Lyapunov function and for polynomial dynamical system

Question

For a nonlinear system, if the orgin is asymptotically stable, then there exist a suitable Lyapunov function, by the famous inverse result of Lyapunov stability. For the case of a polynomial system, if the origin is asymptotically stable, can we assume the form of the Lyapunov function $V(x)$ to have its derivative $\dot{V}(x)$ to be less than or equal to negative sums of squared polynomials?

Answer 1

If the vector field is finite polynomial with only odd exponents and non-negative coefficients, it is not hard to find a suitable Lyapunov map. Otherwise, the conditions for Lyapunov maps are researched-to-be.