Composite values of polynomial power sums

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ANNALES MATHÉMATIQUES

BLAISE PASCAL

Clemens Fuchs & Christina Karolus Composite values of polynomial power sums Volume 26, n^o1 (2019), p. 1-24.

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Publication éditée par le laboratoire de mathématiques Blaise Pascal de l’université Clermont Auvergne, UMR 6620 du CNRS

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Composite values of polynomial power sums

Clemens Fuchs Christina Karolus

Abstract

Let(Gn(x))^∞_n₌₀be ad-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, letm≥2 be a given integer. We ask for n∈Nsuch that the equationGn(x)=g◦his satisfied for a polynomialg∈C[x]with degg=mand some polynomialh∈C[x]with degh>1. We prove that for all but finitely manynthese decompositions can be described in “finite terms” coming from a generic decomposition parameterized by an algebraic variety. All data in this description will be shown to be effectively computable.

1. Introduction and results

LetC[x]be the polynomial ring in the indeterminatex(we remark right away thatCmight be replaced by an algebraically closed field of characteristic 0; however, all polynomials below are assumed to have coefficients inC). Composition of polynomials is a well-defined operation onC[x]. It is associative and has with f(x)=xan identity element, but it is neither commutative nor distributive. There are many reasons to be interested in polynomial composition, e.g. the knowledge of a composition of f ∈C[x]can be of use if one wishes to factor a given polynomial. We illustrate this with a simple example. The polynomial f(x)=x⁸−14x⁴+7 is irreducible over the rationals. However, since f(x)=g(h(x)), whereg(x) = x² −14x+7 and h(x) = x⁴, we can easily determine all solutions of f(x)=0 in radicals by first calculating the rootsyofy²−14y+7 and then determining the correspondingxfromy=x⁴. Even though in the present paper we restrict ourselves to polynomials overC, we also mention that, if for given f ∈k[x]over an arbitrary field kthere is ag ∈k[x]with f ◦girreducible overk, then f is an irreducible polynomial ink[x]as well. Another example is that the decompositions of f exhibit arithmetical properties associated with f, which is used to solve equations of separated variable type (cf. [2, 3]). The invertible elements inC[x]with respect to decomposition are the linear polynomials. We call f(x)=g◦ha non-trivial decomposition if neithergnorhis linear.

Letm≥2 be an integer; we callf(x)=g◦hanm-decomposition if degg=mand we say that f ism-decomposable if anm-decomposition off exists. We call f indecomposable if f admits only trivial decompositions. A pair(g,h) ∈C[x]²is called equivalent to(g⁰,h⁰)

Supported by Austrian Science Fund (FWF) Grant No. P24574.

Keywords:Decomposable polynomials, linear recurrence sequences, Brownawell–Masser inequality.

2010Mathematics Subject Classification:11B37, 12Y05, 11R58.

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if there area,b∈C,a,0 such thatg(x)=g⁰(ax+b),h(x)=(h⁰(x) −b)/a. It is easy to see that every polynomial f ∈C[x]can be decomposed as f(x)= f1◦ f2◦ · · · ◦ fk

with f_iindecomposable. Moreover, this decomposition is unique in the following sense:

if f(x)= f₁⁰◦ · · · ◦ f_l⁰with f_j⁰indecomposable is another decomposition, thenk =l and f₁, . . . ,f_k are obtained applying certain transformations a finite number of times, where in each step a neighboring pair of f₁⁰, . . . , f_l⁰is replaced by another one having the same composition. (This is known as Ritt’s first theorem; cf. [19].) There is a nice algebraic description of decompositions of a polynomial f ∈C[x]since they are (up to equivalence) in one-to-one correspondence to intermediate fields betweenC(x)and C(f(x))(cf. again [19]).

We start with a few general remarks. First, it might happen that for a given f ∈C[x]

and a giveng ∈C[x]we have differenth₁,h₂ ∈C[x]such that f(x)=g◦h₁ =g◦h₂. However, in this caseg(X)=g(Y)has a solutionX =h₁(x),Y =h₂(x). This situation was completely solved in [1]. It follows thatg=g⁰◦x^kfor someg⁰∈C[x],k >1 and h₁andh₂just differ by a constant (to be more precise, by ak-th root of unity). Second, by linear equivalence we can control the leading coefficient and the constant term ofh.

E.g. we may assume thath ∈C[x]is monic and satisfiesh(0)=0 (every other value in Cis fine as well). Moreover, assume that f is given and that we have f(x)=g◦hwith f,g,h ∈C[x]. Leta∈C\{0}be the leading coefficient of f. Then we may also assume thatgis monic by writing f(x)=a(g◦h). Third, assume that f,h ∈C[x]are given. Then there are at most finitely manyg∈C[x]with f(x)=g◦h. This can be seen as follows:

We may assume that f,g,h are all monic. Writeg(x) =(x−b₁). . .(x−b_m), where b1, . . . ,bm∈Care not necessarily distinct. Assume that f(x)=(x−a1). . .(x−an)with a₁, . . . ,a_n ∈C. Theng(h(x))=(h(x) −b₁). . .(h(x) −b_m)=(x−a₁). . .(x−a_n)= f(x).

It follows that there is a partition of the multi-set{a₁, . . . ,an}with equally large blocks (of size degh) that describeguniquely. If we assume thath(0)=0, then theb_iare just the product of all elements in thei-th block. The uniquegcan be found, without calculating the roots of f, by comparing coefficients in f(x)=g◦h(cf. [17]).

In this paper we are interested in non-trivial decompositions (with two factors, an

“inner” and an “outer” factor) of polynomials with coefficients inC. This problem is hard in general since the decompositions of polynomials can be anythinga priori(since conversely, every pair(g,h)gives a polynomialg◦h). Therefore, it is natural to restrict to a subset ofC[x]which is described by a finite amount of data and then to ask whether or not all decompositions in this subset can be describedin finite termsdepending on the data describing the subset. We give a few (important and non-trivial) examples to illustrate this approach.

