On the Jacobian Conjecture and ideal membership for degree $d$-linear maps

TL;DR

This paper investigates degree d-linear polynomial maps satisfying the Jacobian condition, proving certain elements in their inverse coefficients belong to the Jacobian ideal and providing algebraic and combinatorial insights.

Contribution

It introduces new results on ideal membership for inverse coefficients of degree d-linear maps and offers a combinatorial proof of the Cayley-Hamilton theorem.

Findings

Certain inverse coefficients are in the Jacobian ideal

Expressions for elements via Cayley-Hamilton theorem

Gröbner basis computations on related elements

Abstract

We consider polynomial maps, which we call degree $d$-linear maps, that satisfy the Jacobian condition. We prove that certain infinite families of elements, which appear in the coefficients of the formal inverse of such maps, are in the ideal determined by the Jacobian condition. Using the Cayley-Hamilton theorem, we provide expressions for these elements in terms of the generators of that ideal. We also give a combinatorial proof of the Cayley-Hamilton theorem similar to that of Straubing \cite{Straubing}. We also include results of Gr\"obner basis computations regarding other elements.