ( ψ i,j ) + φ p ( ψ i,j − ) = A ψ i,j thus it follows from Lemma 7 that ψ i,j ∈ Q (cid:0)(cid:0) z /d (cid:1)(cid:1) m , as desired. (cid:3) A lower bound for the valuation of a gauge transformation.
Assume thatthe system (1) is regular singular at 0. Let Ψ ∈ GL m ( K ) such that φ p (Ψ) − A Ψ is aconstant matrix. We let d ∈ D denote an integer such that the entries of Ψ belong to Q (( z /d )). WriteΘ( z ) := φ d (Ψ)( z ) = X n ≥ v (Θ) Θ n z n and A ( z ) =: X n ≥ v ( A ) A n z n , where Θ n , A n ∈ M n (cid:0) Q (cid:1) . By definition, v (Θ) = v ( φ d (Ψ)). We give a lower boundfor v (Θ), the valuation at 0 of Θ. It follows from the identity φ p (Θ) = φ d ( A )ΘΛ − that pv (Θ) ≥ v (Θ) + dv ( A ) . Hence v (Θ) ≥ dv ( A ) p − thus v (Θ) ≥ ν d where ν d = ⌈ dv ( A ) / ( p − ⌉ is defined in (2). It follows that we have a decompositionΨ( z ) := P n ≥ ν d Θ n z n/d where Θ ν d might be a zero matrix. Eventually, we just provedthat the valuation at 0 of the entries of φ d (Ψ) are greater than ν d .4. Proof of Theorem 2
This section is devoted to the proof of Theorem 2. If the system is regular singularat 0, we describe how to compute a gauge transformation Ψ such that φ p (Ψ) − A Ψ isconstant with Jordan normal form. Of course, a delicate point in such a construction isthat we have to compute the eigenvalues of this Jordan normal form matrix. Moreover,the gauge transformation Ψ appears with coefficients in an algebraic extension of thefield spanned by the coefficients of the entries of A , which is not optimal.4.1. Equations satisfied by the columns of a gauge transformation.
We recallthat the Mahler system (1) is regular singular at 0 if and only if there exist matricesΨ ∈ GL m ( K ) and Λ ∈ GL m ( Q ) such that φ p (Ψ) Λ = A Ψ . Assume that the system (1) is regular singular at 0 with gauge transformation Ψ anda constant matrix Λ. One has(14) ΨΛ − = A − φ p (Ψ)Then, from Proposition 5, Ψ belongs to GL m ( Q (( z /d ))) for some d ∈ D . We fix suchan integer d and we apply φ d to (14):(15) ΘΛ − = Bφ p (Θ) N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS φ d (Ψ) and B := φ d ( A − ). These matrices have Laurent series developmentΘ( z ) = X n ≥ ν d Θ n z n and B ( z ) = X n ≥ v ( B ) B n z n , where Θ n , B n ∈ M m (cid:0) Q (cid:1) and ν d = ⌈ dv ( A ) / ( p − ⌉ . Remark . We infer from (15) that for all n ∈ Z ,(16) Θ n Λ − = X ( k,l ) : k + pℓ = n B k Θ ℓ . If n > − dv (cid:0) A − (cid:1) / ( p −
1) then the Θ ℓ which are taken into account in the right handside of (16) are the one for which ℓ < n . Therefore, the sequence (Θ n ) n ≥ ν d is uniquelydetermined by the matrices Θ ℓ with ν d ≤ ℓ ≤ µ d , where µ d = ⌈− dv (cid:0) A − (cid:1) / ( p − ⌉ is defined in (2).Assume that the matrix Λ − has a Jordan normal formΛ − := J s ( γ ) J s ( γ ) . . . J s r ( γ r ) where γ , γ , · · · , γ r are algebraic numbers and J s i ( γ i ) is the Jordan block of size s i associated with the eigenvalue γ i . Let θ , , . . . , θ ,s , θ , , . . . , θ ,s , . . . , θ r, , . . . , θ r,s r ∈ Q (( z )) m denote the columns of Θ indexed with respect to the Jordan normal form of Λ − . Weinfer from (15) that the columns θ i,j satisfy(17) γ i θ i, = Bφ p ( θ i, ) γ i θ i,j + θ i,j − = Bφ p ( θ i,j ) j ≥ . Computing the column’s candidates.
In this section we do not assume any-more that the system is regular singular at 0. We fix the integer d ∈ D given byAlgorithm 2. We recall that if the Mahler system (1) is regular singular at 0 then theentries of a gauge transformation Ψ belong to Q (cid:0)(cid:0) z /d (cid:1)(cid:1) . Therefore, instead of lookingfor a gauge transformation with entries in Q (cid:0)(cid:0) z /d (cid:1)(cid:1) , we shall apply φ d to the Mahlersystem and look for a gauge transformation with entries in Q (( z )). Let B := φ d (cid:0) A − (cid:1) . We will show how to build a matrix U with entries in Q (( z )) and with as many linearlyindependent columns as possible, together with a constant square matrix C such that U C = Bφ p ( U ) . Our matrix U may not be a square matrix. As we shall see, it will be if and only ifthe system (1) is regular singular at 0. In that case, setting Ψ := φ /d ( U ) we will have φ p (Ψ) − A Ψ = C − ∈ GL m ( Q ), as wanted. It follows from (17) that to determinethe columns of this matrix U , one has to solve some homogeneous and inhomogeneousMahler systems. In the mean time, we infer from Remark 4 that this can be doneby solving some equations in a finite number of the coefficients of the Puiseux series2 COLIN FAVERJON AND MARINA POULET development of the columns of U . This section is devoted to the construction of thepair ( U, C ).Let M := M d and N := N d be defined as in section 2. We define a map π d : Q (( z )) m → Q m ( µ d − ν d +1) g ( z ) = P n ∈ Z g n z n g ν d ... g µ d .If γ is an eigenvalue of M , for all positive integer j , we consider the Q -vector space V γ,j := ker (cid:16) ( M − γI ) j (cid:17) ∩ j − \ k =0 ker (cid:16) N ( M − γI ) k (cid:17)! . We set V γ, := { } .4.2.1. Solving the homogeneous equations.
Lemma 8.
Let γ ∈ Q . If the equation (18) γ f = Bφ p ( f ) has a nonzero solution then γ = 0 . In that case, the map π d induces a bijection betweenthe Q vector-space of solutions f ∈ Q (( z )) m of (18) and V γ, . Furthermore, given anelement y ∈ V γ, , there is a recurrence relation to determine the coefficients of theLaurent series development of the vector f ∈ Q (( z )) m such that π d ( f ) = y . Note that there might be no nonzero solution of (18), even when γ is a nonzeroeigenvalue of M . More precisely, from this lemma, there exists a nonzero solution of(18) if and only if γ is a nonzero eigenvalue of M which satisfies dim ( V γ, ) ≥ Proof.
