Cauchy–Hadamard theorem for exponential series

In this paper we study the connection between the growth of coefficients of an exponentials series with its convergence domain in finite-dimensional real and complex spaces. Among the first results of the subject is the well-known Cauchy–Hadamard formula.
We obtain exact conditions on the exponentials and a convex region in which there is a generalization of the Cauchy–Hadamard theorem.
To the sequence of coefficients of exponential series we associate a space of sequences forming a commutative ring with unit. The study of the properties of this ring allows us to obtain the results on solvability of non-homogeneous systems of convolution equations.