  Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative
algebra over the field of complex numbers with $m$ idempotents. Let
$e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$
which are linearly independent over the field of real numbers. We consider
monogenic (i.~e. continuous and differentiable in the sense of Gateaux)
functions of the variable $\sum_{j=1}^k x_j\,e_j$, where $x_1,x_2,\ldots,x_k$
are real, and obtain a constructive description of all mentioned functions by
means of holomorphic functions of complex variables. It follows from this
description that monogenic functions have Gateaux derivatives of all orders.
The present article is generalized of the author's paper [1], where mentioned
results are obtained for $k=3$.