Since the matrix B is nonsingular, we must have γ = 0. It follows from theresults in Section 3.3 that any solution in Q (( z )) m of (18) has a valuation at 0 greaterthan ν d . Let f ∈ Q (( z )) m with valuation greater than ν d and set f = X n ≥ ν d f n z n , f n ∈ Q m for n ≥ ν d . The function f satisfies (18) if and only if ∀ n ∈ Z , γ f n = X ( k,l ) : k + pℓ = n B k f ℓ , where we set f n = 0 if n < ν d . This is equivalent to N π d ( f ) = 0 (coming from the cases n < ν d ) M π d ( f ) = γπ d ( f ) (coming from the cases ν d ≤ n ≤ µ d ) ∀ n ≥ µ d + 1 , f n = γ (cid:16)P ( k,l ) : k + pℓ = n B k f ℓ (cid:17) . The first and the second equality imply that π d ( f ) ∈ V γ, . As explained in Remark4, from the third equality in this system, we deduce that the sequence ( f n ) n ≥ ν d isuniquely determined by the vectors f ℓ with ν d ≤ ℓ ≤ µ d , i.e. it is uniquely determinedby π d ( f ). Therefore, f is a solution of (18) if and only if π d ( f ) ∈ V γ, . Furthermore,if f and g are two solutions of (18) such that π d ( f ) = π d ( g ) then f = g . Since any y ∈ V γ, is the image of some f ∈ Q (( z )) m of valuation greater than ν d , the lemma isproved. (cid:3) N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS
Solving the inhomogeneous equations.
Lemma 9.
Let γ ∈ Q ⋆ and j ≥ an integer. Let g ∈ Q (( z )) m be a Laurent powerseries with valuation greater than ν d and such that π d ( g ) ∈ V γ,j . The map π d inducesis a bijection between the affine space of solutions f ∈ Q (( z )) m of (19) γ f + g = Bφ p f , and the affine space of solutions y ∈ V γ,j +1 of ( M − γI ) y = π d ( g ) . Proof.
Set g = X n ≥ ν d g n z n , g n ∈ Q m , From Section 3.3, any solution of (19) has a valuation greater than ν d . Let f = X n ≥ ν d f n z n ∈ Q (( z )) m , f n ∈ Q m . If n < ν d , we set f n := 0 and g n := 0. The power serie f is a solution of (19) if andonly if ∀ n ∈ Z , γ f n + g n = X ( k,l ) : k + pℓ = n B k f ℓ . This is equivalent to
N π d ( f ) = 0 (from the cases n < ν d ) M π d ( f ) = γπ d ( f ) + π d ( g ) (from the cases ν d ≤ n ≤ µ d ) ∀ n ≥ µ d + 1 , f n = γ (cid:16)P ( k,l ) : k + pℓ = n B k f ℓ − g n (cid:17) The second equality implies that ( M − γI ) π d ( f ) ∈ V γ,j . Hence, ( M − γI ) j +1 π d ( f ) = 0and thus π d ( f ) ∈ ∩ jk =1 ker( N ( M − γI ) k ). Eventually, the first equality implies that π d ( f ) ∈ ker( N ) = ker( N ( M − γI ) ). It follows that π d ( f ) ∈ V γ,j +1 . From Remark 4, asequence ( f n ) n ≥ ν d verifying the third equation is uniquely determined by π d ( f ). Thus f is a solution of (19) if and only if π d ( f ) ∈ V γ,j +1 satisfies ( M − γI ) π d ( f ) = π d ( g ),as wanted. Furthermore, π d ( f ) is uniquely determined by f . (cid:3) Computing each column.
We will now select some columns for the matrix U . Lemma 10.
Let γ ∈ Q ⋆ . We have a chain of vector spaces V γ, := { } ⊂ V γ, ⊂ V γ, ⊂ · · · ⊂ V γ,j ⊂ · · · ⊂ Q m ( µ d − ν d +1) . Set a ( γ ) := min { n ∈ N ⋆ | V γ,n = V γ,n +1 } . Then, V γ,j = V γ,a ( γ ) for all j ≥ a ( γ ) . Moreover, for i ∈ { , . . . , a ( γ ) } , if k γ,j := dim( V γ,j ) − dim( V γ,j − ) . then < k γ,a ( γ ) ≤ k γ,a ( γ ) − ≤ · · · ≤ k γ, ≤ k γ, . COLIN FAVERJON AND MARINA POULET
Proof.
Let j ∈ N and v ∈ V γ,j . Then, ( M − γI ) j v = 0. Thus, v ∈ ker( N ( M − γI ) j )and ( M − γI ) j +1 v = 0. This proves the first point. The second point is obvious. Inorder to prove the last point, fix an integer j ∈ { , . . . , a ( γ ) − } and consider the linearmap ψ j : V γ,j +1 → V γ,j /V γ,j − x ( M − γI ) x .Its kernel is V γ,j so the induced map ψ j : V γ,j +1 /V γ,j → V γ,j /V γ,j − is injective. There-fore, taking the dimensions, we have ∀ j ∈ { , . . . , a ( γ ) − } , k γ,j +1 ≤ k γ,j . (cid:3) Lemma 11.
