Decomposability of multivariable polynomials

TL;DR

This paper characterizes the finite support sets of multivariable polynomials over algebraically closed fields that guarantee decomposability, and explores how the irreducibility depends on the field's characteristic.

Contribution

It provides a complete classification of support sets that ensure polynomial decomposability and analyzes the characteristic-dependent nature of irreducibility.

Findings

Identifies all support sets where polynomials are necessarily decomposable.

Shows the irreducibility of polynomials depends on the characteristic of the field.

Highlights the complexity of classifying irreducible polynomials based on support sets.

Abstract

Let $K$ be an algebrically closed field and let $n\geq 1$. If $P\in K[X]=K[X_1,\ldots,X_n]$, $P\neq 0$, we denote by $I(P)$ the support of $P$, which is the finite subset of $\mathbb N^n$ such that $P=\sum_{i\in I(P)}a_iX^i$ with $a_i\in K^*$. (If $i=(i_1,\ldots,i_n)$ 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.