Consider a field $\mathcal{F}$ and any multivariate polynomial over $\mathcal{F}$ with indeterminate $x_1, ...,x_n$ and $a_0 = 0$ (the coefficient for zero degree), denoted by $f(x_1,...,x_n)$. In addition, assign a real-valued norm $||\bullet||$ for $\mathcal{F}$ (as a vector space over itself).
Given a real $r>0$, define $$ \mathcal{X}(r)= \{[x_1,...,x_n] \in \mathcal{B}^n(0,r) \}, $$ where $\mathcal{B}$ denotes an open ball in $\mathcal{F}$.
Then for any non-zero $g \in \mathcal{F}$, does there always exists a real number $r_0>0$, such that
$$ ||f(x_1,...,x_n)|| < ||g||, $$ for any $[x_1,...,x_n]\in \mathcal{X}(r_0)$?
My intuition is that, by making $r$ sufficiently small for a given $g$, the above inequality should be obtainable.