Characterization of the collapsing meromorphic products

Publicacions Matem`atiques, Vol 40 (1996), 331–336. CHARACTERIZATION OF THE COLLAPSING MEROMORPHIC PRODUCTS Alain Escassut and Marie-Claude Sarmant Abstract Let Kbe an algebraically closed complete ultrametric field. Let a∈K,r>0. We consider a meromorphic product F(x)=  n∈N x−an x−bn , where (an)n∈N,(bn)n∈Nare sequences satisfying |bn−a|<rwhenever n∈N, lim n→+∞|bn−a|=r, lim n→∞ an−bn=0 and min m=n |bm−bn|>0. We prove that if Khas characteristic zero, then Fis collapsing if and only if ∞  n=0 (an)j−(bn)j= 0 for every j∈N. Moreover, if Khas characteristic = 0, then there exists a meromorphic product fof the form  n∈N x−cn x−en such that F(x)=(f(x))pwhenever x∈{x∈K||x−a|≥r}if and only if ∞  n=0 (an)j−(bn)j= 0 for every j∈N. Notations and definitions Let Kbe an algebraically closed field, complete with respect to an ultrametric absolute value. Given a set Din K,H(D) denotes the set of the analytic elements in D, i.e., the completion of the algebra R(D) of rational functions with no pole in D, with respect to the topology of uniform convergence. Given a∈Kand r>0, d(a, r) (resp. d(a, r−)) denotes the disk {x∈K||x−a|≤r}(resp. {x∈K||x−a|<r}). We put V=d(a, r−) and E=K\V. A sequence (en)n∈Nin V satisfying lim n→∞ |en−a|=rand min m=n|em−en|>0 will be called a polar sequence associated to V. 332 A. Escassut, M.-C. Sarmant Henceforth, (bn)n∈Nwill denote a polar sequence associated to Vand (an)n∈Nwill denote a sequence in Ksuch that lim n→∞ an−bn=0. For every x∈K\{b0,... ,b n,...}the product Fm= m  n=0 x−an x−bn converges to a limit F(x)= n∈N x−an x−bn . Such a function F(x) defined in K\{b1,... ,b n,...}is called a meromorphic product associated to the sequence (bn)n∈N. The meromorphic product  n∈N x−an x−bn associated to the sequence (bn)n∈Nwill be said to be collapsing if there exists ∈Ksuch that Fsatisfies lim |x−a|→r F(x)=. By [5], [7] it is well known that a meromorphic product fis collapsing if and only if f−1 is vanishing along the increasing filter Fof center 0 and diameter 1, and in particular this requires FtobeaT-filter [4]. Now, the question whether a meromorphic product is collapsing, in connection with the sequences (an)n∈N,(bn)n∈N, is a quite hard question. Here we will give an answer. In particular, this will be used in the study of the homomorphisms from the group of meromorphic products into the circle C(0,1). By [5], [7] we have Lemma a. Lemma a. The following are equivalent. (1) Fis collapsing, (2) lim |x−a|→r F(x)=1, (3) F(x)=1whenever x∈E. Next result is taken from [5]. Theorem 0. Let f∈H(E)satisfy lim |x|→∞ f(x)=1and f−1E<1. Let ∈]0,f−1E[. There exist a polar sequence (en)n∈Nassociated to V, together with a meromorphic product ∞  n=0 x−cn x−en associated to the sequence (en)n∈N, satisfying further |cn−en|<r(f−1E+), and ∞  n=0 x−cn x−en =f(x)whenever x∈E. Collapsing meromorphic produts 333 We notice that for every j∈N∗the series ∞  n=0 aj n−bj nis convergent. Lemma b below is easy and will be used in proving Lemma c. Lemma b. Let λ∈K. The following are equivalent. i) ∞  n=0 aj n−bj n=0for every j∈N∗ ii) ∞  n=0 (an+λ)j−(bn+λ)j=0for every j∈N∗. Lemma c. Fsatisfies F(x)=0for all x∈Eif and only if for every j∈N∗the sequences (an)n∈Nand (bn)n∈Nsatisfy ∞  n=0 (an)j−(bn)j=0. Proof: By Lemma b we may clearly assume a= 0 without loss of generality. Let r∈[r, +∞[ be such that |an|<r for every n∈N, and let E=K\d(0,r −). We can see that F−1 E≤sup n∈N |bn−an| r<1. Hence F Fobviously belongs to H(E). Let g=F F. It is seen that g(x)= ∞  n=0 1 x−an −1 x−bn . For each α,β∈V, and for every x∈E we have 1 x−α−1 x−β= ∞  j=0 αj−βj xj+1 = ∞  j=1 αj−βj xj+1 . Applying this to each term 1 x−an−1 x−bn, we obtain g(x)= ∞  n=0   ∞  j=1 (an)j−(bn)j xj+1   for all x∈E. Now, let us fix x∈E. We see that when jtends to +∞, the convergence of (an)j−(bn)j xj+1 to 0 is uniform with respect to n. Hence we have g(x)= ∞  j=1 ∞  n=0 (an)j−(bn)j xj+1 . 