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

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

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

Thanks in advance.