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Sib. Zh. Vychisl. Mat., 2007, Volume 10, Number 2, Pages 195–208 (Mi sjvm77)  

This article is cited in 2 scientific papers (total in 2 papers)

On precise edges of polynomials

M. A. Novikov

Institute of System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences

Abstract: This paper discusses definitions of precise edges of polynomial functions at infinitely distant points $(x_0,y_0)$. It has been found that the limit equalities at these points are necessary conditions:
$$ \lim_{x\to x_0, y\to y_0}f'_x(x,y)=0,\quad \lim_{x\to x_0, y\to y_0}f'_y(x,y)=0,\quad \lim _{x\to x_0, y\to y_0}(xf'_x(x,y)+yf'_y(x,y))=0. $$
This allows one to obtain both finite and limit solutions of the system of necessary extremum conditions. The most typical properties of the polynomials, which have their precise edges, as well as the largest and the smallest values of polynomials at infinitely distant points have been revealed. An algorithm of finding the precise edges, which is based on constructing a parametric solution for a system of nonlinear equations, has been developed. The problems to be solved are reduced to some simpler, analysis by applying the aids of computer algebraaimed at determination of the largest and the smallest values of polynomials. The corresponding examples are given.

Key words: polynomial, form, infinitely distant point, local extremum, precise edge, the smallest value of polynomial, parametric solution of a system of algebraic equations.

Full text: PDF file (227 kB)
References: PDF file   HTML file
UDC: 517.988.38
Received: 12.05.2006
Revised: 30.05.2006

Citation: M. A. Novikov, “On precise edges of polynomials”, Sib. Zh. Vychisl. Mat., 10:2 (2007), 195–208

Citation in format AMSBIB
\Bibitem{Nov07}
\by M.~A.~Novikov
\paper On precise edges of polynomials
\jour Sib. Zh. Vychisl. Mat.
\yr 2007
\vol 10
\issue 2
\pages 195--208
\mathnet{http://mi.mathnet.ru/sjvm77}
\elib{https://elibrary.ru/item.asp?id=9489353}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Novikov M.A., “Opredeliteli v vychisleniyakh tochnykh granei polinomov”, Sovremennye tekhnologii. Sistemnyi analiz. Modelirovanie, 2009, no. 1, 135–140  elib
    2. Novikov M.A., “O svyazi predelov ekstremumov s tochnymi granyami polinomov”, Sovremennye tekhnologii. sistemnyi analiz. modelirovanie, 2013, no. 1, 33–40  elib
  • Sibirskii Zhurnal Vychislitel'noi Matematiki
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