Initial form of a polynomial I am reading some tropical geometry and came up with the concept of the initial form of a polynomial. The definition says that the initial form of f with respecto to a weight vector $w \in \mathbb{R}^{n+1}$ is \begin{equation} 
in_w(f) = \sum_{\substack{u\in \mathbb{N}^{n+1} \\  
val(c_u) + w\cdot u = W}} \overline{c_ut^{-val(c_u)}}x^u \end{equation} 
However, I don't really see the intuition behind this initial form, could anyone explain this a bit further?

Have a look at Remark 5.7 of Gublers "Guide to tropicalization", I find his approach easier to understand than the definition you seem to have copied from Sturmfels' book.
Initial forms can be used to define the tropicalization of a variety, and are especially important if the field K ist trivially valued.
The set of all weight vectors $w$, such that $in_w(f)$ is not a monomial for $f$ in the Ideal defining a variety $V$ coincides with the set of valuation poins of the K-points in the variety for a non-trivial valuation. The closure of these sets is the tropicalization.
