An introduction to Mahler's method for transcendence and algebraic independence
aa r X i v : . [ m a t h . N T ] S e p An introduction to Mahler’s method for transcendence andalgebraic independence.
Federico PellarinSeptember 16, 2018
Contents Q
33 Transcendence theory in positive characteristic 94 Algebraic independence 20
In his mathematical production, Kurt Mahler (1903-1988) introduced entire new subjects. One ofthem, perhaps chronologically the first one, aimed three fundamental papers [27, 28, 29] and was,later in 1977, baptised “Mahler’s method” by Loxton and van der Poorten. By this locution wemean a general method to prove transcendence and algebraic independence of values of a certainclass of functions by means of the following classical scheme of demonstration whose terminologywill be explained in the present text ( ):(AP) - Construction of auxiliary polynomials ,(UP) - Obtaining an upper bound , by means of analytic estimates ,(NV) - Proving the non-vanishing , by means of zeros estimates ,(LB) - Obtaining a lower bound , by means of arithmetic estimates .For example, with Mahler’s method, and with the help of the basic theory of heights , it is pos-sible to show the transcendence of values at algebraic complex numbers of transcendental analyticsolutions f ( x ) ∈ L [[ x ]] (with L a number field embedded in C ) of functional equations such as f ( x d ) = R ( x, f ( x )) , d > , R ∈ L ( X, Y ) , (1)with d integer, see [27]. We borrowed this presentation from Masser’s article [31, p. 5], whose point of view influenced our point of view.
1n this text, we will also be interested in analogues of these functions over complete, algebraicallyclosed fields other than C and for this purpose it will be advantageous to choose right away anappropriate terminology. Indeed, in the typical situation we will analyse, there will be a base field K , together with a distinguished absolute value that will be denoted by | · | . Over K there will beother absolute values as well, and a product formula will hold. We will consider the completion of K with respect to | · | , its algebraic closure that will be embedded in its completion K with respectto an extension of | · | . The algebraic closure of K alg. , embedded in K , will also be endowed with a absolute logarithmic height that will be used to prove transcendence results. Here, an element of K is transcendental if it does not belong to K alg. .If L is a finite extension of K in K and f ∈ L [[ x ]] is a formal series solution of (1), we willsay that f is a Mahler’s function over K . If f converges at α ∈ K alg. \ { } (for the distinguishedabsolute value), we will say that f ( α ) ∈ K is a Mahler’s value and α is a base point for this value.In spite of the generality of this terminology, in this text we will restrict our attention to the basefields Q , K = F q ( θ ) and C ( t ) where C is the completion of an algebraic closure of the completionof K for the unique extension of the absolute value defined by | a | = q deg θ a , with a ∈ K .The interest of the method introduced by Mahler in [27] is that it can be generalised, as itwas remarked by Mahler himself in [28], to explicitly produce finitely generated subfields of C of arbitrarily large transcendence degree. This partly explains, after that the theory was long-neglected for about forty years, a regain of interest in it, starting from the late seventies, especiallydue to the intensive work of Loxton and van der Poorten, Masser, Nishioka as well as other authorswe do not mention here but that are quoted, for example, in Nishioka’s book [37].In some sense, Mahler’s functions and Siegel’s E -functions share similar properties; large tran-scendence degree subfields of C can also be explicitly constructed by the so-called Siegel-Shidlowskitheorem on values of Siegel E -functions at algebraic numbers (see Lang’s account on the theory in[26]). However, this method makes fundamental use of the fact that E -functions are entire, withfinite analytic growth order. This strong assumption is not at all required when it is possible toapply Mahler’s method, where the functions involved have natural boundaries for analytic contin-uation; this is certainly an advantage that this theory has. Unfortunately, no complex “classicalconstant” (period, special value of exponential function at algebraic numbers. . . ) seems to occuras a complex Mahler’s value, as far as we can see.More recently, a variety of results by Becker, Denis [9, 17, 18, 19, 20] and other authors changedthe aspect of the theory, especially that of Mahler’s functions over fields of positive characteristic.It was a fundamental discovery of Denis, that every period of Carlitz’s exponential function is aMahler’s value, hence providing a new proof of its transcendency. This motivates our choice ofterminology; we hope the reader will not find it too heavy. At least, it will be useful to comparethe theory over Q and that over K .The aim of this paper is to provide an overview of the theory from its beginning (transcen-dence) to its recent development in algebraic independence and its important excursions in positivecharacteristic, where it is in “competition” with more recent, and completely different techniquesinspired by the theory of Anderson’s t -motives (see, for example, the work of Anderson, Brownawell,Papanikolas, Chieh-Yu Chang, Jing Yu, other authors [6, 14, 39] and the related bibliographies).The presentation of the paper essentially follows, in an expanded form, the instructional talkthe author gave in the conference “ t -motives: Hodge structures, transcendence and other motivicaspects”, held in Banff, Alberta, (September 27 - October 2, 2009). The author is thankful tothe organisers of this excellent conference for giving the opportunity to present these topics, andthankful to the Banff Centre for the exceptional environment of working it provided. The author2lso wishes to express his gratitude to B. Adamczewski and P. Philippon for discussions and hintsthat helped to improve the presentation of this text, and to P. Bundschuh, H. Kaneko and T.Tanaka for a description of the algebraic relations involving the functions L r of Section 4.1.1 theyprovided.Here is what the paper contains. In Section 2, we give an account of transcendence theory ofMahler’s values with Q as a base field; this is part of the classical theory, essentially contained inone of the first results by Mahler. In Section 3, we will outline the transcendence theory with,as a base field, a function field of positive characteristic (topic which is closer to the themes ofthe conference). Here, the main two features are some applications to the arithmetic of periodsof Anderson’s t -motives and some generalisations of results of the literature (cf. Theorem 8). InSection 4, we first make an overview of known results of algebraic independence over Q of Mahler’svalues, then we describe more recent results in positive characteristic (with the base field K = F q ( θ ))and finally, we mention some quantitative aspects. The main features of this section are elementaryproofs of two results: one by Papanikolas [39], describing algebraic dependence relations betweencertain special values of Carlitz’s logarithms, and another one, by Chieh-Yu Chang and Jing Yu,describing all the algebraic dependence relations of values of Carlitz-Goss zeta function at positiveintegers.This paper does not contain a complete survey on Mahler’s method. For example, Mahler’smethod was also successful in handling several variable functions. To keep the size of this surveyreasonable, we made the arguable decision of not describing this part of the theory, concentratingon the theory in one variable, which seemed closer to the other themes of the conference. Q The example that follows gives an idea of the method. We consider the formal series: f TM ( x ) = ∞ Y n =0 (1 − x n ) = ∞ X n =0 ( − a n x n ∈ Z [[ x ]]( a n ) n ≥ being the Thue-Morse sequence ( a n is the reduction modulo 2 of the sum of the digits ofthe binary expansion of n and, needless to say, the subscript TM in f TM stands for “Thue-Morse”).The formal series f TM converges in the open unit ball B (0 ,
1) to an analytic function and satisfiesthe functional equation f TM ( x ) = f TM ( x )1 − x , (2)(in (1), d = 2 and R = Y − X so that f TM is a Mahler’s function. In all the following, we fix anembedding in C of the algebraic closure Q alg. of Q . We want to prove: Theorem 1
For all α ∈ Q alg. with < | α | < , f TM ( α ) is transcendental. This is a very particular case of a result of Mahler [27] reproduced as Theorem 2 in the presentpaper. The proof, contained in 2.1.3 uses properties of
Weil’s logarithmic absolute height reviewedin 2.1.2. It will also use the property that f TM is transcendental over C ( x ), proved below in 2.1.1.3 .1.1 Transcendence of f TM . The transcendence of f TM over C ( x ) can be checked in several ways. A first way to proceedappeals to P´olya-Carlson Theorem (1921), (statement and proof can be found on p. 265 of [48]).It says that a given formal series φ ∈ Z [[ x ]] converging with radius of convergence 1, either has { z ∈ C , | z | = 1 } as natural boundary for holomorphy, or can be extended to a rational function ofthe form P ( x )(1 − x m ) n , with P ∈ Z [ x ]. To show that f TM is transcendental, it suffices to prove that f TM is not of the form above, which is evident from the functional equation (2), which implies that f TM has bounded integral coefficients. Indeed, if rational, f TM should have ultimately periodic sequenceof the coefficients. However, it is well known (and easy to prove, see [47, Chapter 5, Proposition5.1.2]) that this is not the case for the Thue-Morse sequence.Another way to check the transcendence of f TM is that suggested in Nishioka’s paper [35].Assuming that f TM is algebraic, the field F = C ( x, f TM ( x )) is an algebraic extension of C ( x ) ofdegree, say n , and we want to prove that this degree is 1. It is possible to contradict this propertyobserving that the extension F of C ( x d ) ramifies at the places 0 and ∞ only and applying Riemann-Hurwitz formula. Hence, f TM is rational and we know already from the lines above how to excludethis case. Here we closely follow Lang [26, Chapter 3] and Waldschmidt [54, Chapter 3]. Let L be a numberfield. The absolute logarithmic height h ( α : · · · : α n ) of a projective point ( α : · · · : α n ) ∈ P n ( L )is the following weighted average of logarithms of absolute values: h ( α : · · · : α n ) = 1[ L : Q ] X v ∈ M L d v log max {| α | v , . . . , | α n | v } , where v runs over a complete set M L of non-equivalent places of L , where d v = [ L v : Q p ] with v | Q = p the local degree at the place v (one then writes that v | p ) ( L v , Q p are completions of L, Q atthe respective places so that if v |∞ , L v = R or L v = C according to whether the place v is real orcomplex), and where | · | v denotes, for all v , an element of v chosen in such a way that the following product formula holds: Y v ∈ M L | α | d v v = 1 , α ∈ L × , (3)where we notice that only finitely many factors of this product are distinct from 1. A common wayto normalise the | · | v ’s is to set | x | v = x if x ∈ Q , x >
0, and v |∞ , and | p | v = 1 /p if v | p .This formula implies that h does not depend on the choice of the number field L , so that wehave a well defined function h : P n ( Q alg. ) → R ≥ . If n = 1 we also write h ( α ) = h (1 : α ). For example, we have h ( p/q ) = h ((1 : p/q )) =log max {| p | , | q |} if p, q are relatively prime and q = 0. With the convention h (0) := 0, this definesa function h : Q alg. → R ≥ (4)4atisfying, for α, β ∈ Q alg. × : h ( α + β ) ≤ h ( α ) + h ( β ) + log 2 ,h ( αβ ) ≤ h ( α ) + h ( β ) ,h ( α n ) = | n | h ( α ) , n ∈ Z . More generally, if P ∈ Z [ X , . . . , X n ] and if α , . . . , α n are in Q alg. , h ( P ( α , . . . , α n )) ≤ log L ( P ) + n X i =1 (deg X i P ) h ( α i ) , (5)where L ( P ) denotes the length of P , that is, the sum of the absolute values of the coefficients of P .Proofs of these properties are easy collecting metric information at every place. More details canbe found in [54, Chapter 3]. Liouville’s inequality , a sort of “fundamental theorem of transcendence”, reads as follows. Let L be a number field, v an archimedean place of L , n an integer. For i = 1 , . . . , n , let α i be an elementof L . Further, let P be a polynomial in n variables X , . . . , X n , with coefficients in Z , which doesnot vanish at the point ( α , . . . , α n ). Assume that P is of degree at most N i with respect to thevariable X i . Then,log | P ( α , . . . , α n ) | v ≥ − ([ L : Q ] −
1) log L ( P ) − [ L : Q ] n X i =1 N i h ( α i ) . The proof of this inequality is again a simple application of product formula (3): [54, Section 3.5].It implies that for β ∈ Q alg. , β = 0, log | β | ≥ − [ L : Q ] h ( β ) . (6)This inequality suffices for most of the arithmetic purposes of this paper (again, see [54] for thedetails of these basic tools). Step (AP) . For all N ≥
0, we choose a polynomial P N ∈ Q [ X, Y ] \ { } of degree ≤ N in both X and Y , such that the order of vanishing ν ( N ) at x = 0 of the formal series F N ( x ) := P N ( x, f TM ( x )) = c ν ( N ) x ν ( N ) + · · · ( c ν ( N ) = 0) . (not identically zero because f TM is transcendental by 2.1.1), is ≥ N .The existence of P N follows from the existence of a non-trivial solution of a homogeneous linearsystem with N linear equations defined over Q in ( N + 1) indeterminates. We will not need tocontrol the size of the coefficients of P N and this is quite unusual in transcendence theory. Step (NV) . Let α be an algebraic number such that 0 < | α | < f TM ( α ) is also algebraic, so that there exists a number field L containing at once α and f TM ( α ).Then, by the functional equation (2), for all n ≥ F N ( α n +1 ) = P N ( α n +1 , f TM ( α n +1 )) = P N (cid:18) α n +1 , f TM ( α )(1 − α ) · · · (1 − α n ) (cid:19) ∈ L.