We use the notations of Lemma 10. For all j ∈ { , . . . , a ( γ ) } , there existsa subspace W γ,j of V γ,j such that V γ,j = V γ,j − ⊕ W γ,j and such that the map W γ,j → W γ,j − x ( M − γI ) x is injective, where we set W γ, := V γ, .Proof. The proof is similar to a proof of Jordan theorem for nilpotent matrices usingYoung tableau. We first choose a basis v ( a ( γ ))1 , . . . , v ( a ( γ )) k γ,a ( γ ) of the vector space W γ,a ( γ ) .Let 0 ≤ ℓ ≤ a ( α ) −
1. Assume that we have chosen a basis v ( a ( γ ) − ℓ )1 , . . . , v ( a ( γ ) − ℓ ) k γ,a ( γ ) − ℓ of W γ,a ( γ ) − ℓ . We set M γ := M − γ i I and v ( a ( γ ) − ℓ − := M γ v ( a ( γ ) − ℓ )1 , . . . , v ( a ( γ ) − ℓ − k γ,a ( γ ) − ℓ := M γ v ( a ( γ ) − ℓ ) k γ,a ( γ ) − ℓ , and we complete it with vectors v ( a ( γ ) − ℓ − k γ,a ( γ ) − ℓ +1 , . . . , v ( a ( γ ) − ℓ − k γ,a ( γ ) − ℓ − to form a basis of thevector space W γ,a ( γ ) − ℓ − . In the end, we have the following Young tableau, where the j th column is a basis of W γ,j . v (1)1 := M a ( γ ) − γ v ( a ( γ ))1 · · · v ( a ( γ ) − := M γ v ( a ( γ ))1 v ( a ( γ ))1 ... · · · ... ... v (1) k γ,a ( γ ) := M a ( γ ) − γ v ( a ( γ )) k γ,a ( γ ) · · · v ( a ( γ ) − k γ,a ( γ ) := M γ v ( a ( γ )) k γ,a ( γ ) v ( a ( γ )) k γ,a ( γ ) v (1) k γ,a ( γ ) +1 := M a ( γ ) − γ v ( a ( γ ) − k γ,a ( γ ) +1 · · · v ( a ( γ ) − k γ,a ( γ ) +1 ... · · · ... v (1) k γ,a ( γ ) − := M a ( γ ) − γ v ( a ( γ ) − k γ,a ( γ ) − · · · v ( a ( γ ) − k γ,a ( γ ) − ... ... v (1) k γ, +1 ... v (1) k γ, (cid:3) N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS { γ , . . . , γ t } be the set of all algebraic numbers γ for which (18) has a solution.From Lemma 8, it is the set of nonzero eigenvalue γ of M which satisfy dim ( V γ, ) ≥ i ∈ { , . . . , t } , we recall that V γ i ,j := ker (cid:16) ( M − γ i I ) j (cid:17) ∩ j − \ k =0 ker (cid:16) N ( M − γ i I ) k (cid:17)! . We decompose V γ i ,a ( γ i ) = a ( γ i ) M j =1 W γ i ,j as in Lemma 11 and for each j we consider the basis (cid:16) v ( j ) γ i , , . . . , v ( j ) γ i ,k γi,j (cid:17) of W γ i ,j constructed as in the Young tableau of Lemma 11. Thus, { v ( j ) γ i ,ℓ : 1 ≤ j ≤ a ( γ i ) , ≤ ℓ ≤ k γ i ,j } is a basis of V γ i ,a ( γ i ) . For every ℓ ∈ { , . . . , k γ i , } we let u (1) γ i ,ℓ ∈ Q (( z )) mν d denote theunique solution of γ i f = Bφ p ( f )such that π d (cid:16) u (1) γ i ,ℓ (cid:17) = v (1) γ i ,ℓ . The existence and unicity of this solution follows fromLemma 8. Now, we define recursively on j ∈ { . . . , a ( γ i ) } vectors u ( j ) γ i ,ℓ ∈ Q (( z )) mν d ,with 1 ≤ ℓ ≤ k γ i ,j , solutions of γ i f + u ( j − γ i ,ℓ = Bφ p ( f )and such that π d (cid:16) u ( j ) γ i ,ℓ (cid:17) = v ( j ) γ i ,ℓ . The existence and unicity of such vectors followsfrom Lemma 9. In fine , we have constructed vectors(20) u (1) γ i , , . . . , u (1) γ i ,k γi, , u (2) γ i , , . . . , u (2) γ i ,k γi, , . . . , u ( a ( γ i )) γ i , , . . . , u ( a ( γ i )) γ i ,k γia ( γi ) , such that π d ( u ( j ) γ i ,ℓ ) = v ( j ) γ i ,ℓ for every ( j, ℓ ) ∈ { , . . . , a ( γ i ) }×{ , . . . , k γ i ,j } . In particular,the family (20) is linearly independent over Q .4.2.4. Building the matrix.
Let U i be the matrix whose columns are the elements in(20) re-ordered using the lexical order on the indices, that is: u (1) γ i , , u (2) γ i , , . . . , u (1) γ i , , u (2) γ i , , . . . , u (1) γ i ,k γi, , . . . As already noticed, the columns of U i are linearly independent. Let C i be the normalJordan form matrix whose Jordan blocks are J γ i ( s i, ) , · · · , J γ i ( s i,k )where we let s i,ℓ denote the number of column vector of the form u ( . ) γ i ,ℓ (that is thenumber of columns in the ℓ th row of the above table). For example, s i, = a ( γ i ). Then, U i C i = Bφ p ( U i ) . COLIN FAVERJON AND MARINA POULET
We let U := ( U | . . . | U t ) denote the matrix whose columns are the ones of the U i and C denote the Jordan normal form matrix whose blocks are C , . . . , C t that is C := C . . . C t . The matrix U is not necessarily a square matrix but it follows from our constructionthat(21) U C = Bφ p ( U ) . Consider a pair of matrices ( U ′ , C ′ ) such that U ′ C ′ = Bφ p ( U ′ ). It follows from Lem-mas 8 and 9 that U ′ has less columns than U . Furthermore, since the vector spaces V γ ,a ( γ ) , . . . , V γ t ,a ( γ t ) are in direct sum, the columns of U - which are the elements of(20) - are linearly independent over Q . It follows from the following lemma that theyare actually linearly independent over Q (( z )) Lemma 12.
Let T be a matrix with entries in Q (( z )) and D be a constant matrixsuch that (22) T D = Bφ p ( T ) . If the columns of T are linearly dependent over Q (( z )) then they are linearly dependentover Q .Proof. We might assume, without loss of generality, that D is upper triangular. Let a ≥ t , . . . , t a are linearly dependent over Q (( z )). Then there exists a column vector g =: ( g , . . . , g a − , , , . . . , ⊥ ∈ Q (( z )) m such that(23) T g = 0 . Mutliplying (22) by φ p ( g ) one obtains(24) T Dφ p ( g ) = Bφ p ( T ) φ p ( g ) = Bφ p ( T g ) = 0 . Since D is upper triangular, only the a first coordinates of Dφ p ( g ) can be nonzero andits a th coordinate is a some eigenvalue η ∈ Q of D . By minimality of a , we infer from(23) and (24) that Dφ p ( g ) = η g . From [Nis97, Thm. 3.1], g ∈ Q m so (23) gives a Q -linearly dependence of the columnsof T , as wanted. (cid:3) A new characterisation of the regular singularity at . The followingproposition states that U is a square matrix if and only if the system (1) is Q (cid:0)(cid:0) z d (cid:1)(cid:1) -equivalent to a constant system. This give a characterisation of Mahler systems thatare regular singular at 0. Proposition 13.
The system (1) is Q (cid:0)(cid:0) z d (cid:1)(cid:1) -equivalent to a constant system if andonly if the number of columns of U is greater than m . In that case, it is equal to m and, setting Ψ := φ /d ( U ) ∈ GL m (cid:0) Q (cid:0)(cid:0) z /d (cid:1)(cid:1)(cid:1) , one has (25) φ p (Ψ) − A Ψ = C − ∈ GL m (cid:0) Q (cid:1) . N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS Proof.