Let n ≥ 2 be a given integer. We consider the set of all polynomials f ∈ C[x]of degree n. Then there is an integer J and for every 1 ≤ j ≤ J an algebraic variety

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V_j ⊂A^n+t^j for some 2≤t_j ≤ndefined overQfor which equations can be written down effectively and there are polynomials fj,hj,g_j with coefficients in the coordinate ring of the variety and depending on integersk₁, . . . ,k_t_j and(l₁, . . . ,l_n) ∈ {0,1, . . . ,n−1}ⁿ such that the following holds: a)g_j◦hj = fj is a polynomial of degreenwith coefficients in the coordinate ring; b) for every point P ∈ V_j(C)and integersk₁, . . . ,k_t_j,l₁, . . . ,l_n one gets a decomposition f_j(P,x)=g_j(P,h_j(P,x)); c) conversely, for every polynomial f ∈C[x]of degreenand every non-trivial decomposition f(x)=g◦hwithg(x)not of the shape(ax+b)^m,m∈N,a,b∈Cthere areP ∈ V_j(C),k₁, . . . ,k_t_j,l₁, . . . ,l_nsuch that f(x)= fj(P,x),g(x)=g_j(P,x),h(x)=hj(P,x). This result (formulated in different forms) can be found in [4, 12, 17].

Let`be a given integer. We consider the set oflacunarypolynomials (with respect to`), that is the set of all polynomials f ∈C[x]with`non-constant terms. Then there are integers p,J depending on ` and for every 1 ≤ j ≤ J an algebraic varietyV_j defined overQand a latticeΛ_j for which equations can be written down explicitly and (Laurent-)polynomials f_j,h_j ∈Q[V_j][z₁^±1, . . . ,z^±1_p ],g_j ∈Q[V_j][z]with coefficients in the coordinate ring of the variety such that the following holds: a)g_j◦hj = fjis a (Laurent- )polynomial with`non-constant terms with coefficients in the coordinate ring; b) for every pointP ∈ V_j(C)and(u₁, . . . ,up) ∈Λ_jone gets a decomposition fj(P,xû¹, . . . ,xû^p)= g_j(P,h_j(P,xû¹, . . . ,xû^p)); c) conversely, for every polynomial f ∈ C[x]with ` non- constant terms and every non-trivial decomposition f(x)=g◦hwithh(x)not of the shapeax^m+b,m ∈N,a,b ∈Cthere is a j, a pointP ∈ V_j(C)and(u₁, . . . ,u_p) ∈Λ_j such that f(x) = f_j(P,xû¹, . . . ,xû^p),g(x) = g_j(P,x),h(x) = h_j(P,xû¹, . . . ,xû^p). This result can be found in [25]; cf. also [24, 26]. (A similar result holds for lacunary rational functions f ∈C(x)by a combination of [10] and [16].)

In the present paper we are interested in another subset ofC[x]namely the subset {G_n(x);n∈N}that consists of elements of a linear recurrence sequence(G_n(x))^∞_n=0of polynomials inC[x]. The sequence is fixed by the recurrence relation and by the initial values. Equivalently, every element of the sequence can be written by a Binet-type formula G_n(x)=a₁α₁ⁿ+· · ·+a_tα_tⁿ, whereα₁, . . . , α_tare the distinct roots of the characteristic polynomial associated to the recurring relation and theai are polynomials in nwith coefficients in the splitting fieldC(x, α₁, . . . , α_t)of degree less than the corresponding multiplicity ofα_ias a root of the characteristic equation. In this way all elements are given by a finite amount of data. Our goal is to describe all decomposableG_n’s in this set and all their decompositions in finite terms, depending only on the given data. To fix terms we shall consider thed-th order linear recurrence sequence(G_n(x))^∞_n=0, given by the relation

G_n+d(x)=A_d−1(x)G_n+d−1(x)+· · ·+A₀(x)G_n(x), (1.1)

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withA₀, . . . ,A_d−1 ∈C[x]and initial termsG₀, . . . ,G_d−1∈C[x]. Denote byα₁, . . . , α_t the distinct characteristic roots of the sequence, that is the characteristic polynomial G ∈C(x)[T]splits as

G(T)=T^d−A_d−1T^d−1− · · · −A₀=(T−α₁)^k¹(T−α₂)^k². . .(T−α_t)^k^t, wherek1, . . . ,kt ∈N. We assume that all roots are simple, i.e.t =d, and that they are polynomials, i.e.α_i ∈C[x]fori=1, . . . ,d. ThenG_n(x)admits a representation of the form

G_n(x)=a₁α₁ⁿ+a₂α₂ⁿ+· · ·+a_dαⁿ_d. (1.2) By assumption we consider the special situation thata₁, . . . ,a_d ∈Candα₁, . . . , α_t ∈C[x].

Finally, we assume that deg(α₁)>deg(α_i)fori>1.

We mention that for binary recurrences the authors together with Kreso proved in [9]

that if G_n(x) = g ◦h, then either degg is bounded independently of n and only in terms of the initial data, unlesshis special (meaning that(g,h)is equivalent to(g⁰,x^m) or(g⁰⁰,T_m(x))where(T_n(x))^∞_n=0denotes the sequence of Chebyshev polynomials and g⁰,g⁰⁰ ∈C[x]) or a technical condition is not verified (see the paper for details). This describes the “outer” decomposition factor in such a decomposition. In view of this result, which we expect (without the technical condition) to hold in general, we restrict ourselves tom-decompositions for an integerm≥2 which we view as fixed from now on.

We further mention that for a given sequence(G_n(x))^∞_n=0the decompositions of the form G_n(x) = G_m◦h for a fixed polynomial h ∈ C[x],degh ≥ 2 were considered in [8, 13, 14]. It was Zannier who proved in general that this equation has only finitely many solutions(n,m),n,m, unless we are in the cyclic or Chebyshev case as above (cf. [23]). This result was made effective in [11]. A further result in this direction can be found in [15].

