Abstract. In this paper a new (symmetric) product for matrices, entries of which are located by pair of multi-indices, and norms of such matrices are introduced. They are used to represent multivariate power series as series in one (vector) variable and to introduce a radius of absolute convergence. For the radius of absolute convergence a formula, similar to ordinary Cauchy-Hadamard formula in one variable case, is given. Some open problems related to the radius of absolute convergence are set.

Received: September 5, 2011

AMS Subject Classification: 32A05, 15A60

Key Words and Phrases: multi-index, multivariate power series

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