  We provide a definition of Multidimensional Chebyshev Systems of order N
which is satisfied by the solutions of a wide class of elliptic equations of
order 2N. This definition generalizes a very large class of Extended Complete
Chebyshev systems in the one-dimensional case. This is the first of a series of
papers in this area, which solves the longstanding problem of finding a
satisfactory multidimensional generalization of the classical Chebyshev systems
introduced already by A. Markov more than hundered years ago, and studied later
by S. Bernstein and M. Krein.
