Decomposability of multivariable polynomials

Abstract

Let $K$ be an algebrically closed field and let $n\ge 1$. If $P\in K[X]=K[X_1,\dots ,X_n]$, $P\ne 0$, we denote by $I(P)$ the support of $P$, which is the finite subset of ${\mathbb{N}}^n$ such that $P={\sum }_{i\in I(P)}{a}_i{X}^i$ with $a_i\in K^{\ast }$. (If $i=(i_1,\dots ,i_n)$ then $X^i:=X_1^{i_1}\cdots X_n^{i_n}$.) We determine all finite, nonempty sets $I\sb\mathbb N^n$ such that every $P\in K[X]$ with $I(P)=I$ is decomposable. We also consider the problem of finding all $I\sb\mathbb N^n$ 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.

22 pages