Since the columns of U are linearly independent over Q (( z )), its number is lessthan m . Assume that U is a square matrix. Then, it is nonsingular, and equality (25)follows from (21) and the system (1) is Q (cid:0)(cid:0) z d (cid:1)(cid:1) -equivalent to a constant system.Assume now that the system (1) is Q (cid:0)(cid:0) z d (cid:1)(cid:1) -equivalent to a constant system withgauge transformation Ψ ∈ GL m (cid:0) Q (cid:0)(cid:0) z /d (cid:1)(cid:1)(cid:1) . Since the columns of Ψ are Q -linearlyindependent, it follows from Remark 4 that their images under π d are Q -linearly inde-pendent. Thus, from (17) they belong to the vector space V γ ,a ( γ ) ⊕ · · · ⊕ V γ t ,a ( γ t ) . Thus, the number of columns of U , which is the dimension of this vector space, isgreater than m , as wanted. (cid:3) Therefore, Proposition 13 gives a first algorithm to test whether or not a system isregular singular at 0: it can be done by computing the solutions of some homogeneousand inhomogeneous Mahler systems. However, this algorithm in not necessarily veryefficient, since one has to find eigenvalues and eigenvectors for M .4.4. An equality of vector space.
We let U denote the Q -vector space spanned bythe columns of U , that is U := span Q n u ( j ) γ i ,ℓ : 1 ≤ i ≤ t, ≤ j ≤ a ( γ i ) , ≤ ℓ ≤ k γ i ,j o , and set(26) V = V γ ,a ( γ ) ⊕ · · · ⊕ V γ t ,a ( γ t ) From Lemma 8 and Lemma 9, π d induces an isomorphism between these two vectorspaces. Recall that we set X := X d := \ n ∈ Z M nd ker( N d ) . We have the following equality of vector spaces.
Lemma 14.
We have V = X . Proof.
We first prove that V ⊂ X . From (26) is it enough to prove that V γ i ,a ( γ i ) ⊂ X for every i ∈ { , . . . , t } . Fix an integer i ∈ { , . . . , t } . We recall that for all k ≥ a ( γ i ), V γ i ,a ( γ i ) = V γ i ,k := ker (cid:16) ( M − γ i I ) k (cid:17) ∩ k − \ j =0 ker (cid:16) N ( M − γ i I ) j (cid:17) . One can check that V γ i ,k = ker (cid:16) ( M − γ i I ) k (cid:17) ∩ k − \ j =0 ker (cid:0) N M j (cid:1) . Therefore, for all n ∈ N , M n V γ i ,a ( γ i ) ⊂ ker( N ). Then, we prove by induction on n ∈ N that for any v ∈ V γ i ,a ( γ i ) , v ∈ M n ker( N ). It is clear for n = 0. Assume that v ∈ M ℓ ker( N ) for all ℓ ≤ n . Since v ∈ V γ i ,a ( γ i ) , for all k ≥ a ( γ i ),( M − γ i I ) k v = 0 . Thus v is a Q -linear combination of M v , . . . , M k v . Moreover,8 COLIN FAVERJON AND MARINA POULET • if j ∈ { , . . . , n + 1 } then M j v ∈ M n +1 ker( N ) because by the inductionhypothesis v ∈ M n +1 − j ker( N ) ; • if j ≥ n + 1, M j v ∈ M n +1 ker( N ) because M j v = M n +1 .M j − ( n +1) v and,previously, we showed that M j − ( n +1) v ∈ ker( N ).Therefore, v ∈ M n +1 ker( N ). It follows that V γ i ,a ( γ i ) ⊂ X .We now prove that X ⊂ V . Let e , . . . , e n denote a basis of X and let E be the m ( µ d − ν d + 1) × n matrix whose columns are the e , . . . , e n . Since X is M -invariant,there exists R ∈ M n (cid:0) Q (cid:1) such that(27) M E = ER.
The matrix R is invertible because ker( M ) ∩ X = 0. Up to change the basis e , . . . , e n we might assume that R has a Jordan normal form. Thus, (27) implies that the setof eigenvalues of R is included in the set of eigenvalue of M . Let J γ ( s ) be the firstJordan block of R . Then, using the fact that the columns of E belong to ker( N ), onecan prove by induction on j ∈ { , . . . , s } that e j ∈ V γ,j . Doing this for all the Jordanblocks of J , we obtain that the columns of E belong to V , as wanted. (cid:3) End of the proof of Theorem 2.
There is not much left to do. Assume thatthe system (1) is regular singular at 0. Then, it follows from Proposition 5 that thereexists a gauge transformation Ψ with entries in Q (cid:0)(cid:0) z d (cid:1)(cid:1) such that φ p (Ψ) − A Ψ is aconstant matrix, where d ∈ D is given by Algorithm 2. Thus, from Proposition 13,the matrix U has m linearly independent columns, that is, dim U = dim V = m . But,from Lemma 14, V = X . Thus X has dimension m .Assume now that X := X d has dimension m for some d ∈ D . Then, from Lemma 14,dim U = dim V = m . Thus, the matrix U has m columns and it follows from Proposi-tion 13 that the system is regular singular and that Ψ = φ /d ( U ) ∈ GL m (cid:0) Q (cid:0)(cid:0) z /d (cid:1)(cid:1)(cid:1) is such that φ p (Ψ) − A Ψ is a constant matrix.5.
The algorithm of Theorem 1
Theorem 2 gives the description of a vector space whose dimension characterises theregular singularity at 0 of a Mahler system. However, if the system is regular singular at0, this theorem does not tell how to build a matrix Ψ such that φ p (Ψ) − A Ψ ∈ GL m (cid:0) Q (cid:1) .Such a construction has been done with the help of the matrix U in section 4. However,the interest of this construction is more heuristic than effective for it necessitates theextraction of roots of polynomials. In this section we describe an algorithm to computethe matrix Ψ which does not require the determination of eigenvalues. In fine , thecoefficients of the Puiseux series defining the matrix Ψ belong to the same numberfield as the coefficients of the rational functions defining the matrix A .5.1. A direct construction of a gauge transformation.
Assume that the Mahlersystem (1) is regular singular at 0. From Theorem 2, there exists an integer d ∈ D such that the dimension of X d is equal to m . Let e , . . . , e m be a basis of X d and let E be the m ( µ d − ν d + 1) × m matrix whose columns are e , . . . , e m . As in (27), we have M d E = ER N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS R ∈ GL m (cid:0) Q (cid:1) . We make a block decomposition of the columns of E into µ d − ν d + 1 matrices E ν d , . . . , E µ d ∈ M m (cid:0) Q (cid:1) : E = E ν d ... E µ d . We then define by induction on n > µ d a matrix E n ∈ M m (cid:0) Q (cid:1) setting(28) E n = X ( k,l ) : k + pℓ = n B k ( d ) E ℓ R − where B ( d ) = P n ∈ Z B n ( d ) z n = φ d ( A ) − . As seen in Remark 4, when n > µ d , thematrices E ℓ contributing to right-hand side of the equality are those for which ℓ < n .Hence the matrices E n are well-defined. If n < ν d , we set E n = 0. We stress that (28)holds for any n ∈ Z : • by construction, it holds when n > µ d ; • when ν d ≤ n ≤ µ d , it follows from the fact that ER = M d E ; • when n < ν d , it follows from the fact that N d E = 0, for X ⊂ ker( N d ).We eventually set U := P n ≥ ν d E n z n . It follows from (28) that(29) U R = B ( d ) φ p ( U ) . Since e , . . . , e m is a basis of X d , the columns of U are linearly independent over Q .It follows from Lemma 12 that they are linearly independent over Q (( z )). Thus, thematrix U is nonsingular. Now, set Ψ = φ /d ( U ). It follows from (29) that(30) φ p (Ψ) − A Ψ = R − ∈ GL m (cid:0) Q (cid:1) . Thus, Ψ is the gauge transformation we are looking for.5.2.