There are a few trivial situations that we have to take into account below. IfG_n(x)= f(βⁿ) with f, β ∈ C[x], then every decomposition f(x) = g ◦ h with degg = m leads to a sought decomposition G_n(x) = g(h(βⁿ)) for every n ∈ N. Observe that this situation might also lead to slightly different decompositions. Assume e.g. that Gn(x) = a1α₁ⁿ+a2,a1,a2 ∈ C, α₁ ∈ C[x]; ifnis a multiple ofm, i.e.n = m`, then G_n(x)=g◦hwithg(x)=a₁x^m+a₂,h(x)=α₁^`. More generally, whenG_n(x)=g(H_n(x)) withg∈C[x],degg=mand(H_n(x))^∞_n₌₀is another linear recurrence sequence inC[x], then obviously we again have a sought decomposition for everyn∈N. Unfortunately it seems that these cases are not exhaustive. There might be many “sporadic” solutions that arise by polynomial-exponential equations that are complicated to control in general.

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We start with the following theorem, which clarifies the structure of the “inner”

decomposition factor that may appear in anm-decomposition of elements in the sequence (G_n(x))^∞_n=0.

Theorem 1.1. Let(G_n(x))^∞_n=0be a non-degenerate simple linear recurrence sequence of orderd ≥2with power sum representationG_n(x)=a₁α₁ⁿ+· · ·+a_dαⁿ_dwitha₁, . . . ,a_d ∈C, α₁, . . . , α_d ∈ C[x]satisfying degα₁ > max{degα₂, . . . ,degα_d}. Moreover, letm ≥ 2 be an integer. Writem0 for the least integer such thatα₁^m⁰^/m ∈ C[x]. Then there is an effectively computable positive constantCsuch that the following holds: Assume that for somen ∈Nwithn >Cwe haveGn(x)=g◦hwithdegg =m,degh >1. Then there arec₁, . . . ,c_l ∈Csuch that

h(x)=c₁γ^`₁+· · ·+c_lγ_l^`,

wherem₀`=nandl ∈Nis bounded explicitly in terms ofm,danddeg(α₁)+· · ·+deg(αd) andγ₁, . . . , γ_l ∈C(x)can be given explicitly in terms ofα₁, . . . , α_d, both independently ofn.

This result should be compared with Proposition 2 in [25].

We illustrate the result with an example. Let (G_n(x))^∞_n₌₀ be given by Gn(x) = x³ⁿ +3(2x²)ⁿ +3(4x)ⁿ+2³ⁿ for all n ≥ 0 and letm = 3. We have α₁ = x³, α₂ = 2x², α3 = 4x, α4 = 8 and m0 = 1. The proof of the theorem shows that we must have n ≤ 30 or h(x) = c₁xⁿ +c₂ with c₁,c₂ ∈ C. Letg(x) = x³. Then g(h(x)) = (c₁xⁿ+c2)³=c₁³x³ⁿ+3c²₁c2x²ⁿ+3c1c₂²xⁿ+c₂³. Comparingg(h(x))withGn(x)shows thatc₁³=1,c₁²c₂=2ⁿ,c₁c₂²=4ⁿ,c₂³=8ⁿ. This defines a subvarietyVofA²×Gm. Up to (possibly) finitely many exceptions for smallnwe haveG_n(x)=g(h(x))=(c₁xⁿ+c₂)³, where(c₁,c₂,n) ∈ V(C).

Observe that m₀ in the theorem can also be described as follows: Write α₁(x) = v(x−v₁)^k¹. . .(x−v_t)^k^t and defineψ byψ^m =α₁. Putd =gcd(k1, . . . ,k_t,m). Then m₀ =m/d. Obviously,ψ^m/d ∈C[x]. Conversely, observe thatm₀is a divisor ofmsince by definition ofm₀the polynomialT^m⁰−ψ^m⁰is the minimal polynomial ofψoverC(x) (cf. Proposition 2.2) and thus dividesT^m−ψ^moverC(x, ψ). Sinceψ^m⁰ ∈C(x), it follows m₀k_i/m∈Nand thusm/m₀dividesk_i(thus gcd(k1, . . . ,k_t,m)=d) fori=1, . . . ,t. The smallest such integer is obtained in the case of equality givingm=m₀das claimed.

The structure of allm-decompositions for a givenm≥ 2 can now be described as follows.

Theorem 1.2. Let(G_n(x))^∞_n=0be a non-degenerate simple linear recurrence sequence of orderd ≥2with power sum representationGn(x)=a1αⁿ

1+· · ·+adαⁿ

dwitha1, . . . ,ad ∈C, α₁, . . . , α_d ∈ C[x]satisfying degα₁ > max{degα₂, . . . ,degα_d}. Moreover, letm ≥ 2 be an integer. Writem₀ for the least integer such thatα₁^m⁰^/m ∈ C[x]. Then there is an

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explicitly computable positive constantC, and a subvarietyVofA^l+m+1×G^tmwitht,l bounded explicitly in terms ofm,danddeg(α₁)+· · ·+deg(α_d)for which a system of polynomial-exponential equations in the polynomial variablesc₁, . . . ,c_l,g₀, . . . ,g_mand the exponential variable`(with coefficients inQ) can be written down explicitly such that the following holds:

(1) DefiningG(x)=g₀x^m+g₁x^m−1+· · ·+g_m∈C[V][x]andH_`=c₁γ₁^`+c₂γ₂^`+

· · ·+clγ^`

l ∈C[V][x], whereγ₁, . . . , γ_l ∈C(x)can be given explicitly in terms of α₁, . . . , α_d, thenG_m₀_`=G◦H_`holds as an equation inxwith coefficients in the coordinate ring ofV. In particular, for any pointP=(c₁, . . . ,cl,g₀, . . . ,g_m, `) ∈ V(C) we get a decomposition G_n(x) = g ◦h, g(x) = G(P,x) ∈ C[x] and h(x)=H_l(P,x) ∈C[x](withn=m₀`).

(2) Conversely, letGn(x)=g◦hbe a decomposition ofGn(x)for somen∈Nwith g,h ∈ C[x],degg = m,degh > 1. Then eithern ≤C or there exists a point P=(g₀, . . . ,g_m,c1, . . . ,cl, `) ∈ V(C)withg(x)=G(P,x)andh(x)=H`(P,x) andn=m₀`.

Remarks and special cases.

