Geometric Study of Complex Analysis

Head investigator Fujimoto studied value-distribution-theoretic properties of meromorphic maps ofC^n n (C) and gave some new results as their applications.He showed that，for 3n +1 hyperplanes h_j 's in P^N(C) located in general position，there exists at most two meromorphic maps f of C^n into P^ n (C) such that the inverse imagesf <-1>(h_j)'s, which are counted with multiplicities truncated by two，coincide with given divisors d_j 's. He also proved that, for 2n +2 hyperplanes h_j，in P^N(C) located in general position, there is some positive integer l_0 such that，if two meromorphic maps f and g of C^n into p_n (C) have the same inverse images counted withmultiplicities truncated by l_0 for each h_j，then f and g are algebraically degenerate.He also studied uniqueness range sets for meromorphicfunctions on C, namely, sets S with the property that the condition f^<-1>(S) = g^<-1>(S)，counted f, g on C.For a finite set S = {a_1, a_2，…，a_q}, he considers the polynomial P(w)：= (w - a_1)（w - a_2）…（w - a_q） and assumes that the derivative P(w) has k distinct zeros d_1，d_h. He showed that, if k <greater than or equal> 4, q > 2k + 6，P(d_l) * P(d_m) for l * m and sigma_l P(d_l) * 0，那是一个独特的范围集，他也给了一些其他的sufficient conditions for a finite set tobe a uniqueness range set。

Head investigator Fujimoto studied value-distribution-theoretic properties of meromorphic maps of C^n into P^N(C) and gave some new results as their applications.He showed that, for 3N +1 hyperplanes H_j's in P^N(C) located in general position, there exists at most two meromorphic maps f of C^n into P^N(C) such that the inverse images f^<-1>(H_j)'s, which are counted with multiplicities truncated by two, coincide with given divisors D_j's. He also proved that, for 2N +2 hyperplanes H_j, in P^N(C) located in general position, there is some positive integer l_0 such that, if two meromorphic maps f and g of C^n into P_N(C) have the same inverse images counted with multiplicities truncated by l_0 for each H_j, then f and g are algebraically degenerate.He also studied uniqueness range sets for meromorphic functions on C, namely, sets S with the property that the condition f^<-1>(S) = g^<-1>(S), counted f, g on C.For a finite set S = {a_1, a_2, ・・・, a_q}, he considers the polynomial P(w) : = (w - a_1)(w - a_2)・・・(w - a_q) and assumes that the derivative P(w) has k distinct zeros d_1, d_2, ・・・, d_h. He showed that, if k <greater than or equal> 4, q > 2k + 6, P(d_l) * P(d_m) for l * m and SIGMA_l P(d_l) * 0, then S is a uniqueness range set. He also gives some other sufficient conditions for a finite set to be a uniqueness range set.