Orbits and tsectors in irregular exceptional directions of full-null degenerate singular point

TL;DR

This paper investigates the complex behavior of orbits near full-null degenerate singular points in analytic vector fields, providing new methods to determine orbit counts and sector types using Newton polygons.

Contribution

It introduces a novel approach employing Newton polygons to analyze nonlinearities and determine orbit and sector counts in high degeneracy cases where traditional blow-up methods fail.

Findings

Developed criteria for orbit and sector counts using Newton polygons.

Derived Newton polygons for multiplication and differentiation of analytic functions.

Provided new methods to analyze irregular exceptional directions in degenerate singular points.

Abstract

Near full-null degenerate singular points of analytic vector fields, asymptotic behaviors of orbits are not given by eigenvectors but totally decided by nonlinearities. Especially, in the case of high full-null degeneracy, i.e., the lowest degree of nonlinearities is high, such a singular point may have irregular exceptional directions and the blow-up technique can be hardly applied, which leaves a problem how to determine numbers of orbits and (elliptic, hyperbolic and parabolic) tangential sectors in this case. In this paper we work on this problem. Using Newton polygons to decompose nonlinearities into principal parts and remainder parts, we convert the problem to the numbers of nonzero real roots of edge-polynomials of principal parts. Computing Newton polygons for multiplication and differentiation of analytic functions and giving Newton polygons for addition, which was not found in literatures, we determine semi-definiteness of the Lie-bracket of principal parts and therefore obtain criteria for those numbers.