Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity

Let $\nabla f$ be a gradient vector field of a weighted homogenous polynomial with isolated critical point at the origin. Let $(w_1,\dotsc,w_n)$ be the weights of $f$. In this paper, we prove that the Łojasiewicz Exponent $\theta$ of $f$ is precisely equal to ${\max_{0\leq i\leq n}}w_i-1$. This means that for some constant $c$, $\vert\nabla f(z)\vert\geq c\vert z\vert^\theta$ in a neighborhood of $0,$ which provides the optimal lower estimate of $\vert\nabla f(z)\vert$.