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

V. N. Nefedov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 28 (2024) no. 4, pp. 615-650

Résumé

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.

Détail

Keywords: polynomials, power series, necessary conditions for an extremum, sufficient conditions for an extremum, quasi-homogeneous forms