Valuations of coefficients of minimal polynomials for tuples

Question

Suppose you are given two valued fields $(K,v) \subseteq (L,w)$ and a tuple $a \in L^n$. What kind of restrictions do we have on the valuation of the coefficients of polynomials $q \in K[x_1,\dots x_n]$ such that $q(a) = 0$ (assuming at least one exists)? What about minimal polynomials, i.e. polynomials with minimal complexity (which one can define for example as a triple $(n,\mathrm{deg}_{x_n},\mathrm{deg})$ where we understand the latter to be the total degree)? We can, for simplicity, assume that we work in equicharacteristic zero.