I have an n-dimensional state vector  x(t) = [x_1(t), x_2(t),... ,x_n(t)] and a polynomial function V(x(t)) that maps x(t) to a nonnegative scalar number, i.e.,  V: \Re^n \rightarrow [0, \infty).

Furthermore, I have that V(x(t)) satisfies the following inequalities:
 c_1 |x(t)|^2 \leq V(x(t)) \leq c_2 |x(t)|^2, where 0 < c_1 \leq c_2 and  | \frac{\partial V}{\partial x}| \leq c_3 |x(t)|, where c_3 > 0.

I would like to know if the degree of my polynomial has to be 2. That is, V(x(t))=x(t)^TPx(t), where P is a positive definite matrix.

Thanks in advance.