5e know that F N ( α n +1 ) = 0 for all n big enough depending on N and α ; indeed, f N is notidentically zero and analytic at 0. Step (UB) . Writing the expansion of f N at 0 f N ( x ) = X m ≥ ν ( N ) c m x m = x ν ( N ) c ν ( N ) + X i ≥ c ν ( N )+ i x i with the leading coefficient c ν ( N ) which is a non-zero rational integer (whose size we do not control),we see that for all ǫ >
0, if n is big enough depending on N, α and ǫ :log | F N ( α n +1 ) | ≤ log | c ν ( N ) | + 2 n +1 ν ( N ) log | α | + ǫ. Step (LB) . At once, by (5) and (6),log | F N ( α n +1 ) | ≥≥ − [ L : Q ]( L ( P N ) + N h ( α n +1 ) + N h ( f ( α ) / (1 − α ) · · · (1 − α n ))) ≥ − [ L : Q ]( L ( P N ) + N n +1 h ( α ) + N h ( f TM ( α )) + n X i =0 h (1 − α n )) ≥ − [ L : Q ]( L ( P N ) + N n +2 h ( α ) + N h ( f TM ( α )) + ( n + 1) log 2) . The four steps allow to conclude: for all n big enough,2 − n − log | c ν ( N ) | + ν ( N ) log | α | + 2 − n − ǫ ≥≥ − [ L : Q ]( L ( P N )2 − n − + 2 N h ( α ) + N − n − h ( f TM ( α )) + ( n + 1)2 − n − log 2) . Letting n tend to infinity and using that ν ( N ) ≥ N (recall that log | α | is negative), we find theinequality N log | α | ≥ − L : Q ] h ( α ) . But the choice of the “auxiliary” polynomial P N can be done for every N >
0. With
N > L : Q ] h ( α ) | log | α || , (7)we encounter a contradiction. For R = N/D ∈ C ( X, Y ) with
N, D relatively prime polynomials in C ( X )[ Y ], we write h Y ( R ) :=max { deg Y N, deg Y D } . With the arguments above, the reader can be easily prove the followingtheorem originally due to Mahler [27]. Theorem 2 (Mahler)
Let L ⊂ C be a number field, R be an element of L ( X, Y ) , d > an integersuch that h Y ( R ) < d . Let f ∈ L [[ x ]] be a transcendental formal series such that, in L (( x )) , f ( x d ) = R ( x, f ( x )) . Let us suppose that f converges for x ∈ C with | x | < . Let α be an element of L such that < | α | < .Then, for all n big enough, f ( α d n ) is transcendental over Q . n big enough, L ( f ( α d n )) alg. = L ( f ( α d n +1 )) alg. . It can happen, under thehypotheses of Theorem 2, that f ( α ) is well defined and algebraic for certain α ∈ Q alg. \ { } . Forexample, the formal series f ( x ) = ∞ Y i =0 (1 − x i ) ∈ Z [[ x ]] , converging for x ∈ C such that | x | < f ( x ) = f ( x )1 − x , (8)vanishes at every α such that α i = 1 / i ≥
0. In particular, f being non-constant and havinginfinitely many zeroes, it is transcendental. By Theorem 2, f (1 /
4) = lim x → / f ( x )1 − x is transcen-dental. Remark 3
Nishioka strengthened Theorem 2 allowing the rational function R satisfying only therelaxed condition h Y ( R ) < d (see [37, Theorem 1.5.1] for an even stronger result). The proof,more involved than the proof of Theorem 2, follows most of the principles of it, with the followingnotable difference. To achieve the proof, a more careful choice of the polynomials P N is needed.In step (AP) it is again needed to choose a sequence of polynomials ( P N ) N with P N ∈ Q [ X, Y ] ofdegree ≤ N in X and Y , such that the function F N ( x ) = P N ( x, f ( x )) vanishes at x = 0 with orderof vanishing ≥ c N for a constant c depending on α and f . Since for h Y ( R ) ≥ d the size of thecoefficients of P N influences the conclusion, the use of Siegel’s Lemma is now needed to accomplishthis choice [53, Section 1.3]. To make good use of these refinements we need an improvement of thestep (NV), since an explicit upper bound like c N log N for the integer k such that F ( x d s ) = 0 for s = 0 , . . . , k is required ( ). In this subsection we discuss about some variants of Mahler’s method and applications to modularfunctions (in 2.2.1). We end with 2.2.2, where we quote a criterion of transcendence by Corvaia andZannier quite different from Mahler’s method, since it can be obtained as a corollary of Schmidt’ssubspace theorem.
We refer to [50] for a precise description of the tools concerning elliptic curves and modular formsand functions, involved in this subsection.Let J ( q ) = 1 q + 744 + X i ≥ c i q i ∈ (1 /q ) Z [[ q ]]be the q -expansion of the classical hauptmodul for SL ( Z ), converging for q ∈ C such that 0 < | q | <
1. The following theorem was proved in 1996; see [8]:
Theorem 4 (Barr´e-Sirieix, Diaz, Gramain and Philibert)
For q complex such that < | q | < , one at least of the two complex numbers q, J ( q ) is transcendental. This is not difficult to obtain; see 2.2.1 below for a similar, but more difficult estimate. stephanese theorem ( ) furnished a positive answer to Mahler’s conjecture on values of themodular j -invariant (see [30]). Although we will not say much more about, we mention that a similarconjecture was independently formulated by Manin, for p -adic values of J at algebraic α ’s, as theseries J also converges in all punctured p -adic unit disks, for every prime p . Manin’s conjecture isproved in [8] as well. Manin’s conjecture is relevant for its connections with the values of p -adic L -functions and its consequences on p -adic variants of Birch and Swinnerton Dyer conjecture.Mahler’s conjecture was motivated by the fact that the function J satisfies the autonomousnon-linear modular equation Φ ( J ( q ) , J ( q )) = 0, whereΦ ( X, Y ) = X + Y − X Y + 1488 XY ( X + Y ) − X + Y ) +40773375 XY + 8748000000( X + Y ) − . Mahler hoped to apply some suitable generalisation of Theorem 2. It is still unclear, at the timebeing, if this intuition is correct; we remark that Theorem 2 does not apply here.The proof of Theorem 4 relies on a variant of Mahler’s method that we discuss now. We firstrecall from [50] that there exists a collection of modular equations Φ n ( J ( q ) , J ( q n )) = 0 , n > , with explicitly calculable polynomials Φ n ∈ Z [ X, Y ] for all n . The stephanese team make use ofthe full collection of polynomials (Φ n ) n> so let us briefly explain how these functional equationsoccur.For q complex such that 0 < | q | < J ( q ) is the modular invariant of an elliptic curve analyticallyisomorphic to the complex torus C × /q Z ; if z ∈ C is such that ℑ ( z ) > e π i z = q , then therealso is a torus analytic isomorphism C × /q Z ≡ C / ( Z + z Z ). Since the lattice Z + nz Z can beembedded in the lattice Z + z Z , the natural map C × /q Z → C × /q n Z amounts to a cyclic isogeny of the corresponding elliptic curves which, being projective smooth curves, can be endowed with Weierstrass models y = 4 x − g x − g connected by algebraic relations independent on the choiceof z . At the level of the modular invariants, these algebraic relations for n varying are precisely themodular equations, necessarily autonomous, defined over Z as a simple Galois argument shows.Assuming that for a given q with 0 < | q | < J ( q ) is algebraic, means that there exists an elliptic curve E analytically isomorphic to the torus C × /q Z , which is definable over a number field(it has Weierstrass model y = 4 x − g x − g with g , g ∈ Q alg. ). The discussion above, with thefact that the modular polynomials Φ n are defined over Z , implies that J ( q n ) is algebraic as well.Arithmetic estimates involved in the (LB) step of the proof of Theorem 4 require a precisecontrol, for J ( q ) algebraic, of the height of J ( q n ) and the degree d n of Q ( J ( q n ) , J ( q )) over Q ( J ( q )).the degree d n can be easily computed counting lines in F p for p prime dividing n ; it thus dividesthe number ψ ( n ) = Q p | n (1 + 1 /p ) and is bounded from above by c n ǫ , for all ǫ >
0. As forthe height h n = h ( J ( q n )), we said that the modular polynomial Φ n is related to a family of cyclicisogenies of degree n connecting two families of elliptic curves. We then have, associated to thealgebraic modular invariants J ( q ) , J ( q n ), two isogenous elliptic curves defined over a number field,and the isogeny has degree n . Faltings theorem asserting that the modular heights of two isogenouselliptic curves may differ of at most the half of the logarithm of a minimal degree of isogeny givesthe bound c ( h ( J ( q )) + (1 /
2) log n ) for the logarithmic height h ( J ( q n )) (this implies the delicateestimates the authors do in [8]). Sometimes, this result is called stephanese theorem from the name of the city of Saint-Etienne, where the authorsof this result currently live. q -expansions of the normalised Eisenstein series E , E of weights 4 , P N ) N ≥ in Z [ X, Y ] \ { } withdeg X P N , deg Y P N ≤ N , such that F N ( x ) := P N ( x, xJ ( x )) vanishes with order ≥ N / x = 0.The (UB) estimate is then exactly as in the Proof of Theorem 2. All the authors of [8] needto achieve their proof is the (NV) step; and it is here that a new idea occurs. They use that thecoefficients of J are rational integers to deduce a sharp estimate of the biggest integer n such that F N ( x ) vanishes at q m for all m = 1 , . . . , n −
1. This idea, very simple and appealing to Schwarzlemma, does not seem to occur elsewhere in Mahler’s theory; it was later generalised by Nesterenkoin the proof of his famous theorem in [33, 34], implying the algebraic independence of the threenumbers π, e π , Γ(1 /
4) and the stephanese theorem ( ). We will come back to the latter result inSection 4. We mention the following result in [16] whose authors Corvaja and Zannier deduce from Schmidt’sSubspace Theorem.
Theorem 5 (Corvaja and Zannier)
Let us consider a formal series f ∈ Q alg. (( x )) \ Q alg. [ x, x − ] and assume that f converges for x such that < | x | < . Let L ⊂ C be a number field and S afinite set of places of L containing the archimedean ones. Let A ⊂ N be an infinite subset. Assumethat:1. α ∈ L , < | α | < f ( α n ) ∈ L is an S -integer for all n ∈ A .Then, lim inf n ∈A h ( f ( α n )) n = ∞ This theorem has as an immediate application with A = { d, d , d , . . . } , d > f ∈ Q alg. [[ x ]] is not a polynomial, converges for | x | < f ( x d ) = R ( x, f ( x )) with R ∈ Q alg. ( X, Y ) with h Y ( R ) < d , then, f ( α d n ) is transcendental for α algebraic with 0 < | α | < n big enough. This implies a result (at least apparently) stronger than Theorem 2;indeed, the hypothesis that the coefficients of the series f all lie in a given number field is dropped. The reduction modulo 2 in F [[ x ]] of the formal series f TM ( x ) ∈ Z [[ x ]] is an algebraic formal series.In this section we will see that several interesting transcendental series in positive characteristic areanalogues of the series satisfying the functional equation (8).Let q = p e be an integer power of a prime number p with e >
0, let F q be the field with q elements. Let us write A = F q [ θ ] and K = F q ( θ ), with θ an indeterminate over F q , and define an We take the opportunity to notice that a proof of an analog of the stephanese theorem for the so-called “Drinfeldmodular invariant” by Ably, Recher and Denis is contained in [1]. | · | on K by | a | = q deg θ a , a being in K , so that | θ | = q . Let K ∞ := F q ((1 /θ )) bethe completion of K for this absolute value, let K alg. ∞ be an algebraic closure of K ∞ , let C be thecompletion of K alg. ∞ for the unique extension of | · | to K alg. ∞ , and let K alg. be the algebraic closureof K embedded in C . There is a unique degree map deg θ : C × → Q which extends the mapdeg θ : K × → Z .Let us consider the power series f De ( x ) = ∞ Y n =1 (1 − θx q n ) , which converges for all x ∈ C such that | x | < f De ( x q ) = f De ( x )1 − θx q (9)(the subscript De stands for Denis, who first used this series for transcendence purposes).For q = 2, we notice that f De ( x ) = P ∞ n =0 θ b n x n , where( b n ) n ≥ = 0 , , , , , , , , , , , , . . . is the sequence with b n equal to the sum of the digits of the binary expansion of n (and whosereduction modulo 2 precisely is Thue-Morse sequence of Section 2.1). It is very easy to show that f De is transcendental, because it is plain that f De has infinitely many zeros θ − /q , θ − /q , . . . (justas the function occurring at the end of 2.1.4). We shall prove: Theorem 6
For all α ∈ K alg. with < | α | < , f De ( α ) is transcendental. The proof of Theorem 6 follows the essential lines of Section 2.1, once the necessary tools areintroduced.
