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By Gravesa W. G., Peckhamb B., Pastorc J.

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Since R N (17) (6), (3), (pOQ0) = R, u p o n J. = 1 (13) and (18) By ; (5) w e get therefore by (PIQI))/(R Q (PQ) = M(R) (15) By (pi+iQi+l) (142 ) we get [ (R n Now R N setting ~(R) , (14) w e get [R/J i : R~ = i (6) and + xiR (17) we h a v e 45 for 0 < i s a . 46 (19) X ( R , J i) = We claim given 2i for 0 ~ i ~ a. that: any ideal J in R with ~(R) c J, upon letting i = ordvJ, (20) we have Namely, 0 < i < a, that i in v i e w is a n (4) and J c R n i = a that then since ~(R) J = J a ; so h e n c e f o r t h ordv~ = i and then = (203 ) c J) (2) and in v i e w 6x I + ~ y (6) w e h a v e y (204 ) e xi c since (R) c J, b y that is an J = Ji" integer with ; by (3) , (6), (17) that (6), ~(R) (13), hence of i < a; w e (2), with ~(R), and can (201 ) w e see take (3) 8 e R\M(R) and hence + ~R (17), and by (15) w e and (pOQ0) = R, b y by the definition of (6), (203 ) w e (201), (13) and we have a = d.

Nonzero is finite, S/M(S) is in- special be a coefficient in i n d e t e r m i n a t e s S - E = S/(y-ax)S Now e homogeneous X,Y, with : R] = and the for a n y poly- canonical local a ¢ k\k I , epimorphism, domain and : E] = e , 31 is a kI = regular [S/(~Q,y-ax)S for coefficients clearly [R/f(y-ax)R case. S/M(S) (x,y) positive and = ords~- Case when in such and th e i n t e q r a l finite P e 90(A) such and X' ([A,~],P) coincide. (A/~)-module. morphism, PROOF. S, domain S = Ap LEMMA. 6) w e a l s o h a v e k(R,f(y-ax)) = [R/f(y-ax)R B y the a b o v e t h r e e d i s p l a y e d : R].

6) we get [R/I : R~ = e + [R/~(R) : R~. Let (8) P0 = R Q M(V). 20 [R/I Since V : k] = e + is r e s i d u a l l y rational hk ( R , I ) §6. Length Let A be Let be of a subring (*) for (Note f: A ~ A every [A/P such (*) is a s u b f i e l d k, is of N P] and let A/C be we obviously have, noetherian C c D 1 (A) the be canonical homomorphic such image. A/C that epimorphism. is Let k that we have k 0 p ~ ~0 and < ~. satisfied A : k] = e + Ik ( R ) . P e ~0([A,C]) : k/(k that over in a o n e - d i m e n s i o n a l a domain noetherian.

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A Bifurcation Analysis of a Differential Equations Model for Mutualism by Gravesa W. G., Peckhamb B., Pastorc J.

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