A set of necessary and sufficient stability conditions for low order two-dimensional polynomials

In this note a set of necessary and sufficient conditions for two-dimensional polynomials which are quadratic in both variables or quartic in one and linear in the other are given. In both cases the important stability condition reduces to checking the nonexistence of positive real roots of a quartic equation. Conditions for the latter have been recently presented by the authors [1]. Furthermore, by appealing to coordinate transformation, the explicit stability conditions are extended to the largest possible class of two-dimensional polynomials. Finally, a sufficient condition for stability is given for any two-dimensional polynomial.