Not all the arguments of 2.1.1 work well to show the transcendence of formal series such as f De ;in particular, the so-called Riemann-Hurwitz-Hasse formula does not give much information forfunctional equations such as f De ( x d ) = af De ( x ) + b with the characteristic that divides d . Sincein general it is hard to detect zeros of Mahler’s functions, we report another way to check thetranscendency of f De , somewhat making use of “automatic methods”, which can also be generalisedas it does not depend on the location of the zeroes. To simplify the presentation, we assume, inthe following discussion, that q = 2 but at the same time, we relax certain conditions so that, in allthis subsection, we denote by ϑ an element of C and by f ϑ the formal series f ϑ ( x ) = ∞ Y n =0 (1 − ϑx q n ) = ∞ X n =0 ϑ b n x n ∈ F [[ x ]] ⊂ C [[ x ]]with F the perfect field ∪ n ≥ F ( ϑ / n ), converging for x ∈ C with | x | <
1. In particular, we have f θ ( x ) = f De ( x / ) ∈ F [ θ ][[ x ]] . We shall prove: 10 heorem 7
The formal series f ϑ is algebraic over F ( x ) if and only if ϑ belongs to F q , embeddedin C .Proof. If ϑ ∈ F q , it is easy to show that f ϑ is algebraic, so let us assume by contradiction that f ϑ is algebraic, with ϑ that belongs to C \ F q .We have the functional equation: (1 − ϑx ) f ϑ ( x ) = f ϑ ( x ) . (10)We introduce the operators f = X i c i x i ∈ F (( x )) f ( k ) = X i c k i x i ∈ F (( x )) , well defined for all k ∈ Z . Since f ( x ) = f ( − ( x ) for any f ∈ F [[ x ]], we deduce from (10) thecollection of functional equations f ( − − k ) ϑ ( x ) (1 − ϑ / k x ) = f ( − k ) ϑ ( x ) , k ≥ . (11)For any f ∈ F [[ x ]] there exist two series f , f ∈ F [[ x ]], uniquely determined, with the propertythat f = f + xf . We define E i ( f ) := f i ( i = 0 , f, g ∈ F [[ x ]], E i ( f + g ) = E i ( f ) + E i ( g ) , ( i = 0 , ,E ( f g ) = E ( f ) E ( g ) + xE ( f ) E ( g ) ,E ( f g ) = E ( f ) E ( g ) + E ( f ) E ( g ) ,E ( f ) = f,E ( f ) = 0 . Therefore, E i ( f g ) = f E i ( g ) , i = 0 , . By (11) we get E ( f ( − k ) ϑ ) = f ( − − k ) ϑ E (1 − ϑ / k x ) = f ( − − k ) ϑ ,E ( f ( − k ) ϑ ) = E (1 − ϑ / k x ) f ( − − k ) ϑ = − ϑ / k f ( − − k ) ϑ , and we see that if V is a F -subvector space of F [[ x ]] containing f ϑ and stable under the action ofthe operators E , E , then V contains the F -subvector space generated by f ϑ , f ( − ϑ , f ( − ϑ , . . . .By a criterion for algebraicity of Sharif and Woodcock [49, Theorem 5.3] there is a subvectorspace V as above, with finite dimension, containing f ϑ . The formal series f ( − k ) ϑ are F -linearlydependent and there exists s > f ϑ , f (1) ϑ , . . . , f ( s − ϑ are F -linearly dependent.Going back to the explicit x -expansion of f ϑ , the latter condition is equivalent to the existenceof c , . . . , c s − ∈ F , not all zero, such that for all n ≥ s − X i =0 c i ϑ i b n = 0 . b : N ∪ { } → N ∪ { } is known to be surjective, so that the Moore determinantdet(( ϑ i j )) ≤ i,j ≤ s − vanishes. But this means that 1 , ϑ, ϑ , . . . , ϑ s − are F -linearly dependent (Goss, [23, Lemma 1.3.3]),or in other words, that ϑ is algebraic over F ; a contradiction which completes the proof that f ϑ and in particular f De are transcendental over F ( x ) (and the fact that the image of b has infinitelymany elements suffices to achieve the proof). A good framework to generalise logarithmic heights to other base fields is that described by Langin [26, Chapter 3] and by Artin and Whaples [7, Axioms 1, 2]. Let K be any field together with aproper set of non-equivalent places M K . Let us choose, for every place v ∈ M K an absolute value | · | v ∈ v and assume that for all x ∈ K × , the following product formula holds (cf. [26] p. 23): Y v ∈ M K | x | v = 1 , x ∈ K × , (12)with the additional property that if α is in K × , then | α | v = 1 for all but finitely many v ∈ M K .Let us suppose that M K contains at least one absolute value associated to either a discrete, or anarchimedean valuation of K . It is well known that under these circumstances [7], K is either anumber field, or a function field of one variable over a field of constants.Given a finite extension L of K , there is a proper set M L of absolute values on L , extendingthose of M K , again satisfying the product formula Y v ∈ M L | α | d v v = 1 , (13)where, if v is the place of K such that w | K = v (one then writes w | v ), we have defined d w = [ L w , K v ],so that P w | v d v = [ L : K ].An analogue of the absolute logarithmic height h is available, by the following definition (see[26, Chapter 3]). Let ( α : · · · : α n ) be a projective point defined over L . Then we define: h ( α : · · · : α n ) = 1[ L : K ] X w ∈ M L d w log max {| α | w , . . . , | α n | w } . Again, we have a certain collection of properties making this function useful in almost everyproof of transcendence over function fields.First of all, product formula (13) implies that h ( α : · · · : α n ) does not depend on the choice ofthe field L and defines a map h : P n ( K alg. ) → R ≥ . We write h ( α ) := h (1 : α ). If the absolute values of M K are all ultrametric, it is easy to prove,with the same indications as in 2.1.2, that for α, β ∈ K alg. × : h ( α + β ) , h ( αβ ) ≤ h ( α ) + h ( β ) ,h ( α n ) = | n | h ( α ) , n ∈ Z . P is a polynomial in L [ X , . . . , X n ] and if ( α , . . . , α n ) is a point of L n , we write h ( P ) for the height of the projective point whose coordinates are 1 and its coefficients. We have: h ( P ( α , . . . , α n )) ≤ h ( P ) + n X i =1 (deg X i P ) h ( α i ) . (14)Product formula (13) also provides a Liouville’s type inequality. Let [ L : K ] sep. be the separabledegree of L over K . Let us choose a distinguished absolute value | · | of L and β ∈ L × . We have:log | β | ≥ − [ L : K ] sep. h ( β ) . (15)The reason of the presence of the separable degree in (15) is the following. If α ∈ L × is separableover K then log | α | ≥ − [ K ( α ) : K ] h ( α ) = − [ K ( α ) : K ] sep. h ( α ). Let β be any element of L × . Thereexists s ≥ α = β p s separable and we get p s log | β | ≥ − [ K ( α ) : K ] h ( α ) = − p s [ K ( β ) : K ] sep. h ( β ). f De We now follow Denis and we take K = K , M K the set of all the places of K and we choose ineach of these places an absolute value normalised so that product formula (12) holds, with thedistinguished absolute value | · | chosen so that | α | = q deg θ α for α ∈ K × .As we already did in 2.1.3, we choose for all N ≥
0, a polynomial P N ∈ K [ X, Y ], non-zero, ofdegree ≤ N in both X, Y , such that the order of vanishing ν ( N ) < ∞ of the function F N ( x ) := P N ( x, f De ( x )) at x = 0 satisfies ν ( N ) ≥ N . We know that this is possible by simple linear algebraarguments as we did before.Let α ∈ K alg. be such that 0 < | α | <
1; as in 2.1.3, the sequence ( P N ) N need not to depend onit but the choice of N we will do does.By the identity principle of analytic functions on C , if ǫ is a positive real number, for n bigenough depending on N and α, l, ǫ , we have F N ( α q n +1 ) = 0 andlog | F N ( α q n +1 ) | ≤ ν ( N ) q n +1 log | α | + log | c ν ( N ) | + ǫ, where c ν ( N ) is a non-zero element of K depending on N (it is the leading coefficient of the formalseries F N ).Let us assume by contradiction that f De ( α ) ∈ K alg. , let L be a finite extension of K containing α and f De ( α ).By the variant of Liouville’s inequality (15) and from the basic facts on the height h explainedabove log | F N ( α q n +1 ) | ≥≥ − [ L : K ] sep. (cid:18) deg θ P N + N h ( α q n +1 ) + N h (cid:18) f De ( α )(1 − θα q ) · · · (1 − θα q n +1 ) (cid:19)(cid:19) . Dividing by
N q n +1 and using that ν ( N ) ≥ N we get, for all n big enough, N log | α | + log | c ν ( N ) | + ǫ ≥≥ − [ L : K ] sep. (deg θ P N N − q − n − + (1 + ( q − q − n − ) / ( q − h ( α ) + q − n − h ( f De ( α )) + ( n + 1) N − q − n − h ( θ )) . n tend to infinity, we obtain the inequality: N log | α | ≥ − [ L : K ] sep. (cid:18) qq − (cid:19) h ( α )for all N >
0. Just as in the proof of Theorem 1, if N is big enough, this is contradictory with theassumptions showing that f De ( α ) is transcendental. The transcendence of values of f De at algebraic series has interesting applications, especially whenone looks at what happens with the base point α = θ − . Indeed, let e π = θ ( − θ ) / ( q − ∞ Y i =1 (1 − θ − q i ) − (16)be a fundamental period of Carlitz’s module (it is defined up to multiplication by an element of F × q ). Then, e π = θ ( − θ ) / ( q − f De ( θ − ) − , so that it is transcendental over K .If α = θ − , h ( α ) = log q so that to show that e π is not in K , it suffices to choose N ≥ q = 2and N ≥ q = 2 in the proof above. Let us look, for q = 2 given, at a polynomial (depending on q ) P ∈ A [ X, Y ] \ { } with relatively prime coefficients in X of degree ≤ X and in Y , such that P ( u, f De ( u )) vanishes at u = 0 with the biggest possible order ν > q ).It is possible to prove that for all q ≥ P = X ( Y − ∈ F q [ X, Y ] . This means that to show that f De ( θ − ) K , it suffices to work with the polynomial Q = Y − ).Indeed, f De ( u ) − − θu q + · · · . Therefore, for n big enough, if by contradiction f De ( θ − ) ∈ K ,then log | f De ( α q n +1 ) − | ≥ − (1 + q/ ( q − h ( α q n +1 ) which is contradictory even taking q = 3, butnot for q = 2, case that we skip.If q = 3, we get a completely different kind of polynomial P of degree ≤ P = 2 + θ + θX + 2 θ X + 2 θ Y + Y + 2 θX Y . It turns out that P ( u, f De ( u )) = ( θ − θ ) u ν + · · · with ν = 36 so that the order of vanishing is three times as big as the quantity expected from thecomputations with q ≥
4: 12 = 3 · ν be? It turns out that this question is impor-tant, notably in the search for quantitative measures of transcendence and algebraic independence;we will discuss about this problem in Section 4.1. Other reasons, related to the theory of Carlitz module, allow to show directly that e π F q (( θ − )) for q > .2 A second transcendence result With essentially the same arguments of Section 3, it is possible to deal with a more general situationand prove the Theorem below. We first explain the data we will work with.Let F be a field and t an indeterminate. We denote by F hh t ii the field of Hahn generalisedseries . This is the set of formal series X i ∈ S c i t i , c i ∈ F, with S a well ordered subset of Q ( ), endowed with the standard addition and Cauchy’s multipli-cation from which it is plain that every non-zero formal series is invertible.We have a field F q hh t ii -automorphism τ : C hh t ii → C hh t ii defined by α = X i ∈ S c i t i τ α = X i ∈ S c qi t i . Assume that, with the notations of 3.1.2, K = C ( t ), with t an independent indeterminate. Let | · | be an absolute value associated to the t -adic valuation and b K the completion of K for thisabsolute value. Let K be the completion of an algebraic closure of b K for the extension of | · | , sothat we have an embedding of K alg. in K . We have an embedding ι : K → C hh t ii (see Kedlaya, [24,Theorem 1]); there exists a rational number c > α is in K and ι ( α ) = P i ∈ S c i t i , then | α | = c − i , where i = min( S ). We identify K with its image by ι . It can be proved that τ K ⊂ K , τ K alg. ⊂ K alg. and τ K ⊂ K .The definition of τ implies immediately that, for all α ∈ K , | τ α | = | α | . (17)We choose M K a complete set of non-equivalent absolute values of K such that the productformula (12) holds. On P n ( K alg. ), we have the absolute logarithmic height whose main propertieshave been described in 3.1.2.There is a useful expression for the height h ( α ) of a non-zero element α in K alg. of degree D . If P = a X d + a X d − + · · · + a d − X + a d is a polynomial in C [ t ][ X ] with relatively prime coefficientssuch that P ( α ) = 0, we have: h ( α ) = 1 D log | a | + X σ : K alg. → K log max { , | σ ( α ) |} ! , (18)where the sum runs over all the K -embeddings of K alg. in K . The proof of this formula follows thesame ideas as that of [54, Lemma 3.10].Let α be in K alg. . From (17) and (18), it follows that: h ( τ α ) = h ( α ) . (19) By definition, every nonempty subset of S has a least element for the order ≤ . K [[ x ]], the F q hh t ii -extension of τ defined inthe following way: f := X i c i x i τ f := X i ( τ c i ) x qi . We can now state the main result of this section.
Theorem 8
Let f ∈ K [[ x ]] be converging for x ∈ K , | x | < , let α ∈ K be such that < | α | < .Assume that:1. f is transcendental over K ( x ) ,2. τ f = af + b , where a, b are elements of K ( x ) .Then, for all n big enough, ( τ n f )( α ) is transcendental over K .Proof. We begin with a preliminary discussion about heights. Let r = r x n + · · · + r n be a polynomialin K [ x ]. We have, for all j ≥ τ j r = ( τ j r ) x q j n + · · · + ( τ j r n ). Therefore, if α is an element of K alg. , we deduce from (14), (19) and from elementary height estimates: h (( τ j r )( α )) ≤ h (1 : τ j r : · · · : τ j r n ) + q j nh ( α ) , ≤ n X i =0 h ( τ j r i ) + q j nh ( α ) ≤ n X i =0 h ( r i ) + q j nh ( α ) , where we have applied (19). Therefore, if a is a rational function in K ( x ) such that ( τ j a )( α ) is welldefined, we have h (( τ j a )( α )) ≤ c + q j c , (20)where c , c are two constants depending on a, α only.The condition on f implies that, for all k ≥ τ k f = f k − Y i =0 ( τ i a ) + k − X i =0 ( τ i b ) k − Y j = i +1 ( τ j a ) (21)(where empty sums are equal to zero and empty products are equal to one). Hence, the field L = K ( α, f ( α ) , ( τ f )( α ) , ( τ f )( α ) , . . . )is equal to K ( α, ( τ n f )( α )) for all n big enough.The transcendence of f implies that a = 0. If α is a zero or a pole of τ k a and a pole of τ k b forall k , then it is a simple exercise left to the reader to prove that | α | = 1, case that we have excluded.Let us suppose by contradiction that the conclusion of the theorem is false. Then, α is nota pole or a zero of τ n a, τ n b , ( τ n f )( α ) is algebraic over K for all n big enough, and L is a finiteextension of K .An estimate for the height of this series can be obtained as follows.16 joint application of (19), (20) and (21) yields: h (( τ k f )( α )) ≤ h ( f ( α )) + k − X i =0 h (( τ i a )( α )) + k − X i =0 h (( τ i b )( α )) ≤ c + c k + c q k . Therefore, if P is a polynomial in K [ X, Y ] of degree ≤ N in X and Y , writing F k for the formalseries τ k P ( x, f ( x )) = P τ k ( x q k , ( τ k f )( x )) ( P τ k is the polynomial obtained from P , replacing thecoefficients by their images under τ k ), we get: h ( F k ( α )) ≤ h ( P τ k ) + q k (deg X P ) h ( α ) + (deg Y P ) h (( τ k f )( α )) ≤ c ( P ) + c N q k , (22)where c is a constant depending on P .Let N be a positive integer. There exists P N ∈ K [ X, Y ] with partial degrees in
X, Y notbigger than N , with the additional property that F N ( x ) := P N ( x, f ( x )) = c ν ( N ) x ν ( N ) + · · · , with ν ( N ) ≥ N and c ν ( N ) = 0.Let us write F ( x ) = P i ≥ c i x i . In ultrametric analysis, Newton polygons suffice to locate theabsolute values of the zeroes of Taylor series. The Newton polygons of the series P i ≥ ( τ k c i ) x i ∈K [[ x ]] for k ≥ P i ≥ ( τ k c i ) α q k i = 0for k big enough. Now, since for k ≥ τ k F N )( x ) = ( τ k c ν ( N ) ) x ν ( N ) q k + · · · , we find, when the logarithm is well defined and by (19), that −∞ < log | ( τ k F N )( α ) | ≤ ν ( N ) q k log | α | + log | c ν ( N ) | + ǫ. (23)On the other side, by (22), h (( τ k F N )( α )) ≤ c ( N ) + c N q k , (24)where c is a constant depending on f, α, N and c is a constant depending on f, α .A good choice of N (big) and inequality (15) with k big enough depending on N give a contra-diction ( ). We look at solutions f ∈ K [[ x ]] of τ -difference equations τ f = af + b, a, b ∈ K ( x ) . (25)Theorem 8 allows to give some information about the arithmetic properties of their values. First application.
