Polarization algebras and their relations

TL;DR

This paper explores the connections between polarization algebras, classical ring theory, and the Jacobian Conjecture, providing a new algebraic perspective and posing open questions in the field.

Contribution

It establishes a correspondence between finite dimensional symmetric algebras and homogeneous polynomial tuples, linking algebraic and geometric problems related to the Jacobian Conjecture.

Findings

Established a correspondence between symmetric algebras and polynomial tuples

Linked Albert's problem with the homogeneous dependence problem

Presented concrete examples and open questions

Abstract

Using an approach to the Jacobian Conjecture by L.M. Dru\.zkowski and K. Rusek 12], G. Gorni and G. Zampieri [19], and A.V. Yagzhev[27], we describe a correspondence between finite dimensional symmetric algebras and homogeneous tuples of elements of polynomial algebras. We show that this correspondence closely relates Albert's problem [10, Problem 1.1], in classical ring theory and the homogeneous dependence problem [13, page 145, Problem 7.1.5], in affine algebraic geometry related to the Jacobian Conjecture. We demonstrate these relations in concrete examples and formulate some open questions.