By J. Van Lint, G. Van Der Geer, Jacobus Hendricus Van Lint, Jacobus H. Van Lint, Gerard Van Der Geer

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As usual it is convenient to collect the information in a generating function. We define the colength series of R to be CLR(Z) = ^%L0ft(h)Zh. For an analytically irreducible ring R we get CLR(Z) = p(Z)/(l - Z). Then p(Z) 6 Z[Z], and p ( l ) = Q(/i), for h > 1R(V/C). The constant fi(/i),/i » 0, of course depends on q — \R/m\. We will determine this dependence in an example. EXAMPLE 2. Let R = k [ [ t 3 , t 4 , t 5 ] } . There are the following shapes of ideals: Ii = shape(-R), Z2 = shape((£ 3 ,£ 5 )), I3 = shape((* 3 ,i 4 )), and finally I4 = sh&pe((t3,t4,t5)).

University Lecture Series, No. : American Mathematical Society, 1995. Moro 2, 00185 Roma, Italy, email: barucci@mat. ) = h, we say that / has colength h. Maximal ideals have colength 1, and there may be many other ideals of finite colength even in non-Noetherian rings. If R is a one-dimensional Noetherian domain, every non-zero ideal has finite colength. We are interested in the class of rings in which there is a finite number of ideals for each finite colength, and how this number grows as a function of h.

G > vanish belongs to V, where K is the algebraic closure of K. Let P(XI, ... ,xn,yi,... , 2/ r ) be such a point. Let Uj be a dth root of Xj for all j = 1,... ,n. Then from the binomial Gi we have j/i = uiua1i'1 •••<'•" (1) for alii = 1 , . . , r, where uji is a d/7;th root of 1. Let r] be a primitive dth root of 1. Then, for alH = 1,... ,r, one has that w, = 77°' for some multiple c; of 7^. , n, a dth root ffj of 1 such that yi = (Oiu1)a^---(dnunT"" for all i = 1,... ,r: then Xj — (9jUj}d 6j € 2Z be such that (2) for j = 1,...