0 A note on holomorphic extensions

We give a criterium of holomorphy for some type formal power series. This gives a stronger form of a Rothstein's type extension theorem for a particular ring of holomorphic functions. We consider the set R ⊂ C[[z 1 , z 2 ]], z 1 , z 2 ∈ C k × C m of formal power series of the form f (z) = f (z 1 , z 2) = n P n (z 2)z n 1 where P n is a polynomial in m variables of total degree d 0 P n ≤ C 0 + C 1 ||n||, for some constants C 0 , C 1 > 0. One easily checks that R is a local sub-ring of C[[z 1 , z 2 ]]. For the notion of Γ-capacity, that generalizes the notion of capacity in one complex variable, we refer to [Ro]. Theorem. Let f (z 1 , z 2) = n P n (z 2)z n 1 be a formal power series of the two complex variables (z 1 , z 2) ∈ C k × C m. We assume that (P n) is a sequence of polynomials in m variables of total degree deg P n ≤ C 0 + C 1 ||n||. We assume that for a set K ∈ C m of positive Γ-capacity, z 2 ∈ K being fixed, the formal power series f (z 1 , z 2) converges. Then for some C 2 > 0, the formal power series f defines a holomorphic function in a neighborhood of the axes {z 1 = 0} of the form, U = {(z 1 , z 2) ∈ C k × C m ; ||z 1 || ≤ C 2 1 + ||z 2 || }. Compare with Rothstein's theorem (see [Siu] p.25). Our theorem is motivated and has applications in problems of holomorphic dynamics and small divisors ([PM]) where power series in the ring A appear naturally.