On the base field of a gauge transformation.
The above construction givesinformation about the field to which the coefficients of the Puiseux series developmentof the entries of Ψ belong.
Proposition 15.
Consider a Mahler system (1) and let K denote a number fieldcontaining the coefficients of the rational functions defining the matrix A . Assumethat (1) is Q (cid:0)(cid:0) z d (cid:1)(cid:1) -equivalent to a constant system, for some integer d ∈ N . Then thesystem is K (cid:0)(cid:0) z /d (cid:1)(cid:1) -equivalent to a constant system. Therefore, any Mahler system defined over some number field K which is regularsingular at 0 is equivalent to a constant matrix with entries in K . The entries of thegauge transformation are Puiseux series with coefficients in K . Proof.
From Theorem 2, the dimension of the Q -vector space X d equals m . Since M d and N d have entries in K , X d is defined over K . Hence, the basis e , . . . , e m of X d canbe chosen in K m ( µ d − ν d +1) . It follows that the matrix R and the matrices E n , n ≥ ν d have their entries in K . As a consequence the matrixΨ( z ) = X n ≥ ν d E n z n/d belongs to GL m (cid:0) K (cid:0)(cid:0) z /d (cid:1)(cid:1)(cid:1) . (cid:3) COLIN FAVERJON AND MARINA POULET
Remark . The proof of Theorem 2 and the construction of the gauge transformationin 5.1 do not rely on the base field on which the entries of A are defined. It does nothave to be a subfield of Q . Actually, this field could even have positive characteristic.5.3. Description of the algorithm.
We now describe precisely the algorithm ofTheorem 1. We first present an algorithm to determine whether or not a Mahlersystem (1) is Q (cid:0)(cid:0) z /d (cid:1)(cid:1) -equivalent to a constant system for some fixed integer d ∈ N . Algorithm 3:
Test to determine if a Mahler system (1) is Q (cid:0)(cid:0) z /d (cid:1)(cid:1) -equivalentto a constant system. Input: A , p , and d . Output:
Whether or not the system (1) is Q (cid:0)(cid:0) z /d (cid:1)(cid:1) -equivalent to a constantsystem. In that case the constant matrix Λ and enough coefficients ofthe Puiseux development of the associated gauge transformation sothat one can compute recursively the whole development.Set m the size of A .Compute ν d , µ d , M d , N d .Set X := ker N d and Y = N d . for i from 0 to m ( µ d − ν d + 1) do X := X ∩ M id X . for i from 0 to m ( µ d − ν d + 1) do Y := Y M d ; X := X ∩ ker( Y ). if dim X = m then From a basis of X , compute R and E ν d , . . . , E µ d as in section 4.Set Λ := R − . return “ True ”, Λ and E ν d , . . . , E µ d .The following lemma implies that, at the end of Algorithm 3, the vector space X isthe same as in Theorem 2. Lemma 16.
Set c d := m ( µ d − ν d + 1) . We have \ − c d ≤ n ≤ c d M nd ker( N d ) = \ n ∈ Z M nd ker( N d ) . Proof.
We denote M := M d and N := N d . Set J n = ∩ nk =0 M − k ker ( N ). It is immediatethat if J n = J n +1 then J ℓ = J n for all ℓ ≥ n . Therefore, ∩ ∞ k =0 M − k ker ( N ) = J c d . Let I n = ∩ nk = − c d M k ker ( N ) = ∩ nk = −∞ M k ker ( N ). Since c d is the size of the matrix M we have ker ( M c d ) = ker (cid:0) M ℓ (cid:1) for every ℓ ≥ c d . Furthermore dim (ker ( N )) ≤ c d .We shall prove that I k = I k +1 for all k ≥ c d . The only nontrivial inclusion is I k ⊂ I k +1 .Fix a integer k ≥ c d . We highlight two facts. • The Q -vector space I k is stable by M . • I k ∩ ker (cid:0) M k (cid:1) = { } . Indeed, if x ∈ I k ∩ ker (cid:0) M k (cid:1) then there exists y ∈ ker ( N )such that x = M y and M k +1 y = M k x = 0. Thus, y ∈ ker (cid:0) M k +1 (cid:1) = ker (cid:0) M k (cid:1) and x = M k y = 0.Let x ∈ I k be nonzero. The vectors x , M x , . . . , M k x ∈ I k are linearly dependant.Thus, there exist a , . . . , a k ∈ Q not all zero such that a x + a M x + · · · + a k M k x = 0 . N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS i := min ( i ∈ { , . . . , k } | a i = 0) and y = − P kj = i +1 a j a i M j − i − x ∈ I k so that M i x = M i +1 y . We have x − M y ∈ I k ∩ ker (cid:0) M k (cid:1) and it follows that x = M y ∈ M I k ∩ I k ⊂ I k +1 , as wanted. (cid:3) Proposition 17.
Algorithm 3 tests whether or not system (1) is Q (cid:0)(cid:0) z /d (cid:1)(cid:1) -equivalentto a constant system.Proof. From Lemma 16, Algorithm 3 computes the vector space X d := \ n ∈ Z M nd ker( N d ) . From Lemma 14, dim X d = dim V = dim U . Thus, it follows from Proposition 13 thatthe system (1) is Q (( z /d ))-equivalent to a constant system if and only if dim X d = m .This is precisely what Algorithm 3 tests. (cid:3) We can then describe an Algorithm for Theorem 1.
Algorithm 4:
Test for the regularity singularity at 0 of a Mahler system
Input: A , p , and n ≥
0, the order of truncation.
Output:
If the system (1) is regular singular at 0 and in that case theconstant matrix Λ to which it is equivalent and a truncation of theassociated gauge transformation Ψ at order n .Compute d with Algorithm 2.Run Algorithm 3 with that d . if Algorithm 3 returns “ True ” thenfor ℓ from µ d + 1 to max { µ d + 1; dn } do Compute E ℓ from (28). endreturn “ The system is regular-singular at 0 ”, Λ and P dnℓ ≥ ν d E ℓ z ℓ/d . endreturn “ The system is not regular singular at 0 ”. Proposition 18.
