On Mahler's Classification of Formal Power Series Over a Finite Field

Gülcan Kekeç

doi:10.1515/ms-2022-0017

Artikel Open Access

On Mahler's Classification of Formal Power Series Over a Finite Field

Gülcan Kekeç

Veröffentlicht/Copyright: 16. Februar 2022

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Mathematica Slovaca Band 72 Heft 1

Abstract

Let K be a finite field, K(x) be the field of rational functions in x over K and K be the field of formal power series over K. We show that under certain conditions integral combinations with algebraic formal power series coefficients of a U₁-number in K are U_m-numbers in K, where m is the degree of the algebraic extension of K(x), determined by these algebraic formal power series coefficients.

2010 Mathematics Subject Classification: Primary 11J61; Secondary 11J82

Keywords: Mahler’s classification of formal power series over a finite field; U-number; transcendence measure

1. Introduction

1.1. The field of formal power series over a finite field

Let K be a finite field with q elements. We denote the ring of polynomials in x with coefficients from K by K[x], the quotient field of K[x] by K(x) and the degree of a non-zero polynomial f(x) in K[x] by deg(f). A non-Archimedean absolute value | · | is defined on K(x) by setting

|0|=0 and f(x)g(x)=qdeg(f)−deg(g),

where f(x) and g(x) are non-zero polynomials in K[x]. The field K of formal power series over K is the completion of K(x) with respect to | · |. We denote the unique extension of | · | to the field K by the same notation | · |. Every non-zero element ξ of K can be represented uniquely as

ξ=∑n=r∞anx−n,

where a_n ∈ K (n = r, r +1, …) with a_r ≠ 0 and r is the rational integer such that |ξ| = q^−r.

An element of K is called an algebraic (resp. a transcendental) formal power series if it is algebraic (resp. transcendental) over K(x). We regard K[x], K(x) and K as the analogues of ℤ, ℚ and ℝ, respectively.

Let P(y) = f₀ + f₁y + ⋯ + f_nyⁿ be a non-zero polynomial in y over K[x]. (Note that f_i ∈ K[x] for i = 0, 1, …, n.) We denote the degree of P(y) with respect to y by deg(P) and the height of P(y) by H(P), where H(P) = max {|f₀|, …, |f_n|}. Let α be an algebraic formal power series and P(y) be its minimal polynomial over K[x]. Then the height H(α) and the degree deg(α) of α are defined by H(P) and deg(P), respectively. Moreover, the roots of P(y) are said to be the conjugates of α over K (x). We refer the reader to Sprindžuk [16] for detailed information on K.

1.2. Mahler’s classification of formal power series over a finite field

In 1932, Mahler [9] introduced a classification of complex numbers. He divided complex numbers into four disjoint classes and called the numbers in these classes A-, S-, T- and U-numbers. Two algebraically dependent complex numbers belong to the same class. The class of A-numbers coincides with the set of complex algebraic numbers. Almost all complex numbers, in the sense of Lebesgue measure on ℂ, are S-numbers. Schmidt [15] proved that there exist uncountably many T-numbers. The class of U-numbers is subdivided into U_m-subclasses (m = 1, 2, 3, …). LeVeque [8] constructed the first explicit examples of U_m-numbers for any positive integer m.

Let p be a prime number. The field ℚp of p-adic numbers is the completion of ℚ with respect to the p-adic absolute value. In 1935, Mahler [10] proposed a classification of p-adic numbers, analogous with his classification of complex numbers. He divided p-adic numbers into four disjoint classes and called the numbers in these classes p-adic A-, S-, T- and U-numbers. Two algebraically dependent p-adic numbers belong to the same class. The class of p-adic A-numbers coincides with the set of p-adic algebraic numbers. Almost all p-adic numbers, in the sense of Haar measure on ℚp, are p-adic S-numbers. Schlickewei [14] proved that there exist p-adic T-numbers. The class of p-adic U-numbers is subdivided into U_m-subclasses (m = 1, 2, 3, …). Alnıaçik [1: Chapter III, Theorem I] constructed the first explicit examples of p-adic U_m-numbers for any positive integer m. We refer the reader to Bugeaud [2] for further information and references on Mahler’s classification of complex numbers and of p-adic numbers.