Assume that in (25), a, b ∈ F q ( t ). Then, since F q ( t ) is contained in the field ofconstants of τ , solutions of this difference equation are related to the variant of Mahler’s methodof Section 3. It is likely that Nishioka’s proof of Theorem 2 can be adapted to strengthen Theorem 8, but we did not enterinto the details of this verification. a = (1 − t − x ) − , b = 0, the equation above has the solution f De2 ( x ) = ∞ Y n =1 (1 − t − x q n ) , which converges for x ∈ K , | x | < x = t , Theorem 8 yields the transcendence of f De2 ( t ) = Q ∞ n =1 (1 − t q n − ) ∈ F q [[ t ]] over F q ( t )and we get (again) the transcendence of e π over K (we also notice the result of the paper [2], whichallows some other applications). More generally, all the examples of functions in [40, Section 3.1]have a connection with this example. Second application.
Theorem 8 also has some application which does not seem to immediatelyfollow from results such as Theorem 2. Consider equation (25) with b = 0 and a = (1 + ϑx ) − ,where ϑ ∈ F q ( t, θ ) is non-zero. We have the following solution of (25) in K [[ x ]]: φ ( x ) = ∞ Y n =0 (1 + ( τ n ϑ ) x q n ) . It is easy to show that φ = P j ≥ c j x j is a formal series of K [[ x ]] converging for x ∈ K , | x | < c j can be computed in the following way. We have c j = 0 if the q -ary ex-pansion of j has its set of digits not contained in { , } . Otherwise, if j = j + j q + · · · + j n q n with j , . . . , j n ∈ { , } , we have, writing ϑ i for τ i r , c j = ϑ j ϑ j · · · ϑ j n n . Therefore, if ψ ( x ) = P ∞ k =0 ϑ ϑ · · · ϑ k x q + ··· + q k , we have φ ( x ) − P ∞ j =0 τ j ψ ( x ).For ϑ = − t − (1 + t/θ ) − , the series φ ( x ) vanishes at every x n = t /q n (1 + t /q n /θ ). The x n ’sare elements of K which are distinct with absolute value < t -adic valuation). Having thus infinitely many zeros in the domain of convergence and not beingidentically zero, φ is transcendental.The series φ converges at x = t . Theorem 8 implies that the formal series φ ( t ) ∈ C (( t ))is transcendental over K . We notice that the arguments of 3.1.1 can be probably extended toinvestigate the transcendence of the series φ associated to, say, ϑ = − (1 + t/θ ) − , case in whichwe do not necessarily have infinitely many zeros. The reason is that, over K (( x )), the F q -linearFrobenius twist F : a a q (for all a ) splits as F = τ χ = χτ where τ is Anderson’s F q (( t ))-linear twist and χ is Mahler’s C (( x ))-linear twist, and most of thearguments of 3.1.1 can be generalised to this setting. τ -difference equations in K [[ x ]]The arguments of the previous subsection deal with formal series in K [[ x ]] = C ( t )[[ x ]]. We haveanother important ring of formal series, also embedded in K [[ x ]], which is C ( x )[[ t ]]. Although thearithmetic of values of these series seems to be not deducible from Theorem 8, we discuss hereabout some examples because solutions f ∈ C ( x )[[ t ]] of τ -difference equations such as τ f = af + b, a, b ∈ C ( x ) (26)are often related to Anderson-Brownawell-Papanikolas linear independence criterion in [6] (see thecorresponding contribution in this volume and the refinement [15]).18ith ζ θ a fixed ( q − − θ , the transcendental formal series Ω ( t ) := ζ − qθ ∞ Y i =1 (cid:18) − tθ q i (cid:19) = ∞ X i =0 d i t i ∈ K alg. [[ t ]] (27)is convergent for all t ∈ C , such that Ω ( θ ) ∈ F × q e π − and satisfies the functional equation Ω ( t q ) = ( t q − θ q ) Ω ( t ) q (28)(see [6]). By a direct inspection it turns out that there is no finite extension of K containing allthe coefficients d i of the t -expansion of Ω ( ). Hence, there is no variant of Mahler’s method whichseems to apply to prove the transcendence of Ω at algebraic elements (and a suitable variant ofTheorem 5 is not yet available).The map τ : C (( t )) → C (( t )) acts on in the following way: c = X i c i t i τ c := X i c qi t i . Let s ( t ) be the formal series τ − Ω − ∈ C [[ t ]] (where τ − is the reciprocal map of τ ). After (28)this function is solution of the τ -difference equation:( τ s )( t ) = ( t − θ ) s ( t ) , (29)hence it is a solution of (26) with a = t − θ and b = 0. Transcendence of values of this kind offunction does not seem to follow from Theorem 8, but can be obtained with [6, Theorem 1.3.2].More generally, let Λ be an A -lattice of C of rank r and let e Λ ( z ) = z Y λ ∈ Λ \{ } (cid:16) − zλ (cid:17) (30)be its exponential function, in Weierstrass product form. The function e Λ is an entire, surjective, F q -linear function. Let φ λ : A → End F q − lin. ( G a ( C )) = C [ τ ] ( ) be the associated Drinfeld module.We have, for a ∈ A , φ Λ ( a ) e Λ ( z ) = e Λ ( az ).Let us extend φ Λ over K by means of the endomorphism τ ( τ t = t ). After having chosen anelement ω ∈ Λ \ { } , define the function s Λ ,ω ( t ) := ∞ X n =0 e Λ (cid:16) ωθ n +1 (cid:17) t n ∈ C [[ t ]] , (31)convergent for | t | < q . We have, for all a ∈ A , φ Λ ( a ) s Λ ,ω = as Λ ,ω , (32)where, if a = a ( θ ) ∈ F q [ θ ], we have defined a := a ( t ) ∈ F q [ t ]. This means that s Λ ,ω , as a formalseries of C [[ t ]], is an eigenfunction for all the F q (( t ))-linear operators φ Λ ( a ), with eigenvalue a , forall a ∈ A .If Λ = e πA , φ Λ is Carlitz’s module, and equation (32) implies the τ -difference equation (29). They generate an infinite tower of Artin-Schreier extensions. Polynomial expressions in τ with the product satisfying τc = c q τ , for c ∈ C . Algebraic independence
In [28], Mahler proved his first result of algebraic independence obtained modifying and generalisingthe methods of his paper [27]. The result obtained involved m formal series in several variables butwe describe its consequences on the one variable theory only. Let L be a number field embeddedin C and let d > Theorem 9 (Mahler)
Given m formal series f , . . . , f m ∈ L [[ x ]] , satisfying functional equations f i ( x d ) = a i f i ( x ) + b i ( x ) , ≤ i ≤ m with a i ∈ L , b i ∈ L ( x ) for all i , converging in the open unit disk. If α is algebraic such that < | α | < , then the transcendence degree over L ( x ) of the field L ( x, f ( x ) , . . . , f m ( x )) is equal tothe transcendence degree over Q of the field Q ( f ( α ) , . . . , f m ( α )) . Mahler’s result remained nearly unobserved for several years. It came back to surface notably thanksto the work of Loxton and van der Poorten in the seventies, and then by Nishioka and several otherauthors. At the beginning, these authors developed criteria for algebraic independence tailored forapplication to algebraic independence of Mahler’s values. Later, criteria for algebraic independenceevolved in very general results, especially in the hands of Philippon. Here follows a particular caseof a criterion of algebraic independence by Philippon The statement that follows merges the results[44, Theorem 2] and [43, Theorem 2.11] and uses the data K , L , | · | , K , A , . . . where K is a completealgebraically closed field in two cases. We examine only the cases in which K is either C or C , butit is likely that the principles of the criterion extend to several other fields, like that of Section 3.2.In the case K = C , L is a number field embedded in C , | · | is the usual absolute value, h is theabsolute logarithmic height of projective points defined over Q alg. , K denotes Q and A denotes Z .In the case K = C , L is a finite extension of K = F q ( θ ) embedded in C , | · | is an absolutevalue associated to the θ − -adic valuation, h is the absolute logarithmic height of projective pointsdefined over K alg. , A denotes the ring A = F q [ θ ] and we write K = K .In both cases, if P is a polynomial with coefficients in L , we associate to it a projective pointwhose coordinates are 1 and its coefficients, and we write h ( P ) for the logarithmic height of thispoint (which depends on P up to permutation of the coefficients). Theorem 10 (Philippon)
Let ( α , . . . , α m ) be an element of K m and k an integer with ≤ k ≤ m . Let us suppose that there exist three increasing functions Z ≥ → R ≥ δ ( degree ) σ ( height ) λ ( magnitude ) and five positive real numbers c , c , c δ , c λ , c σ satisfying the following properties.1. lim n →∞ δ ( n ) = ∞ ,2. lim n →∞ s ( n +1) s ( n ) = c s for s = δ, σ, λ ,3. σ ( n ) ≥ δ ( n ) , for all n big enough, . The sequence n λ ( n ) δ ( n ) k +1 σ ( n ) is ultimately increasing,5. For all n big enough, λ ( n ) k +1 > σ ( n ) δ ( n ) k − ( λ ( n ) k + δ ( n ) k ) . Let us suppose that there exists a sequence of polynomials ( Q n ) n ≥ with Q n ∈ L [ X , . . . , X m ] , with deg X i Q n ≤ δ ( n ) for all i and n , with h ( Q n ) ≤ σ ( n ) for all n , with coefficients integral over A , such that, for all n big enough, − c λ ( n ) < log | Q n ( α , . . . , α m ) | < − c λ ( n ) . Then, the transcendence degree over L of the field L ( α , . . . , α m ) is ≥ k . Theorem 10 can be applied to prove the following result.
Theorem 11
Let us assume that we are again in one of the cases above; K = C or K = C , let L be as above. Let f , . . . , f m be formal series of L [[ x ]] an d > and integer, satisfying the followingproperties:1. f , . . . , f m converge for | x | < ,2. f , . . . , f m are algebraically independent over K ( x ) ,3. For all i = 1 , . . . , m , there exist a i , b i ∈ L ( x ) such that f i ( x d ) = a i ( x ) f i ( x ) + b i ( x ) , i = 1 , . . . , m. Let α ∈ L be such that < | α | < . Then, for all n big enough, f ( α d n ) , . . . , f m ( α d n ) are alge-braically independent over L . This result is, for K = C , a corollary of Kubota’s result [25, Theorem p. 10]. For K = C , it isdue to Denis [18, Theorem 2]. See also [9, 17]. Sketch of proof of Theorem 11 in the case K = C . To simplify the exposition, we assume that L = K .Let N > P N ∈ K [ x, X , . . . , X n ] (thatwe choose) of degree ≤ N in each indeterminate, such that the order ν ( N ) of vanishing at x = 0 ofthe function F N ( x ) = P N ( x, f ( x ) , . . . , f m ( x ))(not identically zero because of the hypothesis of algebraic independence of the functions f i over C ( x )), satisfies ν ( N ) ≥ N m +1 .The choice of the parameter N will be made later. If c ( x ) ∈ A [ x ] is a non-zero polynomial suchthat ca i , cb i ∈ A [ x ] for i = 1 , . . . , m , then we define, inductively, R = P N and R n = c ( x ) N R n − ( x d , a X + b , . . . , a m X m + b m ) ∈ A [ x, X , . . . , X m ] . n big enough de-pending on N, f , . . . , f m , where c , c , c are integer constants depending on f , . . . , f m only:deg X i ( R n ) ≤ N, ( i = 1 , . . . , m ) , deg x ( R n ) ≤ c d n N, deg θ ( R n ) ≤ c d n N,h ( R n ) ≤ c + c d n N, where we wrote c ( N ) = h ( R ); it is a real number depending on f , . . . , f m and N .Since R n ( x, f ( x ) , . . . , f n ( x )) = ( n − Y i =0 c ( x d i ) N ) R ( x d n , f ( x d n ) , . . . , f m ( x d n )) , one verifies the existence of two constants c > c > − c d n ν ( N ) ≤ deg θ ( R n ( α, f ( α ) , . . . , f m ( α ))) ≤ − c d n ν ( N )for all n big enough, depending on N, f , . . . , f m and α .Let us define: Q n ( X , . . . , X n ) = D c d n N R n ( α, X , . . . , X m ) ∈ A [ α ][ X , . . . , X m ] , where D ∈ A \ { } is such that Dα is integral over K . The estimate above implies at once that,for n big enough: deg X i Q n ≤ N,h ( Q n ) ≤ c ( N ) + c d n N, where c ( N ) is a constant depending on f , . . . , f m , N , and α (it can be computed with an explicitdependence on c ( N )) and c is a constant depending on f , . . . , f m , and α but not on N .Finally, Theorem 10 applies with the choices: α i = f i ( α ) , i = 1 , . . . , mk = mλ ( n ) = d n ν ( N ) δ ( n ) = Nσ ( n ) = c d n ν ( N ) , provided that we choose N large enough depending on the constants c , . . . introduced so far. Then,one chooses n big enough (depending on the good choice of N ). Theorem 11 furnishes algebraic independence of Mahler’s values if we are able to check algebraicindependence of Mahler’s functions but it does not say anything on the latter problem; this is notan easy task in general. With the following example, we would like to sensitise the reader to this22roblem which, the more we get involved in the subtleties of Mahler’s method, the more it takes apreponderant place.In the case K = C , we consider the formal series in Z [[ x ]]: L = ∞ Y i =0 (1 − x i ) − , L r = ∞ X i =0 x i r Q i − j =0 (1 − x j ) , r ≥ | x | <
1, to functions satisfying : L ( x ) = (1 − x ) L ( x ) , L r ( x ) = (1 − x )( L r ( x ) − x r ) , r ≥ . By a result of Kubota [25, Theorem 2], (see also T¨opfer, [51, Lemma 6]), if ( L i ) i ∈I (with I ⊂ Z )are algebraically dependent, then they also are C -linearly dependent modulo C ( x ) in the followingsense. There exist complex numbers ( c i ) i ∈I not all zero, such that: m X i =1 c i L i ( x ) = f ( x )with f ( x ) ∈ C ( x ). P. Bundschuh pointed out that L , L , L , . . . are algebraically independent. Toobtain this property, he studied the behaviour of these functions near the the unit circle. For along time the author was convinced of the algebraic independence of the functions L , L , . . . untilvery recently, when T. Tanaka and H. Kaneko exhibited non-trivial linear relations modulo C ( x )involving L , . . . , L s for all s ≥
1, some of which looking very simple, such as the relation: L ( x ) − L ( x ) = 1 − x. Beyond transcendence and algebraic independence, the next step in the study of the arithmetic ofMahler’s numbers is that of quantitative results such as measures of algebraic independence. Veryoften, such estimates are not mere technical refinements of well known results but deep informationon the diophantine behaviour of classical constants; everyone knows the important impact thatBaker’s theory on quantitative minorations of linear forms in logarithms on algebraic groups hadin arithmetic geometry.Rather sharp estimates are known for complex Mahler’s values. We quote here a result ofNishioka [36, 37] and [38, Chapter 12] (it has been generalised by Philippon: [45, Theorem 6]).