Algorithm 4 returns whether or not a system (1) is regular singularat , and in that case, a constant matrix to which the system is K -equivalent and atruncation of an associated gauge transformation at an arbitrary order.Proof. It follows from Proposition 5 that the system is regular singular at 0 if and onlyif it is Q (( z /d ))-equivalent to a constant system where d is defined by Algorithm 2.Assume that the system (1) is regular singular at 0. Then it follows from Proposition17 that Algorithm 3 shall return ”True”. In that case, Ψ( z ) = P ℓ ≥ ν d E ℓ z ℓ/d satisfies(30).Now, assume that the Algorithm 4 returns that the system is regular singular at0. Then Algorithm 3 must have returned ”True”. It thus follows from Proposition 17that the system is indeed regular singular at 0. Thus, from the first part of the proof,Ψ( z ) = P ℓ ≥ ν d E ℓ z ℓ/d satisfies (30). (cid:3) COLIN FAVERJON AND MARINA POULET
When the system (1) is regular singular at 0, Algorithm 4 computes a K -equivalentconstant matrix. Furthermore, Roques [Roq18, § c ∈ Q ⋆ we let e c denote a function suchthat φ p ( e c ) = ce c , and let ℓ denote a function such that φ p ( ℓ ) = ℓ + 1. For example,we can take suitable determinations of log( z ) log( c ) / log( p ) and log (log( z )). Any constantsystem has a basis of solution in Q h ( e c ) c ∈ Q ⋆ , ℓ i . Then, we have the following. Corollary 19.
Consider a system (1) which is regular singular at . From Algo-rithm 4, one can compute a fundamental matrix of solution of (1) with entries in K h ( e c ) c ∈ Q ⋆ , ℓ i . On the complexity of Algorithm 4.
We now propose to discuss the complexityof this algorithm. We start with an observation about the shape of the matrices M d and N d . Definition 3.
Let D = ( D i,j ) ≤ i ≤ r, ≤ j ≤ s be a block matrix with D i,j ∈ M m (cid:0) Q (cid:1) . Wesay that D is a d -gridded matrix if for all ( i , j ) ∈ { , . . . , r } × { , . . . , s } such that D i ,j is nonzero, the matrices D i ,j , D i,j with i i mod d and j j mod d are zeromatrices. Let σ be a permutation of the set { , . . . , d } . We say that σ is associatedwith the d -gridded matrix D if D i,j = 0 for every i, j ∈ { , . . . , d } with j = σ ( i ). Lemma 20.
The multiplication on d -gridded matrices and the computation of a kernelcan be done with complexity O ( d MM( s/d )) where s is an upper bound for the number of rows and the number of columns of theconsidered matrices. Furthermore, the multiplication of two d -gridded matrices withassociated permutation σ and σ is a d -gridded matrix with associated permutation σ ◦ σ .Proof. We describe the computation for the matrix multiplication. A similar compu-tation can be made for the determination of a kernel. Consider two d -gridded matrices D = ( D i,j ) i,j and E = ( E i,j ) i,j and denote by σ D and σ E their associated permutations.For i ∈ { , . . . , d } we let D i ( resp. E i ) denote the block-matrices ( D i + kd,σ D ( i )+ ℓd ) k,ℓ ( resp. ( E i + kd,σ E ( i )+ ℓd ) k,ℓ ). Let F i = D i E σ D ( i ) and set F i =: ( F i,k,ℓ ) k,ℓ its block decom-position. For any integer i , write i = i + kd with i ∈ { , . . . , d } and G i,j := (cid:26) F i ,k,ℓ if j = σ E ◦ σ D ( i ) + ℓd , . Then DE = ( G i,j ) i,j . The matrix ( G i,j ) i,j is itself a d -gridded matrix. Its associated permutation is σ E ◦ σ D .The computation of the product of two permutations of { , . . . , d } has complexity O ( d ). Once σ E ◦ σ D is known, the computation of each matrix F i has complexity O (MM( s/d )). Thus, the computation of DE has complexity O ( d + d MM( s/d )) = O ( d MM( s/d )) . (cid:3) Lemma 21.
The matrices M d and N d are d -gridded matrices. N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS Proof.
Recall that M d := ( B i − pj ( d )) ν d ≤ i,j ≤ µ d and N d := ( ( B i − pj ( d )) v ( B ( d ))+ pν d ≤ i ≤ ν d − ν d ≤ j ≤ µ d if ν d < µ d ∈ M ,m ( µ d − ν d +1) (cid:0) Q (cid:1) if ν d = µ d where φ d ( A − ) := P n B n z n . In particular if d does not divide i − pj then B i − pj = 0.Thus if B i − pj = 0 then B i − pj = 0 for all i such that i i (mod d ). Moreover,since p and d are relatively prime, if B i − pj = 0 then B i − pj = 0 for all j such that j j (mod d ). Associated permutations to these matrices are σ M and σ N such that,for every k ∈ { , . . . , d } , pσ M ( k ) ≡ ( p − − ν d ) + k (mod d ) pσ N ( k ) ≡ v ( B ( d )) + p − k (mod d ) . (cid:3) We can now find an upper bound for the complexity of Algorithm 4.
Proposition 22.
Apart from the computation of the Puiseux development of Ψ , thecomplexity of Algorithm 4 is O (cid:0) MM( m )M( p m (deg( A ) + m )) + p m − mv MM( mv/p ) (cid:1) , where v := − ( v ( A ) + v ( A − ) ≥ .Proof. We follow the script of Algorithm 4. From Proposition 4, the complexity ofAlgorithm 1 is O (MM( m )M( p m (deg( A ) + m ))) . Thus d can be computed from Algorithm 2 with the same complexity. To compute M d and N d one needs to compute the inverse of A and the Laurent series expansionof A − between v (cid:0) A − (cid:1) and ( µ d − pν d ) /d . The computation of the inverse of A maybe done in O (MM( m )M(deg( A ))). Computing n terms of the Laurent expansion of arational function can be done in O (M( n )). Set v := − ( v ( A ) + v ( A − ) ≥
0. One has µ d − pν d d − v (cid:0) A − (cid:1) = O ( v ) . Once d is fixed, the computation of M d and N d can be done with complexity O (MM( m )M(deg( A )) + M( v )) . Computing the intersection of two vector spaces given a basis of each is the same ascomputing a kernel. Since the number of rows and the number of columns of M d and N d is at most O ( mdv ), it follows from Lemmas 20 and 21 that the computation of X d := \ − c d ≤ n ≤ c d M n ker( N ) , necessitates 2 c d + 1 steps with complexity O ( d MM( mv )), where c d := m ( µ d − ν d + 1).Thus, the computation of X d can be done with O ( md v/p MM( mv ))operations. Putting all this together, since d ≤ p m , Algorithm 4 returns if a system isregular singular or not in O (cid:0) MM( m )M( p m (deg( A ) + m )) + p m − mv MM( mv ) (cid:1) . (cid:3) COLIN FAVERJON AND MARINA POULET
If one considers the naive bounds M( n ) = O ( n ) and MM( n ) = O ( n ), the com-plexity of Algorithm 4 is(31) O ( m p m + m p m deg( A ) + m v p m − ) . Remark . In Algorithm 4, we choose to compute first the integer d thanks to theCyclic Vector Lemma, Algorithm 1 and Algorithm 2. Then we run Algorithm 3 withthis precised d . This is not always the fastest way to proceed, since the computationof d necessitates to work with possibly great numbers (see Algorithm 1). However,the cost of manipulating great numbers is hidden by the fact that we describe thecomplexity as the number of arithmetic operations in Q . Another way to proceedwould be to run Algorithm 3 for every d ∈ D . In that case the complexity is O (MM( m )M(deg( A )) + p m − mv MM( mv )) . Using the naive bounds for M( n ) and MM( n ) leads to a complexity in O ( m deg( A ) + p m − m v ) , which can be less than (31), especially for great deg( A ).6. Examples
In this section, we explore the regular singular property of some particular systems.6.1.