In 1978, Bundschuh [3] introduced a classification in K in analogy with Mahler’s classification of complex numbers and of p-adic numbers. This classification is said to be Mahler’s classification of formal power series over a finite field. Formal power series are divided into four disjoint classes with respect to this classification, called A-, S-, T- and U-numbers as follows.

Let ξ∈K be any formal power series over K and let n and H be any positive rational integers. Set

wn(H,ξ)=min{|P(ξ)|:P(y)∈K[x][y],deg(P)≤n,H(P)≤HandP(ξ)≠0},wn(ξ)=lim supH→∞−log wn(H,ξ)log Handw(ξ)=lim supn→∞wn(ξ)n.

We denote by μ(ξ) the smallest integer n such that w_n(ξ) = ∞ if such an integer exists and write μ(ξ) = ∞ otherwise. Then ξ is said to be

an A-number if w(ξ) = 0 and μ(ξ) = ∞,
an S-number if 0 < w(ξ) < ∞ and μ(ξ) = ∞,
a T-number if w(ξ) = ∞ and μ(ξ) = ∞,
a U-number if w(ξ) = ∞ and μ(ξ) < ∞.

Furthermore, a U-number ξ is said to be a U_m-number if μ(ξ) = m.

Recently, Ooto [11] contributed a lot to Mahler’s classification in K, stated as follows. The class of A-numbers coincides with the set of algebraic formal power series (Ooto [11: Proposition A.3]). Two algebraically dependent transcendental formal power series belong to the same class (Ooto [11: Proposition A.4]). Sprindžuk [16: Part II, Chapter 3] proved that almost all formal power series ξ in K, in the sense of Haar measure on K, are S-numbers with w_n(ξ) = n for n = 1, 2, 3, … and hence with w(ξ) = 1. Ooto [11: Theorem 1.1] recently proved that there exist uncountably many T-numbers in K. He constructed explicit examples of T-numbers in K. The existence of U_m-numbers for any positive integer m was first proved by Oryan [12]. His method allows us to construct explicit examples of U_m-numbers in K. Recently, Kekeç [6] and [7] contributed to constructing explicit examples of U_m-numbers in K by making use of the method of Oryan [12: Satz 4 and Satz 5]. We recommend the reader to consult Sprindžuk [16], Bundschuh [3] and Bugeaud [2] for further results and references on Mahler’s classification in K.

1.3. Construction of our main result

In 1979, Alnıaç1k [1: Chapter III, pp. 73–81] proved that under certain conditions rational and integral combinations with p-adic algebraic number coefficients of a p-adic U₁-number are p-adic U_m-numbers, where m is the degree of the p-adic algebraic number field, determined by these p-adic algebraic number coefficients. By this method, Alniaçik proved the existence of p-adic U_m-numbers and gave the first explicit construction of such numbers. In the sense of this result and method of Alniaçik, we prove the following theorem. Our method heavily depends on the method given by Oryan [12: Satz 4 and Satz 5, pp. 57–62] to prove the existence of U_m-numbers in K. We state our main result as follows.

Theorem 1.1.

Letα₀, …, α_k (k ≥ 1) be algebraic formal power series withα_k ≠ 0, and letξbe a U₁-number enjoying such a representation

(1.1)ξ=∑n=0∞anx−un,

wherea_n ∈ K – {0} (n = 0, 1, 2, …) andunn=0∞is a strictly increasing sequence of non-negative rational integers with

(1.2)limn→∞un+1un=∞.

Thenα₀ + α₁ξ + ⋯ + α_kξ^k is a U_m-number, wheremis the degree ofK(x)(α₀, …, α_k) overK(x).