Theorem 12 (Nishioka)
Let us assume that, in the notations previously introduced, K = C .Let L be a number field embedded in C . Let f , . . . , f m be formal series of L [[ x ]] , let us write f ∈ Mat n × ( L [[ x ]]) for the column matrix whose entries are the f i ’s. Let A ∈
Mat n × n ( L ( x )) , b ∈ Mat n × ( L ( x )) be matrices. Let us assume that:1. f , . . . , f m are algebraically independent over C ( x ) ,2. For all i , the formal series f i ( x ) converges for x complex such that | x | < ,3. f ( x d ) = A ( x ) · f ( x ) + b ( x ) . et α ∈ L be such that < | α | < , not a zero or a pole of A and not a pole of b . Then, thereexists a constant c > effectively computable depending on α, f , with the following property.For any H, N ≥ integers and any non-zero polynomial P ∈ Z [ X , . . . , X m ] whose partialdegrees in every indeterminate do not exceed N and whose coefficients are not greater that H inabsolute value, the number P ( f ( α ) , . . . , f m ( α )) is non-zero and the inequality below holds: log | P ( f ( α ) , . . . , f m ( α )) | ≥ − c N m (log H + N m +2 ) . (33)We sketch how Theorem 12 implies the algebraic independence of f ( α ) , . . . , f m ( α ) and g ( α ) with f , . . . , f m , g ∈ Q [[ x ]] algebraically independent over Q [[ x ]] satisfying linear functional equations asin Theorem 11 and g satisfying g ( x d ) = a ( x ) g ( x ) + b ( x ) , with a, b ∈ Q ( x ), A , b with rational coefficients, and α not a pole of all these rational functions. Ofcourse, this is a simple corollary of Theorem 12. However, we believe that the proof is instructive;it follows closely Philippon’s ideas in [45]. The result is reached because the estimates of Theorem12 are precise enough. In particular, the separation of the quantities H and N in (33) is crucial. For the purpose indicated at the end of the last subsection, we assume that α ∈ Q × . This hypothesisin not strictly necessary and is assumed only to simplify the exposition of the proof; by the way,the reader will remark that several other hypotheses we assume are avoidable. Step (AP) . For all N ≥
1, we choose a non-zero polynomial P N ∈ Z [ x, X , . . . , X m , Y ] of partialdegrees ≤ N in each indeterminate, such that, writing F N ( x ) := P N ( x, f ( x ) , . . . , f m ( x ) , g ( x )) = c ν ( N ) x ν ( N ) + · · · ∈ Q [[ x ]] , c ν ( N ) = 0 , we have ν ( N ) ≥ N m +2 (we have already justified why such a kind of polynomial exists).Just as in the proof of Theorem 11 we construct, for each N ≥
1, a sequence of polynomials( P N,k ) k ≥ in Z [ x, X , . . . , X m , Y ] recursively in the following way: P N, := P N ,P N,k := c ( x ) N P N,k − ( x d , a ( x ) X + b ( x ) , . . . , a m ( x ) X m + b m ( x ) , a ( x ) Y + b ( x )) , where c ( x ) ∈ Z [ x ] \ { } is chosen so that ca i , cb j , ca, cb belong to Z [ x ]. The following estimates areeasily obtained: deg Z P N,k ≤ N, for Z = X , . . . , X m , Y, deg x P N,k ≤ c d k N,h ( P N,k ) ≤ c ( N ) + c d k N, where c , c are positive real numbers effectively computable depending on α, f and g , and c ( N ) > N (it depends on the choice of the polynomials P N ).Let us assume by contradiction that g ( α ) is algebraic over the field F := Q ( f ( α ) , . . . , f m ( α )) ,
24f transcendence degree m over Q . We observe that, after the identity principle of analytic functionswe have, for k big enough depending on α, f , g and N : P N,k ( α, f ( α ) , . . . , f m ( α ) , g ( α )) ∈ F × . (34)Let e Q ∈ F [ X ] \ { } be the minimal polynomial of g ( α ), algebraic over F . We can write e Q = a + a X + · · · + a r − X r − + X r with the a i ’s in F . Multiplying by a common denominator, weobtain a non-zero polynomial Q ∈ Z [ X , . . . , X m , Y ] such that Q ( f ( α ) , . . . , f m ( α ) , g ( α )) = 0, withthe property that the polynomial Q ∗ = Q ( f ( α ) , . . . , f m ( α ) , Y ) ∈ F [ Y ] is irreducible. Step (NV) . Let us denote by ∆ k the resultant Res Y ( P N,k , Q ) ∈ Z [ x, X , . . . , X m ]. If δ k :=∆ k ( α, f ( α ) , . . . , f m ( α )) ∈ F vanishes for a certain k , then Q ∗ and P ∗ N,k := P N,k ( α, f ( α ) , . . . , f m ( α ) , Y ) ∈ F [ Y ]have a common zero. Since Q ∗ is irreducible, we have that Q ∗ divides P ∗ N,k in F [ Y ] and P N,k ( α, f ( α ) , . . . , f m ( α ) , g ( α )) = 0;this cannot happen for k big enough by the identity principle of analytic functions (34) so that wecan assume that for k big enough, δ k = 0, ensuring that ∆ k is not identically zero; the estimatesof the height and the degree of ∆ k quoted below are simple exercises and we do not give details oftheir proofs: deg Z ∆ k ≤ c N, Z = X , . . . , X m , deg x ∆ k ≤ c d k N,h (∆ k ) ≤ c ( N ) + c d k N, where c , c , c are positive numbers effectively computable depending on α, f and g , while theconstant c ( N ) depends on these data and on N .Let D be a non-zero positive integer such that Dα ∈ Z . Then, writing∆ ∗ k := D Nd k ∆ k ( α, X , . . . , X m ) , we have ∆ ∗ k ∈ Z [ X , . . . , X m ] \ { } anddeg X i ∆ ∗ k ≤ c N,h (∆ ∗ k ) ≤ c ( N ) + c d k N. Step (LB) . By Nishioka’s Theorem 12, we have the inequality (for k big enough):log | ∆ ∗ k ( f ( α ) , . . . , f m ( α )) | ≥ c N m ( d k N + c ( N )) , (35)where c ( N ) is a constant depending on N .To finish our proof, we need to find an upper bound contradictory with (35); it will be obtainedby analytic estimates as usual. 25 tep (UB) . Looking at the proof of Lemma 5.3.1 of [53], and using in particular inequality (1.2.7)of loc. cit., we verify the existence of constants c , c depending on α, f , g , c ( α, ǫ ) depending on α and ǫ , and c ( N ) depending on α, f , g and N , such that:log | ∆ ∗ k ( f ( α ) , . . . , f m ( α )) | ≤ log( N + c ) + c h ( P ∗ N,k ) + ( N + 1) h ( Q ) +log | c ν N | + ν ( N ) d k log | α | + ǫ ≤ c (log N + d k N ) + c ( N ) − c ( α, ǫ ) d k ν ( N ) ≤ c d k N + c ( N ) − c ( α, ǫ ) d k N m +2 . (36)Finally, it is easy to choose N big enough, depending on c , c , c but not on c , c so that, for k big enough, the estimates (35) and (36) are not compatible: this is due to the particular shape of(33), with the linear dependence in log H . In the sketch of proof of the previous subsection, the reader probably observed a kind of inductionstructure; a measure of algebraic independence for m numbers delivers algebraic independencefor m + 1 numbers. The question is then natural: is it possible to obtain a measure of algebraicindependence for m + 1 numbers allowing continue the process and consider m + 2 numbers? In fact yes , there always is an inductive structure of proof, but no , it is not just a measure for m numbers which alone implies a measure for m + 1 numbers. Things are more difficult than theylook at first sight and the inductive process one has to follow concerns other parameters as well.For instance, the reader can verify that it is unclear how to generalise the arguments of 4.2.1 andwork directly with a polynomial Q which has a very small value at ω = ( f ( α ) , . . . , f m ( α ) , g ( α )).Algebraic independence theory usually appeals to transfer techniques , as an alternative to directestimates at ω . A detour on a theorem of Nesterenko might be useful to understand what is goingon so our discussion now temporarily leaves Mahler’s values, that will be reconsidered in a littlewhile.Precise multiplicity estimates in differential rings generated by Eisenstein’s series obtained byNesterenko, the criteria for algebraic independence by Philippon already mentioned in this paperand a trick of the stephanese team (cf. 2.2.1) allowed Nesterenko, in 1996, to prove the followingtheorem (see [33, 34]): Theorem 13 (Nesterenko)
Let E , E , E the classical Eisenstein’s series of weights 2 , , re-spectively, normalised so that lim ℑ ( z ) →∞ E i ( z ) = 1 (for ℜ ( z ) bounded), let z be a complex numberof strictly positive imaginary part. Then, three of the four complex numbers e π i z , E , E , E arealgebraically independent. Although Eisenstein’s series are not directly related to Mahler’s functions there is a hidden linkand the ideas introduced to prove Theorem 13 influenced Nishioka in her proof of Theorem 33 aswell as other results by Philippon that we will mention below. This is why we cannot keep silenton this aspect.First of all, we recall that in [45], Philippon showed how to deduce the algebraic independence of π, e π , Γ(1 /
4) ( ), a well known corollary of Nesterenko’s Theorem 13, from a measure of algebraic Philippon’s result is in fact more general than the algebraic independence of these three numbers, but less generalthan Nesterenko’s theorem 13, although it historically followed it. Our arguments in 4.2.1 are strongly influenced byit. In [45], Philippon proposes alternative, simpler proofs for Nesterenko’s theorem. π, Γ(1 /
4) by Philibert in [42]. This implication was possible because Philibert’sresult was sharp enough. It is however virtually impossible to deduce Theorem 13 or the quantitativeresult in [33] which can also be deduced from corollary of [45, Theorem 3] ( ) just by usingPhilibert’s result.In the proof of Theorem 33, Nishioka proves (just as Nesterenko does in [33]) a more generalmeasure of the smallness of the values that a polynomial with rational integer coefficients assumesat (1 , f ( α ) , . . . , f m ( α ) , g ( α )), restricting the choice of that polynomial in a given unmixed homo-geneous ideal I . The proof of such a kind of result (cf. [38, Lemma 2.3]) involves induction on thedimension of I .Assuming the existence of an ideal I with minimal dimension “very small” at ω , it is possible,looking at its reduced primary decomposition, to concentrate our attention to I = p prime. The“closest point principle” of [46, p. 89] allows to show the existence, in the projective variety V associated to p , of a point β which is at a very short distance from ω (see also [34, Proposition 1.5]).This shows that in this problem, to measure a polynomial or an ideal at ω it is more advantageousto do it at β ; indeed, all the polynomials of p vanish at β .At this point, it remains to construct an unmixed homogeneous ideal J of dimension dim p − P N,k etc., with the important difference that now, all the estimate depend very much on the choice,that must then use Siegel’s Lemma. Another important tool that has to be used is a multiplicityestimate , that belongs to step (NV), proved by Nishioka [37, Theorem 4.3], that we reproduce here.
