Some necessary and some sufficient conditions for local extrema of polynomials and power series in two variables

This study extends the author's previous works establishing necessary and sufficient conditions for a local extremum at a stationary point of a polynomial or an absolutely convergent power series in its neighborhood. It is known that in the one-dimensional case, the necessary and sufficient conditions for an extremum coincide, forming a single criterion.
The next stage of analysis focuses on the two-dimensional case, which constitutes the subject of the present research. Verification of extremum conditions in this case reduces to algorithmically feasible procedures: computing real roots of univariate polynomials and solving a series of practically implementable auxiliary problems.
An algorithm based on these procedures is proposed. For situations where its applicability is limited, a method of substituting polynomials with undetermined coefficients is developed. Building on this method, an algorithm is constructed to unambiguously verify the presence of a local minimum at a stationary point for polynomials representable as a sum of two $A$-quasihomogeneous forms, where $A$ is a two-dimensional vector with natural components.