Symmetric multilinear forms and polarization of polynomials

TL;DR

This paper explores the generalization of the classical relationship between polynomials and multilinear forms using combinatorial polarization, applicable even in fields where factorials are not invertible.

Contribution

It extends the polarization correspondence to forms in multiple variables without requiring the invertibility of n! in the field.

Findings

Generalizes polarization to n-variable forms

Applicable in fields with non-invertible factorials

Provides a combinatorial approach to multilinear forms

Abstract

We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ variables. The main tool is combinatorial polarization, and the approach is applicable even when $n!$ is not invertible in the underlying field.