Theorem 14 (Nishioka)
Let f , . . . , f m be satisfying the hypotheses of Theorem 12, so that forall P ∈ C [ x, X , . . . , X m ] \ { } , the function F ( x ) := P ( x, f ( x ) , . . . , f m ( x )) has the expansion F ( x ) = c ν x ν + · · · , with c ν = 0 . There exists a constant c > , depending on f , . . . , f m only, with the followingproperty. If P is as above and N := max { , deg x P } and N := max { , deg X P, . . . , deg X m P } ,then ν ≤ c N N m . This result is very similar to Nesterenko’s multiplicity estimate [34, Chapter 10, Theorem 1.1] andagain, its proof essentially follows Nesterenko’s ideas.The ideal J previously mentioned is defined as the ideal generated by p and a polynomialobtained from P N,k by homogenisation, substitution x = α , and a good choice of N, k takinginto account the magnitude of the coefficients of the series f i , g . Indeed, one proves that such apolynomial cannot belong to p . The closest point principle is necessary in this kind of proof.The arguments of the above discussion can be modified to obtain the analog of Theorem 13 forvalues of Mahler’s functions at general complex numbers, obtained by Philippon (cf. [45, Theorem4]). Here, L is again a number field embedded in C and d > Theorem 15 (Philippon)
Under the same hypotheses and notations of Theorem 12, if α is a com-plex number with < | α | < , then, for n big enough, the complex numbers α, f ( α d n ) , . . . , f m ( α d n ) ∈ C generate a subfield of C of transcendence degree ≥ m . A result asserting that, for a polynomial P ∈ Z [ X , . . . , X ] \{ } whose partial degrees in every indeterminate donot exceed N >
H > | P ( e π , π, Γ(1 / | ≥− c ( ǫ )( N + log H ) ǫ , where c is an absolute constant depending on ǫ only. The dependence in ǫ is completelyexplicit. Similar, although simpler arguments are in fact commonly used to obtain measures of transcendence.Several authors deduce them from measures of linear algebraic approximation; see for exampleAmou, Galochkin and Miller [3, 21, 32] ( ). These results often imply that Mahler’s values areMahler’s S -numbers .In positive characteristic, it is well known that separability difficulties occur preventing to deducegood measures of transcendence from measures of linear algebraic approximation ( ). In [20], Denisproves the following result, where d > Theorem 16 (Denis)
Let us consider a finite extension L of K = F q ( θ ) , f ∈ L [[ x ]] , α ∈ L suchthat < | α | < . Let us assume that f is transcendental over C ( x ) , convergent for x ∈ C such that | x | < , and satisfying the linear functional equation f ( x q ) = a ( x ) f ( x ) + b ( x ) , with a, b ∈ L ( x ) .For all n big enough, we have the following property. Let β = α d n . Then, there exists aneffectively computable constant c > depending on β, f only, such that, given any non-constantpolynomial P ∈ A [ X ] , log | P ( f ( β )) | ≥ − deg X ( P ) (deg X ( P ) + deg θ ( P )) . (37)This result yields completely explicit measures of transcendence of e π , of Carlitz’s logarithms ofelements of K , and of certain Carlitz-Goss’s zeta values (see Section 4.3 for definitions).To prove Theorem 16, Denis uses the following multiplicity estimate. Theorem 17
Let K be any (commutative) field and f ∈ K (( x )) be transcendental satisfying thefunctional equation f ( x d ) = R ( x, f ( x )) with R ∈ K ( X, Y ) and h Y ( R ) < d . Then, if P is apolynomial in K [ X, Y ] \ { } such that deg X P ≤ N and deg Y P ≤ M , we haveord x =0 P ( x, f ( x )) ≤ N (2 M d + N h X ( Q )) . It would be interesting to generalise such a multiplicity estimate for several algebraically indepen-dent formal series and obtain a variant of T¨opfer’s [52, Theorem 1] (see [11, 12, 41] to check thedifficulty involved in the research of an analogue of Nesterenko’s multiplicity estimate for Drinfeldquasi-modular forms). This could be helpful to obtain analogues of Theorem 12 for Mahler’s valuesin fields of positive characteristic. Some results hold for series which satisfy functional equations which are not necessarily linear. As first remarked by Lang, for complex numbers, there is equivalence between measures of transcendence andmeasures of linear algebraic approximation in the sense that, from a measure of linear algebraic approximation onecan get a measure of transcendence and then again, a measure of linear algebraic approximation which is essentiallythat of the beginning, with a controllable degradation of the constants. .3 Algebraic independence of Carlitz’s logarithms For the rest of this chapter, we will give some application of Mahler’s method and of Anderson-Brownawell-Papanikolas method to algebraic independence of Carlitz’s logarithms of algebraic ele-ments of C and of some special values of Carlitz-Goss zeta function at rational integers. Results ofthis part are not original since they are all contained in the papers [39] by Papanikolas and [14] byChieh-Yu Chang and Jing Yu. But the methods we use here are slightly different and self-contained.Both proofs of the main results in [39, 14] make use of a general statement [39, Theorem 5.2.2]which can be considered as a variant of Grothendieck period conjecture for a certain generalisationof Anderson’s t -motives, also due Papanikolas. To apply this result, the computation of motivicGalois groups associated to certain t -motives is required.Particular cases of these results are also contained in Denis work [19], where he applies Mahler’smethod and without appealing to Galois theory. Hence, we follow the ideas of the example in 4.1.1and the main worry here is to develop analogous proofs in the Drinfeldian framework. In 4.1.1 theexplicit computation of the transcendence degree of the field generated by L , L , . . . was pointedout as a problem. But we have already remarked there, that if L , L , . . . are algebraic dependent,then they also are C -linearly dependent modulo C ( x ). This property, consequence of a result byKubota, is easy to obtain because the matrix of the linear difference system of equations satisfiedby the L i ’s has the matrix of its associated homogeneous system which is diagonal .For Λ = e πA with e π as in (16), the exponential function e Car := e Λ (30) can be explicitly writtenas follows: e Car ( z ) = X i ≥ z q i [ i ][ i − q · · · [1] q i − , where [ i ] := θ q i − θ ( i ≥ F q -linear surjective function e Car : C → C . The formal series log Car , reciprocal of e Car in 0,converges for | z | < q q/ ( q − = | e π | . Its series expansion can be computed explicitly:log Car ( z ) = X i ≥ ( − i z q i [ i ][ i − · · · [1] . The first Theorem we shall prove in a simpler way is the following (cf. [39, Theorem 1.2.6]):
Theorem 18 (Papanikolas)
Let ℓ , . . . , ℓ m ∈ C be such that e Car ( ℓ i ) ∈ K alg. ( i = 1 , . . . , m ). If ℓ , . . . , ℓ m are linearly independent over K , then they also are algebraically independent over K . Let us write A + = { a ∈ A, a monic } . In [22], Goss introduced a function ζ , defined over C × Z p with values in C , such that for n ≥ ζ ( θ n , n ) = X a ∈ A + a n ∈ K ∞ . In the following, we will write ζ ( n ) for ζ ( θ n , n ). For n ∈ N , let us also write Γ( n ) := Q si =0 D n i i ∈ K , n + n q + · · · + n s q s being the expansion of n − q and D i being the polynomial[ i ][ i − q · · · [1] q i − . It can be proved that z/e Car ( z ) = P ∞ n =0 B n z n Γ( n +1) for certain B n ∈ K . The29o-called Bernoulli-Carlitz relations can be obtained by a computation involving the logarithmicderivative of e Car ( z ): for all m ≥ ζ ( m ( q − e π m ( q − = B m Γ( m ( q −
1) + 1) ∈ K. (38)In particular, one sees that e π q − = ( θ q − θ ) ζ ( q − ∈ K ∞ . We also have the obvious relations: ζ ( mp k ) = ζ ( m ) p k , m, k ≥ . (39)The second theorem we are going to prove directly is: Theorem 19 (Chang, Yu)
The algebraic dependence relations over K between the numbers ζ (1) , ζ (2) , . . . are generated by Bernoulli-Carlitz’s relations (38) and the relations (39). In this subsection we develop an analogue of [51, Lemma 6], for the same purpose we needed it in4.1.1.We consider here a perfect field U of characteristic p > F q and a F q -automorphism τ : U → U . Let U be the subfield of constants of τ , namely, the subset of U whose elements s aresuch that τ s = s .For example, we can consider U = S n ≥ C ( x /p n ) with τ defined as the identity over C , with τ x = x q . Another choice is to consider U = S n ≥ C ( t /p n ), with τ defined by τ c = c /q for all c ∈ C and τ t = t . In the first example we have U = C while in the second, U = S n ≥ F q (( t /p n )).More generally, after 3.2, we can take either U = S n ≥ K ( x /p n ) or the field S n,m ≥ C ( x /p n , t /p m )(which is contained in the previous field) with the corresponding automorphism τ (these settingswill essentially include the two examples above). In the first case, we have U = F q hh t ii , and in thesecond case, we have U = S n ≥ F q (( t /p n )).Let us also consider the ring R = U [ X , . . . , X N ] and write, for a polynomial P = P λ c λ X λ ∈ R , P τ as the polynomial P λ ( τ c λ ) X λ . Let D , . . . , D N be elements of U × , B , . . . , B N be elements of U and, for a polynomial P ∈ R , let us write e P = P τ ( D X + B , . . . , D N X N + B N ) . We prove the following two Propositions, which provide together the analogue in positive char-acteristic of Kubota [25, Theorem 2].
Proposition 20
Let P ∈ R be a non-constant polynomial such that e P /P ∈ R . Then, there existsa polynomial G ∈ R of the form G = P i c i X i + B p such that e G/G ∈ R , where c , . . . , c N ∈ U arenot all vanishing and B ∈ R . If W is the subfield generated by F q and the coefficients of P , thenthere exists M ≥ such that for each coefficient c of G , c p M ∈ W . roof. If P ∈ R is such that e P = QP for Q ∈ R one sees, comparing the degrees of e P and P ,that Q ∈ U and if P is non-zero, Q = 0. The subset of R of these polynomials is a semigroup S containing U . If P ∈ S satisfies e P = QP , then F := ∂P/∂X i belongs to S since e F = D − i QF .Similarly, if P = F p ∈ S with F ∈ S then F ∈ S as one sees easily that in this case, e F = Q /p F .By hypothesis, S contains a non-constant polynomial P . We now show that the polynomial G ∈ S as in the Proposition can be constructed by iterated applications of partial derivatives ∂ = ∂/∂X , . . . , ∂ N = ∂/∂X N and p -root extrations starting from P .Let P be as in the hypotheses. We can assume that P is not a p -th power. We can write: P = X λ =( λ ,...,λ N ) ∈{ ,...,p − } N c λ X λ , c λ ∈ R p . Let M := max { λ + · · · + λ N , c λ = 0 } . We can write P = P + P with P := X λ + ··· + λ N = M c λ X λ . There exists ( β , . . . , β N ) ∈ { , . . . , p − } N with β + · · · + β N = M − P ′ := ∂ β · · · ∂ β N N P = N X i =1 c ′ i X i + c ′ ∈ S \ { } , c ′ , c ′ , . . . , c ′ N ∈ R p , where ∂ β · · · ∂ β N N P = N X i =1 c ′ i X i , ∂ β · · · ∂ β N N P = c ′ . If (case 1) the polynomials c ′ , . . . , c ′ N are all in U , then we are done. Otherwise, (case 2), thereexists i such that c ′ i is non-constant (its degree in X j is then ≥ p for some j ). Now, c ′ i = ∂ i P ′ belongs to ( R p ∩ S ) \ { } and there exists s > c ′ i = P ′′ p s with P ′′ ∈ S which is not a p -thpower. We have constructed an element P ′′ of S which is not a p -th power, whose degrees in X j are all strictly smaller than those of P for all j (if the polynomial depends on X j ).We can repeat this process with P ′′ at the place of P and so on. Since at each stage we geta polynomial P ′′ with partial degrees in the X j strictly smaller than those of P for all j (if P ′′ depends on X j ), we eventually terminate with a polynomial P which has all the partial degrees < p in the indeterminates on which it depends, for which the case 1 holds.As for the statement on the field W , we remark that we have applied to P an algorithmwhich constructs G from P applying finitely many partial derivatives and p -th roots extractionssuccessively, the only operations bringing out of the field W being p -root extractions. Hence, theexistence of the integer M is guaranteed.We recall that U is the subfield of U whose elements are the s ∈ U such that τ s = s . Let V bea subgroup of U × such that V \ V p = ∅ . Proposition 21
Under the hypotheses of Proposition 20, let us assume that for all D ∈ V \ { } ,the only solution s ∈ U of τ s = Ds is zero and that D , . . . , D N ∈ V \ V p . Then, the polynomial G ∈ R given by this Proposition is of the form G = P i c i X i + c with c , . . . , c N ∈ U and c ∈ U . oreover, if c i , c j = 0 for ≤ i < j ≤ N , then D i = D j . Let I be the non-empty subset of { , . . . , N } whose elements i are such that c i = 0 , let D i = D for all i ∈ I . Then, c = τ ( c ) D + 1 D X i ∈I c i B i . Proof.