Remark 1.

By definition, as described in Subsection 1.2, a U₁-number ξ in K is a formal power series ξ in K, enjoying such an approximation

0<ξ−an(x)bn(x)<Han(x)bn(x)−dn (n=1,2,3,…),

where a_n(x), b_n(x) are non-zero polynomials in K[x] with H(a_n(x)/b_n(x)) > 1 for n = 1, 2, 3, … and dnn=1∞ is a sequence of positive real numbers with limn→∞dn=∞. Therefore the representation (1.1) together with (1.2) is not a characterization of a U₁-number. The set of numbers ξ satisfying (1.1) and (1.2) is just a subset of the set of U₁-numbers. Moreover, we see from the definition that a U₁-number in K is the analogue of a Liouville number in ℝ. We refer the reader to Perron [13] for information on Liouville numbers.

Remark 2.

Theorem 1.1 is the K analogue of [1: Chapter III, Theorem I] in p-adic case and of [5: Corollary 1] in real case. Furthermore, we recommend the reader to consult a related work [4] in real case, which yields to construct explicit U_m-numbers in ℝ.

We highlight the following consequence of Theorem 1.1.

Corollary 1.1.1.

Letα∈K−{0}be algebraic overK(x) with deg(α) = m and ξ be a U₁-number satisfying the conditions ofTheorem 1.1. Then, byTheorem 1.1. α + ξandα · ξ are U_m-numbers.

In the next section, we prepare and cite certain lemmas in order to prove Theorem 1.1. In Section 3, we prove Theorem 1.1 and establish explicit examples of U_m-numbers.

2. Auxiliary results

We apply the following four lemmas to prove Theorem 1.1. Lemma 2.1 can be regarded as a K analogue of Lemma 5 in [1: Chapter I].

Lemma 2.1.

Letα₀, …, α_k(k ≥ 1) be algebraic formal power series withα_k ≠ 0. Then for θ inK(x) the algebraic formal power seriesα₀ + α₁θ + … + α_kθ^kis a primitive element ofK(x)(α₀, …, α_k) overK(x) except for only finitely many θinK(x).

Proof. We adapt the method of the proof of Lemma 5 in [1: Chapter I] to the field K(x)¯, where K(x)¯ denotes the algebraic closure of K(x). The field K(x)¯ is the analogue of the field ℚ¯ of complex algebraic numbers. We put

L:=K(x)α0,…,αk

and

R(y)≔α0+α1y+⋯+αkyk.

Let us denote the degree of L over K(x) by m. If m = 1, then Lemma 2.1 obviously holds true. Hence, let us assume that m > 1. We denote by αi(1), …, αi(m) the field conjugates of α_i for L, where i = 0, 1, …, k. These are the conjugates of α_i over K(x), each repeated m/deg(α_i) times. We exclude the values θ ∈ K(x) satisfying R(θ) = 0, if such values of θ exist, since they constitute a finite set. Let θ ∈ K(x) with R(θ) ≠ 0. If R(θ) is not a primitive element of L, then there exist integers s, t in {1, …, m} with s ≠ t such that

(2.1)(R(θ))(s)=(R(θ))(t).

From (2.1),

α0+α1θ+⋯+αkθk(s)=α0+α1θ+⋯+αkθk(t).

This gives

α0(s)+α1(s)θ(s)+⋯+αk(s)θk(s)=α0(t)+α1(t)θ(t)+⋯+αk(t)θk(t)

and hence

α0(s)+α1(s)θ(s)+⋯+αk(s)θ(s)k=α0(t)+α1(t)θ(t)+⋯+αk(t)θ(t)k.

Since θ ∈ K(x), we have θ^(s) = θ^(t) = θ. Thus

α0(s)+α1(s)θ+⋯+αk(s)θk=α0(t)+α1(t)θ+⋯+αk(t)θk.

