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The linearizability (or isochronicity) problem is one of the open problems for polynomial differential systems which is far to be solved in general. A progressive way
to find necessary conditions for linearizability is to compute period constants. In this
paper, we are interested in the linearizability problem of

In the qualitative theory of planar polynomial differential equations, the problem of characterizing isochronous centers has stimulated a great deal of effort but which is also remarkably intractable. The authors of [

The problem of determining isochronous centers for polynomial vector fields with linear part of center type

By means of complex transformation

At times, the problem is restricted to the following polynomial differential system with linear part of

Wang and Liu [

Motivated by the above facts, in this paper, we concentrate on the linearizability problem for the polynomial differential system with a resonant degenerate singular point

(i) When

(ii) When

(iii) When

Consequently, the proposed results are new and extend the existing ones with respect to systems (

The format of this paper is organized as follows. In Section

Most of the calculations in this paper have been done with the computer algebraic system-Mathematica.

System (

We write

For any positive integer

For any positive integer

Then from Lemma

System (

System (

System (

Obviously, system (

Making the transformation

For system (

Denote system (

For

The relations between

Let

Let

where

From expression (

Theorems

We cannot use Theorems

In this section, we investigate the linearizability problem at degenerate singular point for the following septic system:

Factually, the pseudo-isochronous center problem at the degenerate singular point of system (

Firstly, let us see the computational method of singular point quantities at 1 : −1 resonant degenerate singular point.

When

(i) For any positive integer

(ii) If

(iii) If for all

Degenerate singular point of system (

For system (

For all pairs

For any positive integer

Performing the transformation (_{(n = 1)} and renaming

According to Definition

The first 9 singular point quantities at degenerate singular point of system (

From Theorem

For system (

In order to obtain the center conditions of degenerate singular point, we have to find out all the elementary Lie invariants of system (

All the elementary Lie invariants of system (

For system (

If condition (I) is satisfied, system (

We next discuss the linearizability problem of the origin of system (_{(n = 1)} is a homeomorphism, the center conditions of the origin of system (

Substituting center condition (I) into the recursive formulae in Theorem

In the above expression of

From

When center condition (II) holds, the right hand of system (

Putting expression (

The first 6 period constants of the origin of system (

The first 6 period constants at the origin of system (

Being

When

When

Under center condition (II), the origin of system (

When condition (

When condition (

We have from Definition

Degenerate singular point of system (

In the qualitative theory of ODE, the linearizability (or isochronicity) problem is one of the open problems for polynomial differential systems which is far to be solved in general. Up to now, most of the existing literatures are concerned with elementary singular point. A progressive way to find necessary conditions for linearizability is to compute period constants.

The linearizability problem of degenerate singular point is much more difficult. As far as the degenerate case is concerned, the classical methods are invalid. As a result, the literatures about linearizability problem of degenerate singular point are limited. One reason is that the problem itself is very complicated; another is that the problem is subject to limited methods. Therefore, it is completely natural to develop a "new" theory to analyze this dynamical behavior for the planar dynamical systems with degenerate singular points.

In our paper, firstly, the degenerate singular point is taken to the elementary origin by a blow-up; besides, we establish a new recursive algorithm to compute the so-called

The authors are grateful to the anonymous referees for their careful comments and valuable suggestions. Especially, the authors greatly appreciated the help of the referees in posing important questions, which are sure to enhance the readability and interest of the paper and suggest some neat ideas for future studies. This work is supported in part by the National Nature Science Foundation of China (NSFC 11101126 and 11101127) and Scientific Research Foundation for Doctoral Scholars of HAUST (09001524).