The story of this study began when doing a prepa's homework on the Newton-Raphson's method, which consists in finding zero of a real function. This homework interested me a lot, but I didn't have time to continue in this way...

Two years later, when choosing my TIPE, I remenbered about Newton's method, and I could continue my study. This study was more precise than the first, and I discovered that it was a general method to solve systems of non linear equations [1]. A part of my TIPE was to show how difficult it is to initialize the method, and the example was Newton's Method apply to the real polynom x -> x^3-x, which present an interest on a particular segment of initialisation.

Today, three years later, this project of Hydrodynamics Instabilities allows me to extend the study to the associate complex polynom z->z^3-z.