(a) Binary case: Let(G_n(x))^∞_n=0be a non-degenerate binary simple linear recurrence which does not satisfy a recurrence relation of order less than 2; thus, we have G_n(x)=a₁α₁ⁿ+a₂α₂ⁿwitha₁,a₂∈C. We assume thatα₁, α₂ ∈C[x]and degα₁ >

degα₂. Moreover, we assume that one of the conditions of [9, Theorem 2] is satisfied. Then there is an effectively computable constantCand there are finitely many subvarietiesV_i ofA^mⁱ^+1+lⁱ×G^tmⁱ and equationsG_m_0,i_`(x)=G⁽ⁱ⁾◦H_`⁽ⁱ⁾, where degG⁽ⁱ⁾=m_i ≥2, in the coordinate ring ofV_isuch that the following holds: IfGn(x)=g◦hfor somen∈Nandg,h∈C[x],degg,degh >1 with h(x)indecomposable and not of the shapeax^m+b,m∈N,a,b∈C, then either n≤Cor there is aniand aP=(g_i0, . . . ,g_im_i,ci1, . . . ,cili, `) ∈ V_i(C)such that n=m_0,i`,g(x)=G⁽ⁱ⁾(P,x),h(x)=H_`⁽ⁱ⁾(P,x).

(b) When allα_iare monic, then the varieties can be chosen without theGm-part.

(c) Assume thatα₁∈C[x]satisfies 1≤k:=degα₁ ≤m. ThenGn(x)=g◦hwith n>Cimplies thathis of the formc₁+c₂α^deg₁ ^Gⁿ^/(^mk⁾withc₁,c₂ ∈C.

(d) Assume thatα₁ =β^m¹, α₂ =β^m², . . . , α_d =β^m^d withm₁>m₂ ≥ · · · ≥m_d ≥0.

ThenGn(x)= f(βⁿ), where f(x)= f1x^m¹+ f2x^m²+· · ·+fdx^m^d. In this case it follows that eithern ≤Corh(x)=c₁β^k¹^`+· · ·+c_lβ^k^l^` =h⁰(β^`)for some

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h⁰ ∈ C[x]. Thus ifG_n(x) = g◦h = (g◦h⁰)(β^`) = f(β^m⁰^`). Therefore, the problem reduces to find allm-decompositions of the polynomial f ◦x^m⁰. (e) The proof shows that if (G_n(x))^∞_n=0 is defined over a number field, i.e. the

coefficients of the Binet-type equation (1.2) as well as the characteristic roots are polynomials with coefficients in some number fieldK, then all decomposition factorsg,h are defined overK as well. In this case we are interested in decompositions overK, which can be described by the above conclusion of the statements.

(f) We also remark that the above results include a description in finite terms of allm-th powers in a linear recurring sequence of polynomials satisfying the conditions of the theorem (i.e. the sequence is non-degenerate and simple and the characteristic roots are polynomials where one has degree larger than all others).

This follows by fixingg(x)=x^mand then going through the proof of the above theorem.

(g) Finally, we mention that if we know that thec_i can be parametrized by power sums as well (in particular if they are constant) and that we have a decomposition for any`(or for all members along an arithmetic progression), then these families are easy to calculate. This follows since we may identify varying powers by indeterminates (see e.g. [6, Lemma 2.1]) and then use the algorithm in [4]

for polynomials in several variables (actually, we view such a polynomial as a polynomial in one of the variables; the other variables can be embedded intoC so that we may view the polynomial again as an element inC[x]) to determine the decompositions.

The proof of the theorems follows essentially the ideas of [25]. Assume thatG_n(x)= g◦h. This equality is viewed as an equation for the unknown h=h(x); it is a root of g(T) −G_n(x)=0 over the (rational) function fieldC(G_n(x)). Thus we can expandhas a Puiseux series in terms of quantitiesGn(x)^s^/^m,s=1,0,−1, . . ., wherem=degg. Then one uses the multinomial series to expandG_n(x)^s/m for anys; in order to justify this multiple expansion, the “dominant root condition” on the degrees of the characteristic roots is needed. Afterwards we use, as in [25], a function field variant of the Schmidt subspace theorem (Proposition 2.4) proved in [25], to find that eithernis bounded orh can be expressed as given in Theorem 1.1. Using this information, one views thec1, . . . ,cl

as well as the coefficientsg₀, . . . ,g_mofg, while the degree ofgis fixed, as indeterminants and then comparesg◦h withGn(x)for the givenn ∈ N. Using unit equations over function fields, this either implies thatnis bounded or we have two linear recurrences

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that are related (see [20] for this notation). In the latter case, up to a permutation, the characteristic roots have to match up and then, since they are monic polynomials at that point, the coefficients coincide. This gives polynomial-exponential equations that can be written down explicitly and which define a variety. From this the statement follows.

The rest of the paper is organized as follows. In the next section we collect some auxiliary results that will be needed for the proof of the theorems. In Section 3 we give the proofs of Theorem 1.1 and 1.2. In Section 4 we give some more details justifying the remarks and special cases.

2. Auxiliary results

In this section, we recall some basic information and collect some statements, which we will make use of in our proofs later on.

Generally, an algebraic function field F/K is a finite algebraic extension ofK(x), wherexis some element transcendental overK. IfFis itself of the shapeF=K(x), then Fis said to be rational. The rational function field has genusg_F =0. Throughout this paper, we will work over the complex numbersK =C, even though our proofs hold over any other algebraically closed field as well. Then

C(x)= f(x)

g(x); f(x),g(x) ∈C[x], g(x),0

,

i.e.C(x)is the field of fractions ofC[x]. OnC(x)we define valuations as follows. For eacha∈C, letν_a(f)be the unique integer such that f(x)=(x−a)^ν^a^(f⁾p(x)/q(x), where p,q ∈C[x]are such thatp(a)q(a),0. Moreover, with the symbol∞we associate the valuationν∞(f)=degq−degp, where f(x)=p(x)/q(x). Ifν_a(f)>0 for ana∈C,a is called a zero of f, and it is called a pole of f, ifν_a(f) <0. These functions are all (normalized, up to equivalence) valuations onC(x)and for a finite extensionLofC(x) each one of them can be extended to at most[F: C(x)]valuations onF, which again gives all discrete valuations onF. Both, inC(x)and inF, for any f ∈F/Cthe so-called sum formula holds, that is

Õ

ν

ν(f)=0,

where the sum is taken over all valuations on the respective function field. There is a one-to-one relation between valuations and places, namely for any valuationν_aonC(x), a ∈C∪ {∞}, there is a placePa ={f ∈C(x); ν_a(f)>0}(it is the unique maximal ideal of the valuation ringO_P_a ={f ∈ C(x); ν_a(f) ≥ 0}). Therefore, valuations are sometimes introduced in terms of places (and often instead of Pawe simply writea).

