Bazı formel laurent serilerinin transandantlık ölçüsü hakkında
On the measure of transcendence of certain formel laurent series
- Tez No: 39140
- Danışmanlar: PROF.DR. FETHİ ÇALLIALP
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Formel Laurent serileri, Formel Laurent series
- Yıl: 1993
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Bu çalışmada, transandantlığı Wade [4] tarafından isbat edilen bazı formel laurent serileri için transandantlık ölçüsü belirlenmektedir, s > 1 tam sayısı p* kuvvetlerinden farklı olsun. Yani s'nin p'den farklı bir asal sayı böleni bulunsun. GeF[x] sıfırdan farklı sabit tutulan bir polinom ve 8{G) = g olsun. oo i fc=0 serisi ÜT'mn elaman^ olup, transandant olduğu Wade [4] tarafından gösterilmiştir. Bundschuh'un [12] deki bir çalışmasında ki metodlar ve yardımcı teoremler kullanılarak yukarıda tanımlanan u> için. ' T(n,H) = H-ld+1*d-ad?d olarak bir transandant ölçüsü elde edildi. Öte yandan Mahler tasnifinin tanımından u;n(#,w)>#-<n+1)«n-im92n ve dolaysıyla wı»(w) = snq2n + (n + l)qn elde edildi. Yani her n indeksi için Wn{u) < OO I ı olduğu görüldü. Buradan Mahler tasnifi tanımına göre w'mn U sınıfına ait olmadığı fakat ya 5" sınıfına veya T sınıfına ait olduğu gösterildi. iv
Özet (Çeviri)
SUMMARY ON THE MEASURE OF TRANSCENDENCE OF CERTAIN FORMEL LAURENT SERIES Let p a prime number and u > 1 an integer. Let F be a finite field with q = pu elements. We represent the ring of the polinomials with one variable over F with F[x] and its quotient field with F(x). If acFfx] is a non-zero polinomial, denote da its degree. If d = 0, then its degree is defined as dO := - oo. Let a and b (b ^ 0) two polinomials from F[x]' and define a discrete valuation of F(x) as follows ' l-l = a9a~db, V * F(x) can be completed by this valuation and is obtained the field K. Every element u> of K can be uniqly represented by u = y^2,cnx n, c“,eF. 1=fc If u t^ 0, then it exist an keZ for wich Cfc ^ 0. liu) - Q, then all cn are zero. If u) ^ 0, then we have M = <rfc Therefore K is the field obtained by all Formel Laurent series. The classical theory of transcendence number on complex numbers has a similar version on K. Elements of F[x] and F(x) correspond to integer and quotients of the classical theory, respectively. If uj is one of the roots of anon-zero polynomial with coefficients in F[x], then ueK is said to be algebraic over F(x). Otherwise, u is called transcendence over F(x). The studies to find transcendence numbers in K were initiated first by Wade [1-4]. Also Geijsel [5-8] did similar studies. As it is the case in the classical theory of transcendental numbers, it is possible to define a measure of transcendence Attempts to find transcendental numbers belonging to a field K, where K is the completion of a rational function field F[x] whose underlying field F is finite of characteristic p ^ 0 have been initiated by Wade [1-4], and persued further by Geijsel [5-8]. The measure of transcendence is thoroughly studied in the classical theory. For example, the transcendence measure of e has been widely investigated by Mahler [9], Fel'dman [10] and Cijsouw [11]. Example for the transcendence measure in the field K have been given for the first time by Bundschuh [12]. In [12], Bundschuh gave transcendence measures of k=0 ** = !!*,![«].*- (*>0) ' '. (2) Lk = IlkK=1[K} (*>0).,(3) The prerequisities of this work are Theorem 1, Lemma 1, Lemma 2 in section 2. In section 2, we collect also the basic concepts of valuation theory, basic facts about algebraic and transcendental numbers and Mahler's classification. In this work, we determine the transcendence measure of someformal Laurent series whose transcendence has ben established by Wade [4].We take the integer s > 1 distinct from powers of p, so that s has prime divisiors different from p. If GeF[x] is a fixed non-zero polynomial of degree d{G) = g, then the series VIk=0 ^ is an element of K, and Wade showed its transcendence in [4]. Using the methods and lemmas in Bundschuh's article [12], we determine a transcendence measuie of ix>. To this end, we take an, arbitrary non-zero polynomial f(y)r£«y ”(5) i/=0 [aveF[x\\v = 0,1,...,n) whose degree d(P) is less then or equal to n. The height of P is denoted by ! i h{p) = map\au\ = qVk*9^ (6) For the Transcendental element a> of K, we dbfine the positive quantity 1 u)n(H,uj) = min\P{ui)\, (7) d{P) < n h(P)<H If T(n, H) is a function of the variables n, H of u>n{H, u>) which satisfies the inequality un(H,u)>T(n,H),, (8),- for all sufficiently large values of n and H, then T(n,H) is said to be a transcendence measure of w. viiIn this work, we take an arbitrary, non-zero polynomial p{y) = İ2a»yu (9) f=0 {avzF[x]\v = 0,1,..., n) as above, and find the transcendence measure T{n,H) = H-(d+Vqd-ad'I*'i (10) of to, where ' ' ' dqdlogH>9-^- (11) s, This is Theorem 1. Besides, by the definition of Mahler's classification, we find u>n{H,u) > #-(*+D»n-W» ' (12) and consequently wn(w) < snq2n + (n + \)qn ' ' - (13) for all sufficiently large natural numbers n and H. In this connection, we investigated whether the transcendental series u> belongs to the class S, or T or U according to Mahler's classification These investigations show that the transcendental series u> can never belongs to the class U, so that it must belong to the class S or to the class T. vmOn the other hand,let the least n satisfying ion{u>) = oo be denoted' by fi(uj).lî there is no such n, then one may define ft(uj) as oo.In this case, the trascendental number ueR is called as '. S-Laurent series if 1 < a>(oo) < oo and //(a;) = oo T-Laurent series if to(u) = oo and fi(oo) = 00 U-Laurent series if u(u}) = 00 and fi(oo) < 00 Moreover, the U-class may be divided into subclasses. If /u(w) = rn(m > 0) then w is called {/m-Laurent series. Leveque [14] was the first to show that for all m, Um is non-empty in the classical theory but the honour goes to Oryan, [15] if the ground field is K. According to the definition of above classification, the series defined in (4) can not be a U-Laurent series. This fact may be proved by the help of theorem 1. So the series defined in (4) does not belong to the U-class. Hence it belongs to either S or T. ',, IX
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