On the Partial Generalization of the Measure of Transcendence of Some Formal Laurent Series

III this work, we determine the trallscendencp measure oJ the foona,1 Laurellt series" thaL" whose trcl,llSCCllclencc has beC'n est.ablished by L.I. \Vade. Using the methods and lemmas ill P.BulIrlscbull's articl(" mrasure of t.he t.rcllJscelHlence for the above (is dctrrmined (IS On t.he other band, it was proven t.JlaL the t.rnJlSCendellCl: series ~ is not a U but is a oS' or T-llIl111her according to the r'llahler's classification.

Let p a prime number and u ~1 an integer.Let F be a finite field with q = pU elements.We denote the ring of the polynomials with one variable over F by F[x] and its quotient field by F(x) .If a E F[x] is a non-zero polynomial, denote its degree by 8a.If a = 0 , then its degree is defined as 80 := -00.Let  Let J( be the completion of F( x) with respect to this valuation.Every element w of I< can be uniquely represented by 00 w = LCnX-n,cn E F. n=k If w = 0, then all Cn are zero.If w f 0, then there exist an k E Z for wich Ck f O.If w f 0 , then we have Therefore I< is the field of all formal Laurent series.The classical theory of transcendence over complex numbers has a similar version over J( .Elements of F[ x] and F( x) correspond to integers and fractions of the classical theory, respectively.
If w E J( is one of the roots of a non-zero Rolynomial with coefficients in F[x], then w E I< is said to be algebraic over F(x).
Otherwise, w is called transcendental over F( x) .The studies of transcendental numbers in I< were initiated first by Wade [1][2][3][4] .Also Geijsel [5][6][7][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 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] and Fel'dman [10] .Examples for the transcendence measure in the field J( have been given for the first time by Bundschuh [12]. In this work, we determine a transcendence measure of some formal Laurent series whose transcendence has ben established by L.I. Wade  = 0 and 8(Fk) = kqk then the series is an element of J( , and L.I.Wade showed its transcendence in [2].(see Theorem 3.1 and 3.2) Using the methods and lemma.s in Bundschuh's article [12] , we determine a transcendence measure of ~.We take an arbitrary non-zero polynomial According to the above classification, the series defined in (1) can not be a U-Laurent series.This fact may be proved by the help of the Theorem 1.

Theorem 2:
The ~Laurent series defined by (1) doesn't belong to the class U so that it belongs to the class S or to the class T. ( In ( 13) we put e instead of y : Seperate in (12) sum as T1 + T2,where 1) First, we prove that IT11 ~1 .That is, we prove T1 is a polynomial but not equal zel'O.By the definition of F~,obviously T1 is polynomial.Furthermore, for ,B sufficiently large.Therefore, for all sufficiently ,B T1 is not identically zero.So T1 is non-zero polynomial.So it shown that ITti ~1.(where = degFfJ + degAj -degFJ_j(where degr q J = 0) hence rf3qfJ < qfJ(f3 q + q + f3 -d) so we obtain f3 + d < (f3 + 1)q.This inequality is true every time.

IF(OI
a and b (b f: 0) two polinomials from F[x] and define a discrete valuation of F( x) as follows I ~I= 8a-Bb b q .
[2}.If r qk (where r E RealNumbers)and Fk E F[x) is a fixed non-zero polynomial of degree Then there are some elements Ao, AI, ... , Ad E F[xJ ,not all zero satisfying.