Equations of order . Homogeneous equation of order . An homogeneous equation of order 1 is anequation of the form(32) φ p ( y ) = ay where a ∈ Q ( z ), a = 0. Proposition 23.
Any homogeneous equation of order is regular singular at .Proof. Let ν denote the valuation at 0 of a and set ψ = z ν/ ( p − . Then, the system φ p ( y ) = by with b := φ p ( ψ ) − aψ is Fuchsian at 0. Thus, the homogeneous equation(32) is Q (cid:0)(cid:0) z ν/ ( p − (cid:1)(cid:1) -equivalent to a Fuchsian equation at 0, which implies that (32)is regular singular at 0 (see [Roq18, Prop. 34]). (cid:3) Inhomogeneous Mahler equations of order . Consider a inhomogeneous
Mahlerequation of order 1(33) q − + q y + q φ p ( y ) = 0 , with q − , q , q ∈ Q [ z ], q q = 0. The corresponding system has for associated matrix,the matrix A ( z ) = (cid:18) − q q − q − q (cid:19) Proposition 24.
A sufficient condition for the system to be regular singular at isfor (33) to have a solution in K . N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS Proof.
Assume that (33) has a solution f ∈ K . Let g ∈ K and γ ∈ Q ⋆ such that φ p ( g ) γ = − q q g . Then φ p (cid:18) f g (cid:19) (cid:18) γ (cid:19) = A (cid:18) f g (cid:19) , and the system is regular singular at 0. (cid:3) Note that it is not a necessary condition. For example, take q − = q = − q = 1.6.2. An equation of order . Consider the 3-
Mahler equation : z (1 − z + z )(1 − z − z ) φ ( y ) − (1 − z − z − x − z ) φ ( y )+ z (1 + z )(1 − z − z ) y = 0 . The matrix of the 3-Mahler system associated to this equation is A ( z ) := − z (1+ z )(1 − z − z )(1 − z + z )(1 − z − z ) 1 − z − z − x − z z (1 − z + z )(1 − z − z ) ! . We propose to check whether or not the 3-Mahler system associated to this matrix isregular singular at 0. Since we already know an homogeneous linear equation associatedwith this system, it is not necessary to run Algorithm 1. Algorithm 2 applied to thissystem returns d := 2. We now run Algorithm 3 with d = 2. We have v ( A ) = − v ( A − ) = − ν = − µ = 6. We can compute N and M : N := − − − − − − − − − − − − − − − − − − − − − − − − , COLIN FAVERJON AND MARINA POULET M := − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − where we left empty blocks for the ones corresponding to zero-matrices in the 2-griddedblock decomposition of M and N . In that case, the vector space X is spanned by thetranspose of the two linearly independent vectors(0 , , , , , , , , − , , , , , − , , , − , , , , , , , , , , , , , , , , , , , , , − , X has dimension 2 and, from Theorem 2, the system is regular singular at0. One can check that these vectors are eigenvectors of the matrix M for the eigenvalue1. Thus the matrix R is the identity matrix of size 2. In particular, the associatedgauge transformation Ψ is a fundamental matrix of solution because it satisfies φ (Ψ) − A Ψ = I . From these two vectors, we can compute the first terms of the Puiseux developmentof Ψ Ψ = (cid:18) f f f (cid:19) + O ( z / )with f ( z ) = z − / − z / + z / − z / + z / − z / + z / − z / + z / ,f ( z ) = − z + z − z + 2 z − z + 2 z ,f ( z ) = z − / − z / + z / − z / . Remark . Note that this example is the same as the one that illustrates the paper[CDDM18].6.3.
Systems coming from finite deterministic automata.
As mentioned in theintroduction, Mahler systems are related with the automata theory. Indeed, the gener-ating function of an automatic sequence (see [AS03]) is solution of a Mahler equation.Numerous famous automatic sequences are related to homogeneous or inhomogeneousMahler equations of order 1. This is the case of the
Thue-Morse sequence, the regularpaper-folding sequence, the sequences of power of a given integer, the characteristicsequence of triadic Cantor integers - those whose base-3 representation contains no 1.Thus, their associated systems are regular singular at 0.Among the sequences satisfying equations with a greater order, an important one isthe
Baum-Sweet sequence, which is the characteristic sequence of integers whose binarydevelopment have no blocks of consecutive 0 of odd length. The system associated to
N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS
Rudin-Shapiro sequence whose general term is a n = 1 if the number of occurence of two consecutive 1in the binary expansion of n is even a n = − . Its generating power series f := P n ∈ N a n z n satisfies the equation φ (cid:18) f ( z ) f ( − z ) (cid:19) = 12 (cid:18) z − z (cid:19) (cid:18) f ( z ) f ( − z ) (cid:19) This system is not regular singular. Indeed, running Algorithm 3 with d = 3 showsthat the dimension of X is only 1.The regular singular property can be seen as “ normal ” for Mahler systems becausea sufficient condition is to be Fuchsian at 0. However, the generating series of anautomatic sequence satisfy Mahler systems with a very precise shape : A − (0) hasat most one nonzero entry in each column. Among these systems, the Fuchsian onesappear to be the exceptions. Consider for example the following variation of the Baum-Sweet sequence : ( a n ) n ∈ N ∈ { , } N , where a n = 0 if and only if the binary developmentof n contains a block of consecutive 1 with odd length. A Mahler system associatedto this sequence is φ ( Y ) = z z z − Y , which is not regular singular at 0.7.
Open problems
We discuss here some open problems about the regular singularity at 0 of a Mahlersystem.7.1.
The inverse matrix system.