334 A. Escassut, M.-C. Sarmant But now, this holds for any x∈E. Besides, as Fbelongs to H(E), we know that its Mittag-Leffler series [3], [4] is the same in H(E) and in H(E), hence this is the Mittag-Leffler series of Fin H(E). Hence we see that F(x) = 0 if and only if the Mittag-Leffler series of gis identically equal to 0, i.e.: ∞  n=0 (an)j−(bn)j= 0 for every j∈N∗. This ends the proof. Now, we can conclude Theorem 1. Kis supposed to have characteristic zero. Then Fis collapsing if and only if for every j∈N∗the sequences (an)n∈Nand (bn)n∈Nsatisfy (Ej) ∞  n=0 (an)j−(bn)j=0. Proof: Indeed, since Khas characteristic zero, by [1] we know that F(x) is identically zero in Eif and only if F(x) is a constant in E, i.e., Fis collapsing. Theorem 2. Assume Kto be of characteristic p=0. There exists a polar sequence (en)n∈Nassociated to V, and a meromorphic product f(x)= n∈N x−cn x−en , associated to the sequence (en)n∈N, satisfying F(x)= (f(x))pwhenever x∈Eif and only if for every j∈N∗the sequences (an)n∈Nand (bn)n∈Nsatisfy (Ej) ∞  n=0 (an)j−(bn)j=0. Proof: If there exists a meromorphic product fassociated to the sequence (bn)n∈Nsuch that (f(x))p=F(x) for all x∈E, then obviously we have F(x) = 0 for all x∈E, and therefore, by Lemma c, we have (Ej) ∞  n=0 (an)j−(bn)j= 0 for every j∈N∗. Reciprocally, we suppose Relations (Ej) satisfied. By Lemma b we have F(x) = 0 for all x∈E. Hence, there exists g∈H(E) such that (g(x))p=F(x) for all x∈E. Besides, since Fis a meromorphic product associated to the sequence (bn)n∈N, we notice that lim |x|→+∞F(x) = 1. As a consequence, we can choose gsuch that lim |x|→+∞g(x) = 1. Further, it is seen that gp=(g−1) + 1p=(g−1)p+ 1, and therefore we have Collapsing meromorphic produts 335 (1) F−1E=g−1Ep, hence g−1E<1. Let ∈]0,1[. Then by (1) and by Theorem 0 there does exists a polar sequence (en)n∈Nassociated to V, and a meromorphic product fof the form  n∈N x−cn x−en such that f(x)=g(x) whenever x∈E, and such that |en−cn|≤ p F−1E+. This ends the proof. Remark. In [5], and [6] it was shown how one can construct a collapsing meromorphic product, with the help of certain unbounded functions analytic in the disk d(0,r −). Acknowledgement. We are very grateful to Labib Haddad for his contribution to this article. References 1. A. Escassut, Derivative of analytic elements on infraconnected clopen sets, Indag. Math. 51 (1989), 63–70. 2. A. Escassut,T-filtres, ensembles analytiques et transformation de Fourier p-adique, Ann. Inst. Fourier 25(2) (1975), 45–80. 3. A. Escassut and M.-C. Sarmant, Mittag-Leffler series and Motzkin products for invertible analytic elements, Rivista di Matematica Pura ed Applicata 12 (1992), 61–73. 4. M. Krasner, Prolongement analytique uniforme et multiforme dans les corps valu´es complets. Les tendances g´eom´etriques en alg`ebre et th´eorie des nombres, Clermont-Ferrand (1964), 94–141; Centre National de la Recherche Scientifique (Colloques internationaux de C.N.R.S. Paris, 143) (1966). 5. M.-C. Sarmant, Produits M´eromorphes, Bull. Sci. Math. 109 (1985), 155–178. 6. M.-C. Sarmant, Fonctions analytiques et produits croulants, Collectanea Mathematica XXXVI(2) (1985), 199–218. 7. M.-C. Sarmant, Produits meromorphes et prolongement analytique, Th`ese de Doctorat d’Etat, Universit´e Pierre et Marie Curie (1987). 336 A. Escassut, M.-C. Sarmant 8. M.-C. Sarmant, Factorisation en produit m´eromorphe d’un el´em´ent semi-inversible, Bull. Sci. Math. 115 (1991), 379–394. Universit´e Blaise Pascal (Clermont-Ferrand) Laboratoire de Math´ematiques Pures Les C´ezeaux 63177 Aubi`ere Cedex FRANCE Primera versi´o rebuda el 13 de Setembre de 1995, darrera versi´o rebuda el 12 de Gener de 1996