Proposition 20 gives us a polynomial G with e G/G ∈ R , of the form P i c i X i + B p with c i ∈ U not all vanishing and B ∈ R . Let sX pλ be a monomial of maximal degree in B p . Since e G = QG with Q ∈ U × , we have τ s = ( D λ · · · D λ N N ) − p Qs . Moreover, τ ( c i ) = D − i Qc i for all i . Hence, if i is such that c i = 0, r := s/c i satisfies τ r = D i ( D λ · · · D λ N N ) − p r . Now, D i ( D λ · · · D λ N N ) − p = 1(because D i ∈ V \ V p ) and r = 0, that is s = 0. This shows that B ∈ U . Let us suppose that1 ≤ i, j ≤ N are such that i = j and c i , c j = 0. Let us write r = c i /c j ; we have τ r = D j /D i r ,from which we deduce r ∈ U in case D j /D i = 1 and r = 0 otherwise. The Proposition is proveddividing P i c i X i + B p by c j with j = 0 and by considering the relation e P = QP , once observedthat Q = D .We proceed, in the next two subsections, to prove Theorems 18 and 19. We will prove the firsttheorem applying Propositions 20 and 21 to the field U = S n ≥ C ( t /p n ) and then by using thecriterion [6, Theorem 1.3.2] and we will prove the second theorem applying these propositions tothe field U = S n ≥ K alg. ( x /p n ) and then by using Theorem 11. For β ∈ K alg. such that | β | < q q/ ( q − , we will use the formal series in K alg. (( t )) L β ( t ) = β + ∞ X i =1 ( − i β q i ( θ q − t ) · · · ( θ q i − t ) , defining holomorphic functions for | t | < q q with L β ( θ ) = log Car β ( ).We denote by W one of the following fields: K alg. , K alg. ∞ , C . For f = P i c i t i ∈ W (( t )) and n ∈ Z we write f ( n ) := P i c q n i t i ∈ W (( t )), so that f ( − = P i c /qi t i . We have the functional equation L ( − β ( t ) = β /q + L β ( t ) t − θ . The function L β allows meromorphic continuation to the whole C , withsimple poles at the points θ q , θ q , . . . , θ q n , . . . of residue(log Car β ) q , (log Car β ) q D q , . . . , (log Car β ) q n D qn − . . . . (40)Let β , . . . , β m be algebraic numbers with | β | < q q/ ( q − , let us write L i = L β i for i = 1 , . . . , m .Let us also consider the infinite product Ω in (27), converging everywhere to an entire holomorphicfunction with zeros at θ q , θ q , . . . , and write L = − Ω − , which satisfies the functional equation L ( − ( t ) = L ( t ) t − θ , Papanikolas uses these series in [39]. It is also possible to work with the series P ∞ i =0 e Car ((log
Car β ) /θ i +1 ) t i . L ( θ ) = e π , meromorphic with simple poles at the points θ q , θ q , . . . , θ q n , . . . , with residues e π q , e π q D q , . . . , e π q n D qn − , . . . (41)We now prove the following Proposition. Proposition 22
If the functions L , L , . . . , L m are algebraically dependent over K alg. ( t ) , then e π, log Car β , . . . , log Car β m are linearly dependent over K .Proof. The functions L i are transcendental, since they have infinitely many poles. Without loss ofgenerality, we may assume that m ≥ ≤ n ≤ m the functions obtainedfrom the family ( L , L , . . . , L m ) discarding L n are algebraically independent over K alg. ( t ).We now apply Propositions 20 and 21. We take U := S n ≥ C ( t /p n ), which is perfect, and τ : U → U the q -th root map on C (inverse of the Frobenius map), such that τ ( t ) = t ; this is an F q -automorphism. Moreover, we take N = m + 1, D = · · · = D N = ( t − θ ) − ,( B , . . . , B N ) = (0 , β /q , . . . , β /qm ) , and V = ( t − θ ) Z .Let T ⊂ C [[ t ]] be the subring of formal series converging for all t ∈ C with | t | ≤
1, let L be itsfraction field. Let f ∈ L be non-zero. A variant of Weierstrass preparation theorem (see [4, Lemma2.9.1]) yields a unique factorisation: f = λ (cid:18) Y | a | ∞ ≤ ( t − a ) ord a ( f ) (cid:19)(cid:18) ∞ X i =1 b i t i (cid:19) , (42)where 0 = λ ∈ C , sup i | b i | <
1, and | b i | →
0, the product being over a finite index set. Taking intoaccount (42), it is a little exercise to show that U = S i ≥ F q ( t /p i ) and that for D ∈ V \ { } , thesolutions in U of f ( − = Df are identically zero (for this last statement, use the transcendenceover U of Ω ).Let P ∈ R be an irreducible polynomial such that P ( L , L , . . . , L m ) = 0; we clearly have e P = QP with Q ∈ U and Propositions 20 and 21 apply to give c ( t ) , . . . , c m ( t ) ∈ U not all zeroand c ( t ) ∈ U such that c ( t ) = ( t − θ ) c ( − ( t ) + ( t − θ ) m X i =1 c i ( t ) β /qi . (43)We get, for all k ≥ c ( t ) = − m X i =1 c i ( t ) β i + k X h =1 ( − h β q h i ( θ q − t )( θ q − t ) · · · ( θ q h − t ) ! (44)+ c ( k +1) ( t )( θ q − t )( θ q − t ) · · · ( θ q k +1 − t ) . We endow L with a norm k·k in the following way: if f ∈ L × factorises as in (42), then k f k := | λ | .Let g be a positive integer. Then k · k extends in a unique way to the subfield L g := { f : f p g ∈ L} .33f ( f i ) i ∈ N is a uniformly convergent sequence in L g (on a certain closed ball centered at 0) such that k f i k →
0, then f i → g ≥ c ( t ) , c ( t ) , . . . , c m ( t ) ∈ L g . Hence c ( t ) , . . . , c m ( t ) ∈ F q ( t /p g ) and k c i k = 1 if c i = 0. This implies that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 c i β /qi (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ max i {| β /qi |} < q / ( q − . By (43), k c k ≤ q q/ ( q − . Indeed, two cases occur. The first case when k c ( − k ≤ max i {| β /qi |} ; herewe have k c k < q q/ ( q − because k c ( − k = k c k /q by (42) and max i {| β i |} < q q/ ( q − by hypothesis.The second case occurs when the inequality k c ( − k > max {| β /q | , . . . , | β /qm |} holds. In this case,max {k c ( − ( t − θ ) k , k ( t − θ ) P mi =1 c i ( t ) β /qi k} = k c ( − ( t − θ ) k which yields k c k = q q/ ( q − by (43).Going back to (44) we see that the sequence of functions E h ( t ) = c ( h +1) ( t )( θ q − t )( θ q − t ) · · · ( θ q h +1 − t )converges uniformly in every closed ball included in { t : | t | < q q } , as the series defining the functions L i ( i = 1 , . . . , m ) do. We want to compute the limit of this sequence: we have two cases. First case. If k c k < q q/ ( q − , then, there exists ǫ > k c k = q ( q − ǫ ) / ( q − . Then, for all h ≥ k c ( h +1) k = k c k q h +1 = q ( q h +2 − ǫq h +1 ) / ( q − . On the other side: k ( θ q − t )( θ q − t ) · · · ( θ q h +1 − t ) k = | θ | q + ··· + q h +1 = q q ( q h +1 − / ( q − . Hence, k E h k = q qh +2 − ǫqh +1 q − − qh +2 − qq − = q q − ǫqh +1 q − → , which implies E h → P mi =1 c i ( t ) L i ( t ) + c ( t ) = 0. Let g be minimal such that there exists a non-trivial linear relation as above, with c , . . . , c m ∈ U ∩ L g ; we claim that g = 0. Indeed, if g > c , . . . , c m F q and there exists a non-trivial relation P mi =1 d i ( t ) L i ( t ) p g + d ( t ) = 0 with d , . . . , d m ∈ F q [ t ] not all zero, d ( t ) ∈ C ( t ) and max i { deg t d i } minimal, non-zero. But letting the operator d/dt act on this relation we get a non-trivial relation with strictly lower degree because dF p /dt = 0,leading to a contradiction.Hence, g = 0 and c , . . . , c m ∈ F q ( t ). This also implies that c ∈ C ; multiplying by a commondenominator, we get a non-trivial relation P mi =1 c i ( t ) L i ( t ) + c ( t ) = 0 with c , . . . , c m ∈ F q [ t ] and c ∈ C ( t ). The function c being algebraic, it has finitely many poles. This means that m X i =1 c i ( t ) L i ( t )has finitely many poles but for all i , L i has poles at θ q , θ q , . . . with residues as in (40), which impliesthat P mi =1 c i ( t ) L i ( t ) has poles in θ q , θ q , . . . . Since the functions c i belong to F q [ t ], they vanish only34t points of absolute value 1, and the residues of the poles are multiples of P mi =0 c i ( θ ) q k (log Car β i ) q k ( k ≥
1) by non-zero factors in A . They all must vanish: this happens if and only if m X i =1 c i ( θ ) log Car β i = 0 , where we also observe that c i ( θ ) ∈ K ; the Proposition follows in this case. Second case.
Here we know that the sequence E h converges, but not to 0 and we must computeits limit. Let ν be in C with | ν | = 1. Then, there exists µ ∈ F alg. q × , unique such that | ν − µ | < λ ∈ C is such that | λ | = q q/ ( q − , there exists µ ∈ F alg. q × unique with | λ − µ ( − θ ) q/ ( q − | < q q/ ( q − . (45)We have: c ( t ) = λ Y | a |≤ (cid:16) t /p g − a (cid:17) ord a c X i ≥ b i t i/p g , with λ ∈ C × , the product being finite and | b i | < i so that k c k = | λ | .Let µ ∈ F alg. q × be such that (45) holds, and write: c ( t ) = ( λ − µ ( − θ ) q/ ( q − ) Y | a |≤ (cid:16) t /p g − a (cid:17) ord a c X i ≥ b i t i/p g ,c ( t ) = µ ( − θ ) q/ ( q − Y | a |≤ (cid:16) t /p g − a (cid:17) ord a c X i ≥ b i t i/p g , (46)so that c ( t ) = c ( t ) + c ( t ), k c k < q q/ ( q − and k c k = q q/ ( q − . For all h , we also write: E ,h ( t ) = c ( h +1)1 ( t )( θ q − t )( θ q − t ) · · · ( θ q h +1 − t ) , E ,h ( t ) = c ( h +1)2 ( t )( θ q − t )( θ q − t ) · · · ( θ q h +1 − t ) . Following the first case, we easily check that E ,h ( t ) → { t : | t | < q q } . It remains to compute the limit of E ,h ( t ).We look at the asymptotic behaviour of the images of the factors in (46) under the operators f f ( n ) , n → ∞ . The sequence of functions (1 + P i ≥ b i t i/p g ) ( n ) converges to 1 for n → ∞ uniformly on every closed ball as above. Let E be the finite set of the a ’s involved in the finiteproduct (46), take a ∈ E . If | a | <
1, then a ( n ) → t /p g − a ) ( n ) → t /p g . If | a | = 1, there exists µ a ∈ F alg. q × such that | a − µ a | < n a > s →∞ a ( sn a ) = µ a ,whence lim s →∞ ( t /p g − a ) ( sn a ) = t /p g − µ a .Let us also denote by ˜ n > µ q ˜ n = µ . Let N be thelowest common multiple of ˜ n and the n a ’s with a varying in E . Then the sequence of functions: Y | a |≤ (cid:16) t /p g − a (cid:17) ord a c X i ≥ b i t i/p g ( Ns ) , s ∈ N Z ∈ F alg. q ( t /p g ).For n ∈ N , let us write: V n ( t ) := µ q n ( − θ ) q n +1 / ( q − ( θ q − t )( θ q − t ) · · · ( θ q n +1 − t ) . We have:( − θ ) q/ ( q − n +1 Y i =1 (cid:18) − tθ q i (cid:19) − = ( − q/ ( q − θ q/ ( q − θ ( q + ··· + q n +1 ) n +1 Y i =1 ( θ q i − t ) − = ( − q/ ( q − θ q n +2 / ( q − n +1 Y i =1 ( θ q i − t ) − . Hence, lim n →∞ θ q/ ( q − / (( θ q − t )( θ q − t ) · · · ( θ q n +1 − t )) − = Ω ( t ) − from which we deduce thatlim s →∞ E ,sN ( t ) = c ( t ) L ( t ) with c ∈ F alg. q × ( t /p g ). We have proved that for some c , . . . , c m ∈ F q ( t /p g ) , c ∈ F q ( t /p g ) × and c ∈ K alg. ( t /p g ), P mi =0 c i L i + c = 0. Applying the same tool used inthe first case we can further prove that in fact, g = 0. If c is not defined over F q , then applyingthe operator f f ( − we get another non-trivial relation c ′ + P mi =1 c i L i = c ′ with c ′ ∈ K alg. ( t )and c ′ ∈ F × q ( t ) not equal to c ; subtracting it from the former relation yields L ∈ K alg. ( t ) whichis impossible since Ω is transcendental over C ( t ). Hence c belongs to F q ( t ) too. Multiplyingby a common denominator in F q [ t ] and applying arguments of the first case again (by using theexplicit computation of the residues of the poles of L at θ q , θ q , . . . ), we find a non-trivial relation c ( θ ) e π + P mi =1 c i ( θ ) log Car β i = 0. Proof of Theorem 18 . If ℓ ∈ C is such that e Car ( ℓ ) ∈ K alg. , then there exist a, b ∈ A , β ∈ K alg. with | β | < q q/ ( q − such that ℓ = a log Car β + b e π . This well known property (also used in [39], seeLemma 7.4.1), together with Theorem 3.1.1 of [6], implies Theorem 18. Let s ≥ n denote the s -th Carlitz’s polylogarithm by:Li s ( z ) = ∞ X k =0 ( − ks z q k ([ k ][ k − · · · [1]) s , so that Li ( z ) = log Car ( z ) (the series Li s ( z ) converges for | z | < q sq/ ( q − ).For β ∈ K alg. ∩ K ∞ such that | β | < q sq/ ( q − (a discussion about this hypothesis follows in4.3.5), we will use as in [19] the series F s,β ( x ) = e β ( x ) + ∞ X i =1 ( − is e β ( x ) q i ( x q − θ ) s · · · ( x q i − θ ) s , where e β ( x ) is the formal series in F q ((1 /x )) obtained from the formal series of β ∈ F q ((1 /θ )) byreplacing θ with x , an independent indeterminate.36et us assume that x ∈ C , with | x | >
1. We have, for i big enough, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e β ( x ) q i ( x q − θ ) s · · · ( x q i − θ ) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | x | q i deg θ β − q qi − q − , so that the series F s,β ( x ) converges for | x | > | β | < q qq − and x is not of the form θ /q i .We have the functional equations: F s,β ( x q ) = ( x q − θ ) s ( F s,β ( x ) − e β ( x )) , moreover, F s,β ( θ ) = β j + ∞ X i =1 ( − i β q i ( θ q − θ ) s · · · ( θ q i − θ ) s = Li s ( β ) . Therefore, these series define holomorphic functions for | x | > q /q an allow meromorphic contin-uations to the open set { x ∈ C, | x | > } , with poles at the points θ /q i . We have “deformed”certain Carlitz’s logarithms and got in this way Mahler’s functions (except that the open unit diskis replaced with the complementary of the closed unit disk, but changing x to x − allows us to workin the neightbourhood of the origin).Let J be a finite non-empty subset of { , , . . . } such that if n ∈ J , p does not divide n .Let us consider, for all s ∈ J , an integer l s ≥ β s, , . . . , β s,l s ∈ K alg. ∩ K ∞ with | β s,i | < q qs/ ( q − ( i = 1 , . . . , l s ). We remark that if s is divisible by q − r > q /q theproduct: ( − x ) sq/ ( q − ∞ Y i =1 (cid:18) − θx q i (cid:19) − s converges uniformly in the region { x ∈ C, | x | ≥ r } to a holomorphic function F s, ( x ), which is the( q − K ((1 / ( − x ) / ( q − )), hence in K ((1 /x )) (compare with thefunction of 3). Moreover, F s, ( θ ) = e π s . Proposition 23
If the functions ( F s,β s, , . . . , F s,β s,ls ) s ∈J are algebraically dependent over K alg. ( x ) ,there exists s ∈ J and a non-trivial relation l s X i =1 c i F s,β s,i ( x ) = f ( x ) ∈ K alg. ( x ) with c , . . . , c l s ∈ K alg. if q − does not divide s , or a non-trivial relation: l s X i =1 c i F s,β s,i ( x ) + λF s, ( x ) = f ( x ) ∈ K alg. ( x ) with c , . . . , c l s , λ ∈ K alg. if q − divides s . In both cases, non-trivial relations can be found with c , . . . , c l s , λ ∈ A . roof. Without loss of generality, we may assume that J is minimal so that for all n ∈ J and i ∈ { , . . . , l n } the functions obtained from the family ( F s,β s, , . . . , F s,β s,ls ) s ∈J discarding F n,β n, are algebraically independent over K alg. ( x ).We want to apply Propositions 20 and 21. We take U := S n ≥ K alg. ( x /p n ), and τ : U → U theidentity map on K alg. extended to U so that τ ( x ) = x q . We also take:( X , . . . , X N ) = ( Y s, , . . . , Y s,l s ) s ∈J , ( D , . . . , D N ) = (( x q − θ ) s , . . . , ( x q − θ ) s | {z } l s times ) s ∈J , ( B , . . . , B N ) = ( g β s, ( x ) , . . . , ^ β s,l s ( x )) s ∈J . We take V = ( x q − θ ) Z . We have U = K alg. and for D ∈ V \ { } , the solutions of f ( x q ) = Df ( x )are identically zero as one sees easily writing down a formal power series for a solution f ∈ U .Let P ∈ R be an irreducible polynomial such that P (( F s,β s, , . . . , F s,β s,ls ) s ∈J ) = 0; we clearlyhave e P = QP with Q ∈ U and Propositions 20 and 21 apply. They give s ∈ J , c , . . . , c l s ∈ K alg. not all zero and c ∈ U such that c ( x ) = c ( x q )( x q − θ ) s − x q − θ ) s l s X i =1 c i g β s,i ( x ) . (47)We now inspect this relation in more detail. To ease the notations, we write l s = m and F i ( x ) := F s,β s,i ( i = 1 , . . . , m ). Since e β ( x ) q = e β ( x q ) for all β ∈ K , from (47) we get, for all k ≥ c ( x ) = − m X i =1 c i e β i ( x ) + k X h =1 ( − hs e β i ( x ) q h (( x q − θ )( x q − θ ) · · · ( x q h +1 − θ )) s ! (48)+ c ( x q k +1 )(( x q − θ )( x q − θ ) · · · ( x q k +1 − θ )) s . By Proposition 20, there exists
M > c ( x ) q M ∈ K alg. ( x ), which implies that c ( x q M ) ∈ K alg. ( x ). By equation (48) we see that c ( x ) ∈ K alg. ( x ).We write c ( x ) = P i ≥ i d i x − i with d i ∈ K alg. . The sequence ( | d i | ) i is bounded; let κ be anupper bound. If x ∈ C is such that | x | ≥ r > q /q with r independent on x , then | c ( x ) | =sup i | d i || x | − i ≤ κ sup i | x | − i ≤ κ | x | deg x c . Moreover, for | x | > r with r as above, | x | q s > | θ | = q forall s ≥ | x q s − θ | = max {| x | q s , | θ |} = | x | q s . Hence we get: | ( x q − θ )( x q − θ ) · · · ( x q k +1 − θ ) | = | x | q + q + ··· + q h +1 = | x | q ( qk +1 − q − . Let us write: R k ( x ) := c ( x q k +1 )(( x q − θ )( x q − θ ) · · · ( x q k +1 − θ )) s . We have, for | x | ≥ r > q /q and for all k : | R k ( x ) | ≤ κ | x | q k +1 deg x c − sq ( qk +1 − q − . (49)38ince | β i | < q sq/ ( q − , deg x e β i ( x ) < sq/ ( q −
1) for all i . In (47) we have two cases: one ifdeg x ( c ( x q ) / ( θ − x q ) s ) ≤ max i { deg x e β i } , one if deg x ( c ( x q ) / ( θ − x q ) s ) > max i { deg x e β i } . In thefirst case we easily see that deg x c < sq/ ( q −
1) (notice that deg x c ( x q ) = q deg x c ). In the secondcase, deg x c = q deg x c − sq which implies deg x c = sq/ ( q − First case.