If (2.1) were satisfied by infinitely many θ in K(x), then the last equality would turn to an identity and we would have

(2.2)αi(s)=αi(t)(i=0,1,…,k).

Let β be a primitive element of L over K(x). Then deg(β) = m and the field conjugates of β for L are distinct from each other. We have

β=Sα0,α1,…,αk,

where S(y₀, y₁, …, y_k) is a rational function in y₀, y₁, …, y_k with coefficients from K(x). Then we obtain

β(ν)=Sα0(ν),α1(ν),…,αk(ν)

for ν = 1, 2, …, m. By (2.2), this would give us

β(s)=β(t).

But this is impossible since β is a primitive element of L. Therefore (2.1) can hold true for only finitely many θ in K(x). This completes the proof of Lemma 2.1.

Lemma 2.2.

(Oryan [12: Hilfssatz 4]). Let β₁, …, β_nbe in a finite extension of degree moverK(x). Then

Hβ1+⋯+βn≤Hβ12m2⋯Hβn2m2

and

Hβ1⋯βn≤Hβ12m2⋯Hβn2m2.

In classical case, it is well-known that the inequality |β| ≤ H(β) + 1 holds for a complex algebraic number β, where H(β) denotes its usual height and | · | denotes the usual absolute value on ℂ The following lemma is the K analogue of this well-known result, whose proof can be done easily by following the lines in Waldschmidt [17: 3.5 Liouville’s Inequalities, p. 82] and using the property of non-Archimedean absolute value.

Lemma 2.3.

Let β be an algebraic formal power series. Then

|β|≤H(β).

Lemma 2.4 (Oryan [12], Hilfssatz 2).

LetP(y) andQ(y) be polynomials overK[x] with degreesn ≥ 1 andm ≥ 2, respectively. If the polynomialsP(y) andQ(y) are relatively prime overK[x] andα is a root of Q(y), then

|P(α)|≥H(P)−m+1H(Q)−n.

3. Proof of Theorem 1.1

We write

ξ=ξn+ρn (n=0,1,2,…),

where

ξn=∑h=0nahx−uh and ρn=∑h=n+1∞ahx−uh (n=0,1,2,…).

It follows that

(3.1)|ξ|=ξn=q−u0≤1 (n=0,1,2,…)

and

(3.2)ρn=q−un+1<1 (n=0,1,2,…).

We put

R(y)≔α0+α1y+⋯+αkyk and γ≔R(ξ).

We want to show that γ is a U_m-number. Set

γ=Rξn+ρn=γn+ρnδn,

where

γn=Rξn (n=0,1,2,…)

and

δn=α1+α22ξn+ρn+⋯+αkk1ξnk−1+k2ξnk−2ρn+⋯+ρnk−1

for n = 0, 1, 2, …. Let

αi≕q−ti (i=1,…,k) and t≔min0,t1,…,tk.

Then, using (3.1), (3.2) and the property of non-Archimedean absolute value, that is, |α + β| ≤ max {|α|, |β|} holds for α,β∈K, we get

(3.3)δn≤q−t (n=0,1,2,…).

Note that γ_n ∈ K(x)(α₀, …, α_k) for n = 0, 1, 2, …. We deduce from Lemma 2.1 that deg(γ_n) = m and further from Lemma 2.2, the fact H(ξ_n) = q^u_n (n = 0, 1, 2, …) that

(3.4)Hγn≤c0un

hold for sufficiently large n, where c₀ is a real constant which depends only on m, q, k and the algebraic formal power series α_i (i = 0, 1, …, k) with c₀ > q.

We shall first show that γ is a U-number with μ(γ) ≤ m. Let A_n(y) = a_n0 + a_n1y + ⋯ + a_nmy^m be the minimal polynomial of γ_n over K[x]. Since A_n(γ_n) = 0, we have

An(γ)=Anγn+ρnδn=ρnσn,

where

σn=an1δn+⋯+anmm1γnm−1δn+m2γnm−2ρnδn2+⋯+ρnm−1δnm.