We writePFfor the set of places of the fieldF. Now letF⁰be an algebraic extension of

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the function field F. Then a placeP⁰ ∈PF⁰is said to lie overP ∈PF, ifP ⊂ P⁰. We writeP⁰|Pin this case. Then there is an integere=e(P⁰|P), 1≤e≤ [F⁰: F], called the ramification index ofP⁰overP, such thatν_P⁰(x)=e·ν_P(x)for allx∈ F. We say thatP⁰|Pis ramified ife(P⁰|P) >1 and unramified otherwise. Ife(P⁰|P)=[F⁰: F], there is exactly one placeP⁰∈F⁰lying aboveP∈FandP⁰is said to be totally ramified.

The placesP⁰∈PF⁰lying aboveP∈PFcorrespond to the extensions of the respective valuationν_PinF. Denote byF_Pthe residue class fieldO_P/P. We shall need the following two statements, which can be found in [21].

Proposition 2.1. LetF/Cbe a function field in one variable andϕ∈F[T], ϕ(T)=a_nTⁿ+a_n−1Tⁿ⁻¹+· · ·+a₁T+a₀.

If there is a placeP∈PFsuch thatν_P(a_n)=0,ν_P(a_i) ≥0fori=1, . . . ,n−1,ν_P(a₀)<0 andgcd(n, ν_P(a₀)) =1, thenϕ(T)is irreducible inF[T]. Furthermore, ifF⁰ =F(y), whereyis a root ofϕ(T), thenPhas a unique extensionP⁰∈PF⁰ande(P⁰|P)=n.

Proposition 2.2. LetF/Cbe a function field in one variable. Suppose thatu∈Fsatisfies u,w^dfor allw∈Fandd|n,d >1. LetF⁰=F(z)withzⁿ=u. ThenF⁰is said to be a Kummer extension ofFand we have:

(1) The polynomialϕ(T)=Tⁿ−uis the minimal polynomial ofzoverF(in particular, it is irreducible overF). The extensionF⁰/Fis Galois of degreen; its Galois group is cyclic, and all automorphisms ofF⁰/Fare given byσ(z)=ζz, where ζ ∈Cis ann-th root of unity.

(2) LetP ∈PF andP⁰ ∈PF⁰be an extension ofP. Letr_P :=gcd(n, ν_P(u)). Then e(P⁰|P)=n/r_P.

(3) Denote byg(resp.g⁰) the genus ofF/C(resp.F⁰/C). Then g⁰=1+n(g−1)+1

2 Õ

P∈PF

(n−r_P)degP.

Our strategy also involves the use of height functions in function fields. Define the projective heightHofu₁, . . . ,u_n∈F/C, wheren≥2 and not allu_iare zero, via

H (u₁, . . . ,un)=−Õ

ν

min(ν(u₁), . . . , ν(u_n)).

Also, for a single element f ∈F^∗, set H (f)=−Õ

ν

min(0, ν(f)).

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In both cases the sum is taken over all discrete valuationsνonF. Note thatν(f),0 only for a finite number of valuationsνand thatH (f)=Í

νmax(0, ν(f))if f ∈F^∗, by the sum formula. For f =0, we defineH (f)=∞. We state some basic properties of the projective height, cf. [9].

Lemma 2.3. Denote as above byHthe projective height onF/C. Then for f,g∈F^∗the following properties hold:

(1) H (f) ≥0andH (f)=H (1/f),

(2) H (f) − H (g) ≤ H (f +g) ≤ H (f)+H (g), (3) H (f) − H (g) ≤ H (fg) ≤ H (f)+H (g), (4) H (fⁿ)=|n| · H (f),

(5) H (f)=0⇔ f ∈C^∗,

(6) H (A(f))=degA· H (f)for anyA∈C[T]\ {0}.

The following proposition is an important ingredient for the proof of our first theorem.

It can be seen as a function field analogue of the Schmidt subspace theorem, modelled by Zannier, cf. [25].

Proposition 2.4(Zannier). LetF/Cbe a function field in one variable, of genusg, let ϕ₁, . . . , ϕ_n∈Fbe linearly independent overCand letr∈ {0,1, . . . ,n}. LetSbe a finite set of places ofFcontaining all the poles ofϕ₁, . . . , ϕ_nand all the zeros ofϕ₁, . . . , ϕ_r. Putσ=Ín

i=1ϕ_i. Then Õ

ν∈S

ν(σ) − min

i=1,...,nν(ϕ_i)

≤ n

2

(|S|+2g−2)+

n

Õ

i=r+1

deg(ϕ_i).

Recall that for a finite setSof places (or valuations, respectively) ofF, an element f ∈Fis called anS-unit, if it has zeros and poles only at places inS, i.e. the set ofS-units is given by

O^∗_S={f ∈L;ν(f)=0 for allν<S}.

We will also use the following result due to Brownawell and Masser [5] taken from [16], giving an upper bound for the height of S-units, which arise as a solution of certain S-unit-equations.

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Proposition 2.5(Brownawell–Masser). LetF/Cbe a function field in one variable of genusg. Moreover, for a finite setSof discrete valuations, letu1, . . . ,unbeS-units, not all constant, and

1+u1+u2+· · ·+un =0, where no proper subsum of the left side vanishes. Then it holds

i=1,...,nmax H (u_i) ≤ 1

2n(n−1)(|S|+2g−2).