Let A ∈ GL m ( z ) and p ≥ p -Mahler system with matrix A is Fuchsian at 0, then the p -Mahler system with matrix A − is also Fuchsian at 0 (and hence, regular singular at 0). This property doesnot extend to regular singular systems. For example, if A denotes the matrix of theregular singular system in subsection 6.2, the 3-Mahler system associated with A − is not regular singular at 0. We ask the following question. Is there a characterisation of matrices A such that the p -Mahler systems associatedwith both A and A − are regular singular at ? Changing the Mahler operator.
Assume that a system is Fuchsian at 0. Ifwe change the integer p then the system remains Fuchsian at 0 (hence regular singularat 0). This property does not extend to regular singular systems. Indeed, the 3-Mahler system of subsection 6.2 is regular singular at 0, while the 2-Mahler systemwith the same matrix is not. Similarly, the p -Mahler system associated to this matrixis not regular singular when p ∈ { , . . . , } (and probably beyond). Similarly, thecompanion system associated with the p -Mahler equation( z + z ) φ p ( y ) + ( − /z − z − z + z ) φ p ( y ) + (1 − z ) y = 0 , is regular singular at 0 for p = 2 and p = 4 but not for p ∈ { , , , . . . , } (andprobably beyond). It seems that for a matrix A ∈ GL m (cid:0) Q ( z ) (cid:1) the p -Mahler system8 COLIN FAVERJON AND MARINA POULET associated with A is either regular singular at 0 for every integers or for finitely many(possibly none) integers p ≥ Is that true that only these two situations may occur ?
Acknowledgement. — The authors would like to thank Thomas Dreyfus for his lightsabout the paper [CDDM18]. They also thank Boris Adamczewski and Julien Roquesfor the precious discussions about this work.
References [AF17] B. Adamczewski, C. Faverjon,
M´ethode de Mahler: relations lin´eaires, transcendance et appli-cations aux nombres automatiques , Proc. London Math. Soc. (2017), 55–90.[AF20] B. Adamczewski, C. Faverjon,
Mahler’s method in several variables and finite automata ,preprint 2020, arXiv:2012.08283 [math.NT], 52p.[AS03] J.-P. Allouche and J. Shallit, Automatic sequences. Theory, applications, generalizations, Cam-bridge University Press, Cambridge, 2003.[Bar89] M. Barkatou,
On the Reduction of Linear Systems of Difference Equations , Proceedings ofISAAC’89 (1989) 1–6.[Bar95] M. Barkatou,
A rational version of Moser’s algorithm , Proceedings of ISAAC’95 (1995), 297–302.[BBP08] M. Barkatou, G. Broughton, E. Pfluegel,
Regular systems of linear functional equations andapplications , Proceedings of the International Symposium on Symbolic and Algebraic Computa-tion (2008), 15–22.[BCR13] J. Bell, M. Coons, E. Rowland,
The Rational-Transcendental Dichotomy of Mahler Functions ,J. of Integer Seq. (2013), Article 13.2.10, 11p.[Bir13] G.D. Birkhoff, Singular points of ordinary linear differential equations , Trans. Amer. Math.Soc. (1913), 134–139.[Bir30] G.D. Birkhoff, Formal theory of irregular linear difference equations , Acta Math. (1930),205–246.[BP96] M. Bronstein, M. Petkovsek, An introduction to pseudo-linear algebra , Theoretical ComputerScience (1996), 3–33.[CDDM18] F. Chyzak, T. Dreyfus, P. Dumas, M. Mezzarobba,
Computing solutions of linear Mahlerequations , Mathematics of Computation (2018), 2977–3021.[Cob68] A. Cobham, On the Hartmanis-Stearns problem for a class of tag machines , ConferenceRecord of 1968 Ninth Annual Symposium on Switching and Automata Theory, Schenectady,New York (1968), 51–60.[Dum93] P. Dumas,
R´ecurrences mahl´eriennes, suites automatiques, ´etudes asymptotiques , Th`ese,Universit´e de Bordeaux I, Talence (1993).[Hil87] A. Hilali,
Solutions formelles de syst`emes diff´erentiels lin´eaires au voisinage d’un point sin-gulier , Th`ese de doctorat, Universit´e Joseph-Fourier - Grenoble I (1987).[HW86] A. Hilali, A. Wazner
Formes super-irr´eductibles des syst`emes diff´erentiels lin´eaires , Nu-merische Mathematik (1986). 429–449.[Mah29] K. Mahler, Arithmetische Eigenschaften der L¨osungen einer Klasse von Funktionalgleichun-gen , Math. Ann. (1929), 342–367.[Mah30a] K. Mahler, ¨Uber das Verschwinden von Potenzreihen mehrerer Ver¨anderlichen in speziellenPunktfolgen , Math. Ann. (1930), 573–587.[Mah30b] K. Mahler,
Arithmetische Eigenschaften einer Klasse transzendental-transzendente Funktio-nen , Math. Z. (1930), 545–585.[MF80] M. Mend`es France, Nombres alg´ebriques et th´eorie des automates , Enseign. Math. (1980),193–199.[Mos59] J.Moser, The order of a singularity in Fuchs’ theory , Mathematische Zeitschrift. (1959),379-398.[Nis97] Ku. Nishioka, Mahler functions and transcendence , Lecture Notes in Math. , Springer-Verlag, Berlin (1997).[Phi15] P. Philippon,
Groupes de Galois et nombres automatiques , J. Lond. Math. Soc. (2015),596–614. N ALGORITHM TO DETERMINE REGULAR SINGULAR MAHLER SYSTEMS [Pou20] M. Poulet, A density theorem for the difference Galois group of regular singular Mahler sys-tems , preprint 2020, arXiv:2012.14659 [math.NT], 35p.[Pra83] C. Praagman,
The formal classification of linear difference operators , Indagationes Mathe-maticae (proceedings) (1983), 249–261.[vdPS03] M. van der Put, M.F. Singer, Galois theory of linear differential equations , Grundlehren derMathematischen Wissenschaften, , Springer-Verlag, Berlin (2003).[Ran92] B. Rand´e, ´Equations Fonctionnelles de Mahler et Applications aux Suites p-r´eguli`eres , Th`esede doctorat, Universit´e de Bordeaux I, Talence (1992).[Roq18] J. Roques,
On the algebraic relations between Mahler functions , Trans. Amer. Math. Soc. (2018), 321–355.[Roq20] J. Roques,
On the local structure of Mahler systems
Inter. Math. Res. Not. (2020) 1–21.[Sau00] J. Sauloy, Syst`emes aux q-diff´erences singuliers r´eguliers : classification, matrice de connexionet monodromie , Annales de l’Institut Fourier (2000), 1021–1071. Univ Lyon, Universit´e Claude Bernard Lyon 1, CNRS UMR 5208, Institut CamilleJordan, F-69622 Villeurbanne, France
Email address : [email protected] Univ Lyon, Universit´e Claude Bernard Lyon 1, CNRS UMR 5208, Institut CamilleJordan, F-69622 Villeurbanne, France
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