By Kenji Nagasaka

Those lawsuits encompass invited papers by means of mathematicians reviewing the hot growth in analytic quantity idea and comparable subject matters. Papers on Diophantine approximations, zeta services, Dirichlet L-functions, general numbers, dispersion of multidimensional sequences, and Diophantine equations also are awarded.

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**Example text**

Cr(z) = f3o,rZo + · · · + f3d,rZd independantes a coefficients dans Q, sur l'espace tangent Tg 1 :5 i :5 k, soit u; E Ta;(C),u; f. 0 tel que 'Yi = expa;(u;) ap Soit K le corps engendre sur K 0 par tous les coefficients de n et toutes les coordonnees projectives des 'Yb · · · ,'"fk· Dans §3(4 f3I. Cr(1 :5 T :5 t), et on considere une {'7I = 1, '72• · · · , '7D} formee d'elements de }'ensemble a -t Notons c5 le degre de K 0 sur Q. Soit {(t. v} une base de K sur Q formee d'elements {r1i(;;1 j ::; c5 }.

I be a nonzero element assumption of (2) there is an unique monomial ~!.. which has L(~) in the monomials in f with nonzero coefficients. g where g = 1 + h with h = ~L(i)>o a(i. + io) g' = 1 + (-h) 1 + (-h) 2 + ... is L- finite and the inverse of g, so f has Remark 1. Under the assumptions a;> 0 (i = 1, ... ,m), KL Q(k[(x~>···•xm]]). ···,xm]]) is contained in the smal in which all exponents of each element are contained in some c respectively. (cf. ·!! 2. Let ~ = (6, ... ,= ("'t, ... , "'m) be respectively nand m-tup terminates respectively, and let A= (a;,i) be an (m,n) matrix wit ai,i· We use the following notation Then it is clear that ~~'A= (~A)~' for integer vector p = (p1, ...

Halbritter and M. Pohst, On the computation of the v functions of totally real cubic fields, J. Number Th. 36(199 7. H. Rademacher and E. l6,( Mathematical Association of America, D. C. ,1973). 8. T. Shintani, On a Kronecher limit formula for real quadratic Sci. Univ. Tokyo24(1977),167-199. 9. D. Zagier,Higher dimensional Dedekind sums, Math. An ,149-172. This page is intentionally left blank Takasbi Harase Department of Mathematics, Tokyo Institute of Technolog Oh-okayama, Meguro-ku, Tokyo 152, Japan ABSTRACT The definition of extended formal power series rings and generalizations o results are given.