On precise edges of polynomials

M. A. Novikov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 10 (2007) no. 2, pp. 195-208

Résumé

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.