Decomposability of multivariable polynomials

Let $K$ be an algebrically closed field and let $n\ge 1$. If $P\in K[X]=K[X_1,\ldots,X_n]$,$P\neq 0$,wedenoteby$I(P)$thesupportof$P$, which is the finite subset of ${\mathbb{N}}^n$ such that $P=\sum_{i\in I(P)}a_iX^i$with$a_i\in K^*$\mathrm{.}(If$i=(i_1,\ldots,i_n)$then$X^i:=X_1^{i_1}\cdots X_n^{i_n}$\mathrm{.})Wedetermineallfinite,nonemptysets$I\sb\mathbb N^n$suchthatevery$P\in K[X]$with$I(P)=I$ is decomposable. We also consider the problem of finding all $I\text{sb}{\mathbb{N}}^n$ such that every $P\in K[X]$ with $I(P)=I$ is irreducible. We do not solve this problem, which is very unlikely to have a simple answer. We show however that the answer depends on the characteristic of $K$ and we determine the nature of this dependence.