Methods and algorithms for determining the main quasi-homogeneous forms of polynomials and power series

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, 125993, Russia

^* e-mail: nefedovvn54@yandex.ru

Abstract

Methods are proposed that allow one to determine the special forms of polynomials and power series used in solving a number of practical problems. The most important of them are the construction of necessary and sufficient conditions for an extremum for polynomials and power series, as well as checking matrices for D-stability arising in the study of ecosystem stability. This special forms (the so-called main quasi-homogeneous polynomial forms) are generalizations of the concept of a homogeneous polynomial form. They correspond to the sum of the terms of the polynomial belonging to some face of the Newton polytope of this polynomial. In some cases, the main quasi-homogeneous polynomial forms necessary for research can also be determined for power series (in particular, when constructing necessary and sufficient conditions for an extremum). In the case of polynomials, two cases are investigated separately: the selection of all the main forms of the polynomial and the selection of the main forms corresponding to the faces of the Newton polytope in its “southwestern” part (such forms also can be distinguished for an arbitrary power series), since both cases have their practical applications. Practically applicable methods are described for each of these cases. Several methods are considered sequentially (starting with a simple enumeration and ending with a method with a significant reduction in the number of options in the enumeration). The last (most economical) method is described as a practically realizable algorithm. A practically realizable rather economical algorithm for solving an auxiliary problem is described—finding the set of corner points of the Newton polytope.