It follows from Lemma 2.3, (3.2), (3.3) and (3.4) that

σn≤c1un,

where c₁ is a real constant with c₀ < c₁, thus from (3.2) that

An(γ)=ρnσn≤q−un+1c1un

and hence from (3.4) that

(3.5)0<An(γ)≤Hγn−sn

hold for sufficiently large n, where

sn=un+1unlogqlogc1−1 and limn→∞sn=∞.

Since ξ is a transcendental formal power series, note that A_n (γ) = A_n (R(ξ)) is not zero. The sequence An(y)n=0∞ has infinitely many different terms. For otherwise An(y)n=0∞ would have finitely many different terms, that is, there would exist a repeating polynomial A(y) among these terms. Namely, there would exist a subsequence Anj(y)j=0∞ of An(y)n=0∞ such that A_n_j (y) = A(y) for j = 0, 1, 2, …. Since A_n_j (y) is the minimal polynomial of γ_{n_j} = R(ξ_n_j) for j = 0, 1, 2, …, we would obtain

ARξnj=0 (j=0,1,2,…).

Letting j tend to infinity and observing that limn→∞ξn=ξ, this would give us A (R(ξ)) = 0 which contradicts with the fact that ξ is a transcendental formal power series. Therefore the sequence An(y)n=0∞ must have infinitely many different terms. This implies that

limsupn→∞HAn=∞.

For otherwise the sequence HAnn=0∞ would be bounded above. Since A_n(y) (n = 0, 1, 2, …) are polynomials over K[x] with deg(A_n) = m for sufficiently large n and k is finite, the sequence An(y)n=0∞ would have finitely many different terms, contrary to the fact that An(y)n=0∞ has infinitely many different terms. Thus lim supn→∞ H(An)=∞ must hold. Then the sequence HAnn=0∞ has a subsequence HAnii=0∞ such that

1<HAn1<HAn2<HAn3<⋯, limi→∞HAni=∞.

Therefore we infer from (3.5) and H(γ_{n_i}) = H(A_{n_i}) (i = 0, 1, 2, …) that

0<Ani(γ)≤HAni−sni

for sufficiently large i. Since deg(A_{n_i}) = m, this implies that γ is a U-number with

(3.6)μ(γ)≤m.

Now we shall show that μ(γ) ≥ m. If m = 1, then μ(γ) ≥ m. Let m > 1 and B(y) = b₀ + b₁y + ⋯ + b_gy^g be any polynomial over K[x] with deg(B) = g and 1 ≤ g ≤ m – 1. Then, for any positive rational integer ν,

(3.7)B(γ)=Bγν+ρνδν=Bγν+ρνθν,

where

θν=b1δν+⋯+bgg1γνg−1δν+g2γνg−2ρνδν2+⋯+ρνg−1δνg.

Similarly as in the previous paragraph, we obtain for sufficiently large ν

θν≤c1uν and ρνθν≤c1−uνsν.

There exists a real constant c₂ with 0 < c₂ < 1 such that

sν≥c2uν+1uν

and hence

(3.8)ρνθν≤c1−c2uν+1

hold for sufficiently large ν. Since deg(γ_ν) = m, it follows that B(γ_ν) ≠ 0. Thus, by Lemma 2.4 and (3.4), we have for sufficiently large ν

(3.9)Bγν≥H(B)−(m−1)Hγν−(m−1)≥H(B)−(m−1)c1−uν(m−1).

We choose two real numbers λ and η such that

(3.10)λ>2(m−1)c2

and

(3.11)η>(m−1)(λ+1)c2.

The inequality

(3.12)η<uν+1uν

is verified for sufficiently large ν.

Let B(y) be a polynomial over K[x] with 1 ≤ deg(B) ≤ m – 1 and with sufficiently large height H(B). Let n be the unique positive rational integer satisfying

c1un≤H(B)<c1un+1.