In our proof we will use an expansion of the polynomialhinGn=g◦has a Puiseux series. Therefore, we give a quick review on formal power series, Laurent series and Puiseux series (cf. [22] and [18]). Formally, a (complex) polynomial is a sequence (a₀,a₁, . . .), where a_i ∈ Cand where there exists ann ∈ Nsuch thata_j =0 for all j ≥n. Such an element is associated with the finite suma₀+a₁x+· · ·+a_nxⁿ, wherex is an indeterminate identified with the element(0,1,0,0, . . .). Together with the usual addition and multiplication this gives theC-algebra of (complex) polynomialsC[x]. If we also allow sequences with infinite support, we obtain the algebra of formal power series, denoted byC[[x]], that is the set

C[[x]]={(a₀,a₁,a₂, . . .); a_i ∈C}=

a₀+a₁x+a₂x²+. . .; a_i ∈C ,

where addition and multiplication are defined just in the same way as for polynomials. Note that the notation as an infinite sum is meant only formally, i.e. questions of convergence are disregarded.C[[x]]is an integral domain and the units inC[[x]]are precisely the elements with non-zero constant term. The field of fractions ofC[[x]]is the field of formal Laurent series, denoted byC((x)). It is the localization ofC[[x]]with respect to the ideal(x)and its elements are given by the set

C((x))={(a_m,a_m+1,a_m+2, . . .); m∈Z, a_i ∈C}

=

amx^m+a_m+1x^m+1+a_m+2x^m+2+. . .; m∈Z, ai ∈C ,

so a Laurent series is the sum of a formal power series plus possibly a finite number of terms with negative exponent. Defining ord(f)to be the smallestisuch thata_i ,0 if f ,0, and ord(0)=∞gives a discrete valuation onC((x)). The valuation ring is given by C[[x]]and the residue field isC. Moreover,C((x))is a field (in general,K((x))is a field ifK is a field), which is the completion with respect to this valuation topology of the field C(x)of rational functions.

As a generalization, the field of formal Puiseux series is obtained by allowing also fractional exponents, i.e. Laurent series inC((x^1/n))for somen∈N. More precisely, the

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field of formal Puiseux series is given by C((x^1/∞))=

∞

Ø

n=0

C((x^1/n)).

The valuation ord naturally extends to this field and it is complete with respect to the induced topology. The classical Newton–Puiseux theorem shows thatC((x¹^/∞))is an algebraic closure of the field of formal Laurent seriesC((x)).

If F/Cis a function field in one variable of degree[F:C(x)] =n, then there is a primitive elementy∈FoverC(x)which satisfies an irreducible equation

f(x,Y)=Yⁿ+c₁(x)Yⁿ⁻¹+· · ·+c_n(x)=0 (2.1) with coefficientsc_i(x)inC(x). Without loss of generality, we can also assume thec_i(x) to be polynomials. For the expansion ofhas a Puiseux series, we rely on the following classical theorem, cf. [7].

Proposition 2.6(Puiseux’s Theorem). LetF/Cbe a function field in one variable of degree[F:C(x)]=n. Then there are1≤r≤nnatural numberse_isatisfying

e₁+e₂+· · ·+e_r =n

which have the following meaning: the irreducible equation(2.1)satisfied by an arbitrary functionyinFhas for solutions therseries

y_i =

∞

Õ

k=νi

a_ikx⁻^k^/^eⁱ, a_iν_i ,0, i=1,2, . . . ,r. (2.2) With a primitivee_i-th root of unityζ form

y_{i j} =Õ

k

a_ikζ^jkx⁻^k^/^eⁱ, j=0, . . . ,e_i−1;

then f(x,Y)is identical with

f(x,Y)=Ö

i,j

(Y−y_{i j}). (2.3)

The coefficientsa_ik are constant, i.e.a_ik ∈C, and their images under isomorphisms ofC give permutations of the y_{i j}in(2.3).

We remark that in the above theoremris the number of placesPi|P∞for the unique infinite placeP∞∈P_C(x), whereP_i ∈PF. Moreover,ν_i =ν_P_i(y)is the valuation ofyat P_iande_i =e(P_i|P∞)is the ramification index ofP_ioverP∞.

Finally, we state the following little lemma that will be useful in the proof.

Lemma 2.7. Let f ∈C[x]. Thenf(1/y) ∈C(y)with a pole only aty=0. Moreover, the order of vanishing aty=0of f(1/y)is equal to−degf.

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Proof. Write f(x)= f₀(x−a₁)^k¹. . .(x−a_t)^k^t withk₁+· · ·+k_t =degf. Puty=1/x.

Then f(1/y)= f(x) =y^−(k¹⁺^···^+k^t⁾f0(1−a1y)^k². . .(1−aty)^k^t, which shows that the multiplicity ofy=0 as a pole of f(1/y)inC(y)is equal to degf. This is the claim.

3. Proof of Theorem 1.1 and 1.2

3.1. Proof of Theorem 1.1

Letg∈C[x]be a polynomial of degree degg=m≥2 and(G_n(x))^∞_n=0as given in the theorem.

Let b₁, . . . ,b_d ∈Cbe the leading coefficients ofα₁, . . . , α_d respectively. We write α_iⁿ=bⁿ_iβ_iⁿfori=1, . . . ,dand therefore have

G_n(x)=a₁bⁿ₁β₁ⁿ+· · ·+a_dbⁿ_dβⁿ_d,

wherea₁, . . . ,a_d,b₁, . . . ,b_d ∈Candβ₁, . . . , β_d ∈C[x]have leading coefficients equal to 1 and satisfy degβ₁>max{degβ₂, . . . ,degβ_d}.

LetKbe the rational function fieldK=C(x)and letzbe a root ofϕ(T)=g(T) −x= g₀T^m+· · ·+g_m ∈ K[T]. At the infinite placeP =P∞we haveν∞(g₀) =ν∞(g_i)=0 for i = 1, . . . ,m−1 and ν∞(g_m) = ν∞(g(0) −x) = −1. Also, gcd(degϕ, ν∞(g_m)) = gcd(m,−1)= 1. Therefore, by Proposition 2.1,ϕ(T)is irreducible overC(x)and P∞

has a unique extensionP_∞⁰0 ∈PL, whereL=C(x)(z)=C(x,z). By Puiseux’s Theorem (Theorem 2.6), it follows that there is an expansion ofzof the form

z=

∞

Õ

k=ν∞0

uk(p^m 1/x)^k

=u−1(^mp

1/x)⁻¹+u0(^mp

1/x)⁰+u1(p^m

1/x)¹+u2(^mp

1/x)²+. . .

