  The present paper is a continuation of the author's previous works, in which
necessary and sufficient local extrema at a stationary point of a polynomial or
a power series (and thus of an analytic function) are given. It is known that
for the case of one variable, the necessary and sufficient conditions of the
extremum are closing, i.e., they can be formulated as a single condition. The
next most complicated case is the case with two variables, which is the one
considered in this paper. In this case, many procedures, to which the
verification of necessary and sufficient conditions is reduced, are based on
the computation of real roots of a polynomial from one variable, as well as on
the solution of some other rather simple practically realizable problems. An
algorithm based on these procedures is described. Nevertheless, there are still
cases where this algorithm "doesn't work". For such cases we propose the method
of "substitution of polynomials with uncertain coefficients", using which, in
particular, we have described an algorithm that allows us to unambiguously
answer the question about the presence of a local minimum at a stationary point
for a polynomial that is the sum of two A-quasi-homogeneous forms, where A - is
a two-dimensional vector, whose components are natural numbers.
