On product decomposition

TL;DR

This paper investigates the algebraic conditions and computational methods for decomposing finite sets in algebraic geometry into products of simpler sets under linear transformations, focusing on uniqueness and special cases.

Contribution

It provides an algebraic characterization of product decompositions of finite sets, especially for equal-sized components, and explores the computational aspects and uniqueness of such decompositions.

Findings

Characterization of product decompositions via algebraic conditions.

Proof of essential uniqueness for decompositions into equal-sized components.

Analysis of computational problems related to the decomposition process.

Abstract

Given a finite set $W$ in $\bar{k}^n$ where $\bar{k}$ is the algebraic closure of a field $k$ one would like to determine if $W$ can be decomposed as $\prod_{i=1}^n V_i$ where $V_i \subset \bar{k}$ under a linear transformation, that is, $W\stackrel{\lambda}{\to} \prod_{i=1}^n V_i$ where $\lambda\in Gl_n (\bar{k})$. We assume that $W$ is presented as $W=Z(\mathcal{F})$, the zero set of a polynomial system $\mathcal{F}$ in $n$ variables over $k$. We study algebraic characterization of such product decomposition. For decomposition into component sets of the same cardinality we obtain a stronger characterization and show that the decomposition in this case is essentially unique (up to permutation and scalar multiplication of coordinates). We investigate computational problems that arise from the decomposition problem.