Email this article Some necessary and some sufficient conditions for local extrema of polynomials and power series in two variables
- Authors: Nefedov V.N1
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Affiliations:
- Moscow Aviation Institute (National Research University)
- Issue: Vol 28, No 4 (2024)
- Pages: 615-650
- Section: Differential Equations and Mathematical Physics
- URL: https://journals.rcsi.science/1991-8615/article/view/311015
- DOI: https://doi.org/10.14498/vsgtu2103
- EDN: https://elibrary.ru/KECQQD
- ID: 311015
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Abstract
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.
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##article.viewOnOriginalSite##About the authors
Viktor N Nefedov
Moscow Aviation Institute (National Research University)
Author for correspondence.
Email: nefedovvn54@yandex.ru
ORCID iD: 0000-0001-6053-2066
https://www.mathnet.ru/person63464
Cand. Phys. & Math. Sci., Associate Professor; Associate Professor; Dept. of Mathematical Cybernetics
Russian Federation, 125993, Moscow, Volokolamskoe shosse, 4References
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