If c1un≤H(B)<c1un+1/λ holds, then it follows from (3.7), (3.8) and (3.9) for ν = n that

B(γ)=Bγn+ρnθn,Bγn≥H(B)−2(m−1),ρnθn<H(B)−λc2,

Then, by (3.10), we obtain |ρ_n||θ_n| < |B(γ_n)|. So, using the property of non-Archimedean absolute value, that is, |α + β| = max {|α|, |β|} is verified for α,β∈K when |α| ≠ |β|, we get

(3.13)|B(γ)|=Bγn≥H(B)−2(m−1).

If c1un+1/λ≤H(B)<c1un+1 holds, then it follows from (3.7), (3.8), (3.9) and (3.12) for ν = n + 1 that

B(γ)=Bγn+1+ρn+1θn+1,Bγn+1≥H(B)−(m−1)(λ+1),ρn+1θn+1≤c1−c2un+2<c1−c2ηun+1<H(B)−ηc2.

Then, by (3.11), we obtain |ρ_n+1||θ_n+1| < |B(γ_n+1)|. Hence,

(3.14)|B(γ)|=Bγn+1≥H(B)−(m−1)(λ+1).

We infer from (3.13) and (3.14) that

|B(γ)|≥H(B)−(m−1)(λ+1)

for all polynomials B(y) over K[x] with 1 ≤ deg(B) ≤ m – 1 and with sufficiently large height H(B). This implies that

(3.15)μ(γ)≥m.

By (3.6) and (3.15), we conclude that μ(γ) = m. Then γ = α₀ + α₁ξ + ⋯ + α_kξ^k is a U_m-number. This completes the proof of Theorem 1.1.

Remark 3.

Using Lemma 2.4 and (3.4), we can rewrite (3.9) exactly as

(3.16)Bγν≥H(B)−(m−1)Hγν−deg(B)≥H(B)−(m−1)c1−uνdeg(B).

If we apply (3.16) in place of (3.9) in the proof of Theorem 1.1, then we see that

|B(γ)|≥H(B)−(m−1)−λdeg(B)

holds true for all polynomials B(y) over K[x] with 1 ≤ deg(B) ≤ m – 1 and with sufficiently large height H(B). This yields that

wt(γ)≤m−1+λ⋅t for t=1,…,m−1

when m ≥ 2.

Remark 4.

Let us denote by K¯ the completion of the algebraic closure of K with respect to | · |. We regard the field K¯ as the analogue of the field ℂ of complex numbers. The height, the degree and the conjugates of an element α in K¯, which is algebraic over K(x) are defined exactly as in Subsection 1.1. Due to Bundschuh [3], Mahler’s classification is carried from K to K¯. A-, S-, T-, U- and U_m-numbers in K¯ are defined as in Subsection 1.2. (We just begin with ξ∈K¯ in place of ξ∈K) Moreover, we observe that Lemma 2.1, Lemma 2.2, Lemma 2.3 and Lemma 2.4 hold true not only for algebraic formal power series in K but also for elements in K¯, which are algebraic over K(x). Then we see that Theorem 1.1 is also valid for the elements α₀, …, α_k (k ≥ 1) in K(x)¯ with α_k ≠ 0. (Note that an element of K(x)¯ need not be in K, but it is certainly in K¯.) In this case, the occurring U_m-number is in K.

Remark 4 enables us to give the following example to illustrate Theorem 1.1.

Example 1.

Let m be any positive rational integer. In Theorem 1.1, let us take α0=⋯=αk=xm, where xm is defined as a root of the polynomial y^m – x, and ξ=∑n=0∞x−(n+1)!. Then α₀ + α₁α + ⋯ + α_kξ^k is a U_m-number.

(Communicated by István Gaál)

Acknowledgement

The author would like to thank the referees for their helpful suggestions.