=u−1x^1/m+u0+u1x^−1/m+u2x^−2/m+. . . , (3.1)

where theu_j ∈Cdepend only ong. Note that we have used thate_i(∞⁰)=min Theorem 2.6, by Proposition 2.1 and thatν∞⁰(z)=−1. This can be seen as follows. Sincezis a root of ϕ(T), we have thatg(z)=x, henceν∞⁰(g₀z^m+· · ·+g_m−1z+g(0))=ν∞⁰(x)=m·ν∞(x)=

−m. By integrality we haveν∞⁰(z)<0, so using the strict triangle inequality we get m·ν∞⁰(z)= min

i=1,...,m(i·ν∞⁰(z))= min

i=1,...,m(ν∞⁰(g_m−izⁱ))=−m.

This expansion is understood as an equality in the algebraic closure of the field of formal Laurent seriesC((x)), which itself is the usual metric completion ofK(which has the property that an infinite sum converges if and only if each fixed power ofxappears in

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only finitely many terms). Hence, ifG_n(x)=g◦h, substitutingG_n(x)forxandh(x)for z, this yields an expansion forh(x)of the form

h(x)=u₋₁G_n(x)¹^/^m+u₀+u₁G_n(x)⁻¹^/^m+u₂G_n(x)⁻²^/^m+. . . , (3.2) for a suitable choice of them-th root ofG_n(x), where the coefficientsu_i ∈Cdepend only ong. As in [25], we expand the different rootsG_n(x)^s/mfors∈ {1,0,−1, . . .}using the multinomial theorem and equation (1.2) to get

G_n(x)^s/m=(a₁α₁ⁿ+a₂α₂ⁿ+· · ·+a_dα_dⁿ)^s/m

=a^s/m₁ α₁^ns/m

1+a₂ a1

α₂ α₁

n

+· · ·+a_d a1

α_d α₁

ns/m

=a^s/m₁ α₁^ns/mÕ

h¯

bh¯a₂^h². . .a^h_d^da^−(h₁ ²⁺^···^+h^d⁾

α₂^h². . . α^h_d^dα₁^−(h²⁺^···^+h^d⁾n

, where ¯h =(h₂, . . . ,h_d)runs throughN^d⁻¹. Now, we put y = 1/x. Then we see that h(x)=h(1/y)can be written as an infinite sum

h(1/y)=u−1Gn(1/y)^1/m+u0+u1Gn(1/y)^−1/m+u2Gn(1/y)^−2/m+. . . of termst(x)=t_h₂_,...,h_d_,s(1/y)of the shape

c·α₁(1/y)^ns/m−n(h²⁺^···^+h^d⁾·α₂(1/y)^nh². . . α_d(1/y)^nh^d, c∈C. (3.3) Observe that as an element ofC((y))the rational functionh(1/y)equalsy⁻^degh+. . ., since by Lemma 2.7 it can be written as y⁻^deg^h times a polynomial in y starting with a non-zero constant term (which is a unit in the ring C[[y]]). (Here we assume, as we may, that h is monic; otherwise the series would start with y⁻^degh times the leading coefficient ofh.) A similar consideration shows that each of the terms on the right hand side, written down explicitly in (3.3), up to a non-zero constant equals y⁽⁻^degα¹^(s/m−(h²⁺^···^+h^d^)−h²^degα²^{−···−h}^d^degα^d⁾ⁿ times a power series that is a unit in the ringC[[y]]. The “smallest” such term (i.e. the term with smallest order as a Laurent series) appears precisely withs=1 andh2 =· · ·=hd=0 (since by assumption degα₁ >degα_i for i ≥ 2), from which we see that the right hand side starts with y⁻ⁿ^deg^α¹^/m up to a constant inC. In view of degG_n = ndegα₁ = mdegh =deggdegh =deg(g◦h) this perfectly makes sense. We also remark that only terms withs=0,1 contribute to h(1/y)=y^−deg^h+· · ·+h₀, whereh₀ =0=h(0), since terms withs<0, up to a constant, start withy^|s^|^deg^α¹^n/m. We still have to show that the infinite sum converges as an element ofC((y)). This is clear since fors→ −∞the power ofygoes to∞. We give an alternative proof: We show that for an arbitraryJ∈N, there is an upper bound on the numberLof quantitiest₁(1/y), . . . ,t_L(1/y)in the expansion ofh(1/y)that satisfyν₀(t_i(1/y))<nJ,

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whereν₀denotes the order onC((y))(observe thatnis considered at this point to be fixed such thatGn(x)=g◦h). For a term of the shape (3.3), its order is given by

ν₀(t(1/y))=n·h s

m− (h₂+· · ·+h_t)

·ν₀(α₁(1/y))+h₂ν₀(α₂(1/y))+. . .

· · ·+h_dν₀(α_d(1/y))i

=n·hs

m·ν₀(α₁(1/y))+h₂(ν₀(α₂(1/y) −ν₀(α₁(1/y)))+. . .

· · ·+h_d(ν₀(α_d(1/y)) −ν₀(α₁(1/y)))i

. (3.4)

Note again that by Lemma 2.7 we have ν₀(α_i(1/y)) =ν₀(α_i(x)) =−degα_i. Since by assumption degα₁ >degα_i fori ≥2, it follows that degα₁ ≥ degA0/d. In the case s≤0 we therefore find

ν0(t(1/y))=n h−s

m degα1+h2(degα1−degα2)+· · ·+hd(degα1−degαd)i

≥n −s

m ·degA₀

d +h₂+· · ·+h_d

.

We observe that ifν0(t_i(1/y))<nJ, we must have thats∈ {0,−1, ...,−Jmd/degA0+1}, since otherwise we would get

ν₀(t(1/y)) ≥n −s

m ·degA₀

d +h₂+· · ·+h_d

≥n Jmd

degA₀ ·degA₀ md

=nJ.

To estimate the number of possible(d−1)-tuples (h₂, . . . ,hd) ∈ N^d−1 for each s ∈ {0,−1, . . . ,−Jmd/degA₀+1}, it obviously must hold that

hi <J− |s|degA0

md ≤J, so for each suchsthere are at mostJ^d−1such(h₂, . . . ,hd).

