Real normal form of a binary polynomial at a second-order critical point

A real polynomial in two variables is considered. Its expansion near the zero critical point begins with a third-degree form. The simplest forms to which this polynomial is reduced with the help of invertible real local analytic changes of coordinates are found. First, for the cubic form, normal forms are obtained using linear changes of coordinates. Altogether, there are three of them. Then three nonlinear normal forms are obtained for the complete polynomial. Simplification of the calculation of a normal form is proposed. A meaningful example is given.