REFERENCES

[1] ALNIAҪIK, K.: On the subclasses U_min Mahler’s classification of the transcendental numbers, İstanbul Üniv. Fen Fak. Mecm. Ser. A 44(1979), 39–82.Suche in Google Scholar

[2] BUGEAUD, Y.: Approximation by Algebraic Numbers, Cambridge Tracts in Math. 160, Cambridge University Press, Cambridge, 2004.10.1017/CBO9780511542886Suche in Google Scholar

[3] BUNDSCHUH, P.: Transzendenzmasse in Körpern formaler Laurentreihen, J. Reine Angew. Math. 299/300(1978), 411–432.10.1515/crll.1978.299-300.411Suche in Google Scholar

[4] CHAVES, A. P.—MARQUES, D.: An explicit family of Um-numbers, Elem. Math. 69(2014), 18–22.10.4171/EM/240Suche in Google Scholar

[5] CHAVES, A. P.—MARQUES, D.—TROJOVSKÝ, P.: On the arithmetic behavior of Liouville numbers under rational maps, Bull. Braz. Math. Soc. New Series 52(2021), 803—813.10.1007/s00574-020-00232-7Suche in Google Scholar

[6] KEKEҪ, G.: U-numbers infields of formal power series over finite fields, Bull. Aust. Math. Soc. 101(2020), 218–225.10.1017/S0004972719000832Suche in Google Scholar

[7] KEKEҪ, G.: On transcendental formal power series over finite fields, Bull. Math. Soc. Sci. Math. Roumanie 63(111) (2020), 349–357.Suche in Google Scholar

[8] LEVEQUE, W. J.: On Mahler’s U-numbers, J. London Math. Soc. 28(1953), 220—229.10.1112/jlms/s1-28.2.220Suche in Google Scholar

[9] MAHLER, K.: Zur Approximation der Exponentialfunktion und des Logarithmus I, II, J. Reine Angew. Math. 166(1932), 118–150.10.1515/crll.1932.166.137Suche in Google Scholar

[10] MAHLER, K.: Über eine Klasseneinteilung der p-adischen Zahlen, Mathematica (Leiden) 3(1935), 177—185.Suche in Google Scholar

[11] OOTO, T.: The existence of T-numbers in positive characteristic, Acta Arith. 189(2019), 179—189.10.4064/aa180325-24-8Suche in Google Scholar

[12] ORYAN, M. H.: Über die Unterklassen U_m der Mahlerschen Klasseneinteilung der transzendenten formalen Laurentreihen, İstanbul Üniv. Fen Fak. Mecm. Ser. A 45(1980), 43—63.Suche in Google Scholar

[13] PERRON, O.: Irrationalzahlen, Walter de Gruyter & Co., Berlin, 1960.10.1515/9783110836042Suche in Google Scholar

[14] SCHLICKEWEI, H. P.: p-adic T-numbers do exist, Acta Arith. 39(1981), 181—191.10.4064/aa-39-2-181-191Suche in Google Scholar

[15] SCHMIDT, W. M.: T-numbers do exist. In: Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/1969), Academic Press, London, 1970, pp. 3—26.Suche in Google Scholar

[16] SPRINDŽUK, V. G.: Mahler’s Problem in Metric Number Theory. Transl. Math. Monogr. 25, Amer. Math. Soc., Providence, R.I., 1969.10.1090/mmono/025Suche in Google Scholar

[17] WALDSCHMIDT, M.: Diophantine Approximation on Linear Algebraic Groups. Grundlehren Math. Wiss. 326, Springer, Berlin-Heidelberg-New York, 2000.10.1007/978-3-662-11569-5Suche in Google Scholar

Received: 2020-09-27

Accepted: 2021-03-31

Published Online: 2022-02-16

Published in Print: 2022-02-16

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/ms-2022-0017

Schlagwörter für diesen Artikel

Mahler’s classification of formal power series over a finite field; U-number; transcendence measure

Creative Commons

BY 4.0