The Lie derivative of a differentiable function $g: \mathbb{R}^n \rightarrow \mathbb{R}$ along a vector field $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is
$\mathcal{L}_f(g)(x) = \left<\left(\begin{matrix}f_1(x) \\ \ldots \\ f_n(x)\end{matrix}\right),\left(\begin{matrix}\frac{\partial g}{\partial x_1} \\ \ldots \\ \frac{\partial g}{\partial x_n}\end{matrix}\right)\right>$
where $\left<\cdot,\cdot\right>$ is the inner product on $\mathbb{R}^n$.
Suppose we have a polynomial $p$ such that $\mathcal{L}_f(p) = \alpha \mathcal{L}_f(p)$ with $\alpha \in \mathbb{R}$ and $\alpha < 0$. Does this imply that $|p(x_0)| \geq |p(x)|$ and that $p(x)$ converges to $0$ as $x$ changes according to the vector field $f$ with $x_0$ being the initial value for $x$?
Bit late, but assuming you meant to write $\mathcal{L}_f(p)=\alpha p$ (I.e., the definition of a Darboux polynomial), then yes that inequality is correct. In the constant cofactor case, solving $\dot{p}=\alpha p$ for $p$ gives $p=C_0e^{\alpha t}$, which strictly monotonically converges to $0$ as $t\rightarrow\infty$ for $\alpha<0$ as your inequality implies.