Exceptional values of entire functions of finite order in one of the variables

Let F(z, w) be a holomorphic function in C-n x C of finite order in w with n >= 2. Let S2 be the set of points z E C-n where F(z, w) is a non-constant function omitting a value pi(z). Near a finite accumulation point z(0) of Omega, we prove in the main result (Theorem 1) that Omega is a local analytic set and pi(z) is holomorphic, and show the existence of a proper globally analytic set Delta of C(n )such that either Omega subset of Delta or Omega = C-n \ Delta, being possible in the last case to also determine F(z, w) in terms of pi(z). We apply this result to several problems. First, we extend a Theorem due to Nishino about exceptional values when near z(0) dimension of Omega is n and assure the existence of a meromorphic function alpha(z) in Cn such that pi(z) = alpha(z) except at points where alpha(z) has poles or F(z, w) is constant (also being F(z, w) a polynomial in w if alpha(z) is infinity). After, we prove that Omega is a local analytic set in C-n and the existence of a proper analytic subset E of C-n such that Omega subset of E or Omega = C-n\E. Finally, we generalize a Lelong-Gruman Theorem about the set of points z where pi(z) = 0.(c) 2023 Elsevier Masson SAS. All rights reserved.