  In 1977, Yu. V. Matiyasevich proposed a formula expressing the chromatic
polynomial of an arbitrary graph as a linear combination of flow polynomials of
subgraphs of the original graph. In this paper, we prove that this
representation is a particular case of one (easily verifiable) formula, namely,
the representation of the characteristic polynomial of an arbitrary matroid as
a linear combination of characteristic polynomials of dual matroids. As an
application, we represent the flow polynomial of a complete graph with $n$
vertices as the sum of elementary terms with respect to all partitions of
positive integer $n$. Since the growth rate of the number of all partitions is
less than exponential, this technique allows us to evaluate the flow polynomial
for values of $n\approx 50$. We also get an explicit expression for the
characteristic polynomial of the matroid dual to the matroid of the projective
geometry over a finite field. We prove, in particular, that major coefficients
of all these polynomials coincide with the beginning of the row in the Pascal
triangle, whose number equals the quantity of elements in the corresponding
matroid. At the end part of the paper, we consider one more approach, which
allows us to obtain the same results of application of our main theoren by
using properties of the Tutte polynomial and the classical Rota formula for
coefficients of the characteristic polynomial of a matroid. In addition, we
describe the connection between the matroid variant of the Matiyasevich formula
and convolution formulas for Tutte polynomials.
