## degree of polynomial functions that satisfy a set of inequalities

Hi,

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.