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

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