Question

Given a polynomial $f \in \mathrm{C} [y_{0}, y_{1}, \cdots, y_{n-1} ]$, where $n$ is a natural number with $n >1$. Define a derivation $\delta$, s.t., it is linear and $\delta(y_{a}*y_{b})= y_{a+1}*y_{b}+ y_{a}*y_{b+1}, all a, b=1, \cdots, n-1 $. For example, if $f=3*y_{1}*y_{4}-y_{5},$ then, we have $\delta (f)=3*y_{2}*y_{4}+3*y_{1}*y_{5}-y_{6}$. Then, the question is : given a point $ z=(z_{0}, \cdots, z_{n})$, with $z\neq 0$, is there any relation between $|f(z)|$ and $|\delta (f)|$ ? (here, $|\ |$ denotes the absolute value defined by $|z|= \sum_{k=0}^{n} z_{k}$).\

One of my motivation of this problem is that I expect to use the result combining with W.Dale.Brownawell's result on a analytic description of Nullstellensatz Theorem to test a problem in differential algebra, however, I do not know how to get any useful result on the relation of $|f(z)|$ and $|\delta (f)|$ mentioned above. c.f.link text for the differential algebra problem.