Here, there exists ǫ > x c = ( sq − ǫ ) / ( q − | x | ≥ r > q /q ): | R k ( x ) | ≤ κ | x | sq − ǫq − q k +1 − sq ( qk +1 − q − ≤ κ | x | sq − ǫqk +1 q − and the sequence of functions ( R k ( x )) k converges uniformly to zero in the domain { x, | x | ≥ r } forall r > q /q . Letting k tend to infinity in (48), we find P i c i F i ( x ) + c ( x ) = 0; that is what weexpected. Second case.
In this case, the sequence | R k ( x ) | is bounded but does not tend to 0. Notice that thiscase does not occur if q − s , because c ∈ K alg. ( x ) and its degree is a rationalinteger. Hence we suppose that q − s .Let us write: c ( x ) = λx sq/ ( q − + X i>sq/ ( q − d i x i , with λ ∈ K alg. × . We have lim k →∞ P i>sq/ ( q − d i x q k i (( x q − θ )( x q − θ ) · · · ( x q k +1 − θ )) s = 0(uniformly on | x | > r > q /q ), as one verifies following the first case.For all k ≥ − x ) sq/ ( q − k +1 Y i =1 (cid:18) − θx q i (cid:19) − s = ( − sq/ ( q − x sq/ ( q − x s ( q + ··· + q k +1 ) k +1 Y i =1 ( x q i − θ ) − s = ( − sq/ ( q − x sq k +2 / ( q − k +1 Y i =1 ( x q i − θ ) − s . Hence we have lim k →∞ λx sq/ ( q − / (( x q − θ )( x q − θ ) · · · ( x q k +1 − θ )) s = λF s, ( x ) and P i c i F i ( x ) + λF s, ( x ) + c ( x ) = 0.We now prove the last statement of the Proposition: this follows from an idea of Denis. Theproof is the same in both cases and we work with the first only. There exists a ≥ p a -th powers of c , . . . , c l s are defined over the separable closure K sep of K . The trace K sep → K can be extended to formal series K sep ((1 /x )) → K ((1 /x )); its image does not vanish.We easily get, multiplying by a denominator in A , a non-trivial relation X i b i F i ( x ) q a + b ( x ) = 0with b i ∈ A and b ( x ) ∈ K alg. ( x ). If the coefficients b i are all in F q , this relation is the p a -th powerof a linear relation as we are looking for. If every relation has at least one of the coefficients b i not39elonging to F q , the one with max i { deg θ b i } and a minimal has in fact a = 0 (otherwise, we applythe operator d/dθ to find one with smaller degree, because dg p a /dθ = 0 if a > Proposition 24
Let L ⊂ K alg. be a finite extension of K . We consider f , . . . , f m holomorphicfunctions in a domain | x | > r ≥ with Taylor’s expansions in L ((1 /x )) . Let us assume that thereexist elements a i , b i ∈ L ( x ) ( i = 1 , . . . , m ) such that f i ( x ) = a i ( x ) f i ( x q ) + b i ( x ) , ≤ i ≤ m. Let α be in L , | α | > r , such that for all n , α q n is not a zero nor a pole of any of the functions a i , b j .If the series f , . . . , f m are algebraically independent over K alg. ( x ) , then the values f ( α ) , . . . , f m ( α ) are algebraically independent over K . The next step is the following Proposition.
Proposition 25
If the numbers ( Li s ( β s, ) , . . . , Li s ( β s,l s )) s ∈J are algebraically dependent over K alg. ,there exists s ∈ J and a non-trivial linear relation l s X i =1 c i Li s ( β s,i ) = 0 with c , . . . , c l s ∈ A . If q − does not divide s , or a non-trivial relation: l s X i =1 c i Li s ( β s,i ) + λ e π s = 0 with c , . . . , c l s , λ ∈ A if q − divides s .Proof. By Proposition 24, the functions F s,i ( s ∈ J , ≤ i ≤ l s ) are algebraically dependent over K alg. ( t ). Proposition 23 applies and gives s ∈ J as well as a non-trivial linear dependence relation.If q − s , by Proposition 23 there exists a non-trivial relation l s X i =1 c i F s,β s,i ( x ) = f ( x ) ∈ K alg. ( x )with c , . . . , c l s ∈ A . We substitute x = θ in this relation: l s X i =1 c i Li s ( β s,i ) = f ∈ K alg. . After [5] pp.172-176, for all x ∈ C such that | x | < q qs/ ( q − , there exist v ( x ) , . . . , v s − ( x ) ∈ C x = exp s v ( x )... v s − ( x )Li s ( x ) , exp s being the exponential function associated to the s -th twist of Carlitz’s module. Moreover:exp s c j v ( β s,j )... c j v s − ( β s,j ) c j Li s ( β s,j ) = φ ⊗ s Car ( c j ) β s,j ∈ ( K alg. ) s , j = 1 , . . . , n s , where φ ⊗ s Car ( c j ) denotes the action of the s -th tensor power of Carlitz’s module. By F q -linearity,there exist numbers w , . . . , w s − ∈ C such thatexp s w ... w s − c ∈ ( K alg. ) s . Yu’s sub- t -module Theorem (in [55]) implies the following analogue of Hermite-Lindemann’s Theo-rem. Let G = ( G sa , φ ) be a regular t -module with exponential function e φ , with φ ( g ) = a ( g ) τ + · · · ,for all g ∈ A . Let u ∈ C s be such that e φ ( u ) ∈ G sa ( K alg. ). Let V the smallest vector subspaceof C s containing u , defined over K alg. , stable by multiplication by a ( g ) for all g ∈ A . Then the F q -subspace e φ ( V ) of C s equals H ( C ) with H sub- t -module of G .This result with G the s -th twist of Carlitz’s module and e φ = exp s implies the vanishing of c and the K -linear dependence of the numbersLi s ( β s, ) , . . . , Li s ( β s,l s ) . If q − s then by Proposition 23 there exists a non-trivial relation l s X i =1 c i F s,β s,i ( x ) + λF s, ( x ) = f ( x ) ∈ K alg. ( x )with c , . . . , c l s , λ ∈ K . We substitute x = θ in this relation: l s X i =1 c i Li s ( β s,i ) + λ e π s = f ∈ K alg. . The Proposition follows easily remarking that, after [5] again, there exist v , . . . , v s − ∈ C such that = exp s v ... v s − e π s . roof of Theorem 19. To deduce Theorem 19 from Proposition 25 we quote Theorem 3.8.3 p. 187of Anderson-Thakur in [5] and proceed as in [14]. For all i ≤ nq/ ( q −
1) there exists h n,i ∈ A suchthat if we set P n := X i φ ⊗ n Car ( h n,i ) θ i , then the last coordinate P n is equal to Γ( n ) ζ ( n ) (where Γ( n ) denotes Carlitz’s arithmetic Gammafunction). Moreover, there exists a ∈ A \ { } with φ ⊗ n Car ( a ) P n = 0 if and only if q − n .This implies that Γ( n ) ζ ( n ) = [ nq/ ( q − X i =0 h n,i Li n ( θ i ) . The numbers h n,i are explicitly determined in [5]. In particular, one has ζ ( s ) = Li s (1) , s = 1 , . . . , q − . We apply Proposition 24 and Proposition 25 with J = J ♯ ∪ { q − } , J ♯ being the set of all theintegers n ≥ p, q − n , l q − = 1, β q − , = 1, and for s ∈ J ♯ ,( β , . . . , β l s ) = ( θ i , . . . , θ i ms ) , where the exponents 0 ≤ i < · · · < i m s ≤ sq/ ( q −
1) are chosen so that ζ ( s ) ∈ K Li s (1) + · · · + K Li s ( θ [ sq/ ( q − ) = K Li s ( θ i ) ⊕ · · · ⊕ K Li s ( θ i ms ) . Remark 26
With β ∈ K as above, we can identify, replacing θ with t − , the formal series F β ∈ C (( x )) with a formal series F ∗ β ∈ F q [ t ][[ x ]] ⊂ K [[ x ]] (as in 3.2), over which the operator τ definedthere acts. Carlitz’s module φ Car : θ θτ + τ acts on F q [ t ][[ x ]] and it is easy to compute theimage of F ∗ β under this action. from this we get: F Φ Car ( θ ) β ( x ) = θF β ( x ) + ( x − θ ) β ( x ) , which implies that, for all a ∈ A , F Φ Car ( a ) β ( x ) ∈ aF β ( x ) + F q ( θ, x ). In some sense, the functions F β are “eigenfunctions” of the Carlitz module (a similar property holds for the functions L β and thefunctions (31), which also have Mahler’s functions as counterparts. The fact that we could obtain Theorems 18 and 19 in a direct way should not induce a false optimismabout Mahler’s approach to algebraic independence; the matrices of the linear τ -difference equationsystems involved are diagonal and we benefitted of this very special situation. In the general case,42t seems more difficult to compute the transcendence degree of the field generated by solutions f = t ( f , . . . , f m ) ∈ K (( x )) of a system like: f ( x d ) = A ( x ) · f ( x ) + b ( x )(see for example, the difficulties encountered in [37, Section 5.2]). One of the reasons is thattannakian approach to this kind of equation is, the time being, not yet explored.This point of view should be considered since it has been very successful in the context of t -motives as in Papanikolas work [39], which is fully compatible with Galois’ approach. We couldexpect, once the tannakian theory of Mahler’s functions is developed enough, to reach more generalresults by computing dimensions of motivic Galois groups (noticing the advantageous fact that thefield of constants is here algebraically closed).However, there is an important question we shall deal with: is any “period” of a trivially analytic t -motive (in the sense of Papanikolas in [39]) a Mahler’s value? For example, in 4.3.4, we madestrong restrictions on the β ∈ K alg. so that finally, Denis Theorem in [19] is weaker than PapanikolasTheorem 18. Is it possible to avoid these restrictions in some way?
We presently do not have a completely satisfactory answer to this question, but there seem tobe some elements in favour of a positive answer. We will explain this in the next few lines.The method in 4.3.4 of deforming Carlitz logarithms log
Car ( β ) into Mahler’s functions requiresthat the β ’s in K alg. correspond to analytic functions at infinity. If β = P i c i θ − i lies in K ∞ = F q ((1 /θ )), the series e β ( x ) := P i c i x − i converges for x ∈ C such that | x | > β = e β ( θ ).This construction still works in the perfect closure of the maximal tamely ramified extension F of F q (( x − )) but cannot be followed easily for general β ∈ K alg. \ K ∞ . Artin-Schreier’s polynomial X p − X − θ does not split over F . Hence, if ξ ∈ K alg. is a root of this polynomial (it has absolutevalue | ξ | = q /p ), the construction fails with the presence of divergent series.Let us consider ℓ = log Car ( ξ ) and ℓ = log Car ( θ ) = log Car ( ξ p ) − ℓ . It is easy to show that ℓ , ℓ are K -linearly independent. By Theorem 18, ℓ , ℓ are algebraically independent over K . Thediscussion above shows that it is virtually impossible to apply Proposition 24 with the base point α = θ .We now show that it is possible to modify the arguments of 4.3.4 and apply Proposition 24 withthe base point α = ξ .To do so, let us consider, for β ∈ K , the formal series e F β ( x ) = e β ( x ) + ∞ X n =1 ( − n e β ( x q n ) Q nj =1 ( x q j +1 − x q j − θ ) . If | β | < q q q − and if | x | > x
6∈ { ξ /q j + λ, j ≥ , λ ∈ F q } , these series converge. In particular,under the condition on | β | above, they all converge at x = ξ since they define holomorphic functionson the domain { x ∈ C, | x | > q /pq } , which contains ξ . More precisely, the value at x = ξ is: e F β ( ξ ) = log Car ( β ( ξ )) . We have the functional equations e F β ( x q ) = ( θ − x q + x q )( e F β ( x ) − β ( x ))43hich tells us that the functions e F β define meromorphic functions in the open set { x ∈ C, | x | > } .With all these observations, it is a simple exercise to apply Proposition 24 with α = ξ and showthe algebraic independence of ℓ , ℓ .The reader can extend these computations and show the algebraic independence of other log-arithms of elements of K alg. . However, the choice of the base point α has to be made cleverly,and there is no general recipe yet. Here, the occurrence of Artin-Schreier extensions is particu-larly meaningful since it is commonly observed that every finite normal extension of F q ((1 /θ )) iscontained in a finite tower of Artin-Schreier extensions of F q ((1 /θ /n )) for some n [24, Lemma 3]. Remark 27
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