Reducing the number of variables of a polynomial

Abstract

In this paper, we consider two basic questions about presenting a homogeneous polynomial f: how many variables are needed for presenting f? How can one find a presentation of f involving as few variables as possible? We give a complete answer to both questions, determining the minimal number of variables needed, NEssVar(f), and describing these variables through their linear span, EssVar(f). Our results give rise to effective algorithms which we implemented in the computer algebra system CoCoA.