On Convergence Sets of Formal Power Series

Abstract

The (projective) convergence set of a divergent formal power series $f(x_1,...,x_n)$ is defined to be the image in $\PP^{n-1}$ of the set of all $x\in {\mathbb{C}}^n$ such that $f(x_{1}t,...,x_{n}t)$, as a series in $t$, converges absolutely near $t=0$. We prove that every countable union of closed complete pluripolar sets in $\PP^{n-1}$ is the convergence set of some divergent series $f$. The (affine) convergence sets of formal power series with polynomial coefficients are also studied. The higher-dimensional results of A. Sathaye, P. Lelong, N. Levenberg and R.E. Molzon, and of J. Ribón are thus generalized.

17 pages

References in corpus (1)

• A note on holomorphic extensions