Necessary and sufficient conditions of extremum for polynomials and power series in the case of two variables

Abstract

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.