Ifs=1 we have that ν₀(t(1/y))=n

−degα₁

m +h₂(degα₁−degα₂)+· · ·+h_d(degα₁−degα_d)

≥n

−degα₁

m +h₂+· · ·+h_d

≥n

−degA₀

m +h₂+· · ·+h_d

, hence, ifs=1, we must haveh_i <J+^deg_m^A⁰. Then the number of possible(d−1)-tuples (h₂, . . . ,h_d)is not greater than(J+^deg_m^A⁰)^d−1.

We conclude that we may write

h(x)=h(1/y)=t₁(1/y)+· · ·+t_L(1/y)+ Õ

ν0(t(1/y)) ≥J n

t(1/y), (3.5)

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whereLis bounded above by L ≤

J+degA0

m d−1

+J^d md

degA₀. (3.6)

This now justifies the above formal expansions. They are well-defined in the ringC((y)). We now distinguish between two cases, namely that {t₁, . . . ,t_L,h(x)} is linearly dependent or linearly independent overC, respectively (we will often simply writeti

instead oft_i(x)).

Case 1. Let us assume that the set{t₁, . . . ,t_L,h(x)}is linearly independent overC. With the intention of applying Proposition 2.4, let F =C(y, α₁(1/y)^1/m)and write ϕ₁ = −t₁(1/y), . . . , ϕ_L = −t_L(1/y) and ϕ_L+1 = h(1/y). Also, set σ = Í_L+1

i=1 ϕ_i = Íν0(t(1/y)) ≥J nt(1/y). Observe thatϕ(T)=T^m⁰−α₁(1/y)^m⁰^/mis the minimal polynomial ofα₁(1/y)^1/moverC(y), where we use that the definition ofm₀forα₁as a polynomial in x over C implies that α₁(1/y)^m⁰^/^m is not a power of an element in C(y) for a smaller power. Since this is a Kummer extension we can apply Proposition 2.2 ([21, Theorem III.7.3]). Therefore, we get that only places inFabove 0,∞and the inverses of non-zero roots of α1 (as a polynomial in C[x]) ramify. Thus for the genus g_F of F we find 2g_F −2 ≤ m₀degα₁ ≤ mdegα₁. We define S to be the set of zeros and poles of the t1, . . . ,tL together with the poles of h(1/y). Observe that h(1/y) has poles at most at places above 0,∞. Therefore S may contain at most the places above 0,∞and the inverses of the non-zero roots of α₁, . . . , α_d. This gives at most m₀(2+degα₁+· · ·+degα_d) ≤m(2+degA₀)elements inS.

Note that for any placePinFabove 0 inC(y)we have that ν_P(σ)=ν_P

Õ

ν0(t(1/y)) ≥J n

t(1/y)

=e(P|0) ·ν₀ Õ

ν0(t(1/y)) ≥J n

t(1/y)

≥Jn.

Clearly,P ∈S. We will also need to give an upper bound on the degree degh(1/y)= [F : C(h(1/y))] = H (h(1/y)). Note thatH (h(1/y)) = (degh)H (1/y) = (degh)[F : C(1/y)]=(degh)[F:C(x)]=m₀degh≤mdegh.Hence, degh(1/y)=m₀degh.

By Proposition 2.4 we find that Õ

ν∈S

(ν(σ) − min

i=1,...,L+1ν(ϕ_i)) ≤ 1

2L(L+1)(|S|+2g_F−2)+degh(1/y)

≤ 1

2L(L+1)2m0(degA₀+1)+m₀degh. (3.7) On the other hand, sinceσ=Í_L+1

i=1 ϕ_i, it follows thatν(σ) −min_i=1,...,_L+1ν(ϕ_i) ≥0 for every valuation ν ∈ S. Moreover, for ν_P(σ) ≥ Jn and mini=1,...,L+1ν_P(ϕ_i) ≤

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ν_P(h(1/y)) ≤m₀ν₀(h(1/y)))=−m₀degh(by Lemma 2.7), we see that Jn+m₀degh≤ν_P(σ) − min

i=1,...,L+1ν_P(ϕ_i) ≤Õ

ν∈S

(ν(σ) − min

i=1,...,L+1ν(ϕ_i)).

We conclude that

Jn+m₀degh ≤Õ

ν∈S

(ν(σ) − min

i=1,...,L+1ν(ϕ_i)) (3.8)

≤L(L+1)m0(degA₀+2)+m₀degh, and therefore, fornwe get the upper bound

n≤mL(L+1)(degA₀+2)/J (3.9)

(recall that forJ∈None may take any natural number to get an upper bound, and that Lis bounded above by a constant depending only onJ,m=deggand the recurrence sequence, but not onn).

Case 2. Let us now consider the second case, namely that the set{t₁, . . . ,t_L,h(x)}is linearly dependent overC.

We may assume that thet₁, . . . ,t_Lare linearly independent, since otherwise we just group together the terms in question properly. Therefore, in a relation of linear dependency, h(x)must appear and we may writeh(x)as a linear combination oft₁(x), . . . ,t_L(x), i.e.

there arew_i ∈Csuch that

h(x)=

L

Õ

i=1

w_it_i(x). (3.10)

Recall, that thet_i’s are all of the shape (3.3), wheresand theh_i’s are elements of a finite set of numbers. After possibly renumbering the terms, we may assume thatw_i,0 exactly fori=1, . . . ,l≤Land we get a power sum representation ofh(x)of the shape

h(x)=w₁d₁δⁿ₁ +· · ·+w_ld_lδ_lⁿ,

where we can control theδ_i’s, since they are elements of a finite set, namely δ_i ∈ {α₁^s/m−(h²⁺^···^+h^t⁾α₂^h². . . α^h_d^d; s∈A, h_i ∈B},

where A={0,1}andB ={0,1, . . . ,J+degA₀/m}(for Aonly termst_i withs=0,1 contribute, as we have already observed above). However, note that we have no control over the coefficientsw_idi ∈C.

We pause a moment to investigate this relation. Remember thatm₀was defined to be the least integer such thatα^m₁⁰^/m ∈C(x). Thus we havem₀=[F:C(x)]. Observe thatm₀