A chain rule formula for higher derivations and inverses of polynomial maps

TL;DR

This paper develops a characteristic-independent chain rule formula for higher derivations, enabling the computation of polynomial automorphism inverses in any characteristic, extending classical calculus concepts to algebraic settings.

Contribution

It introduces a new chain rule formula for higher derivations valid in all characteristics and applies it to invert polynomial automorphisms, generalizing existing methods.

Findings

Derived a chain rule formula for higher derivations in arbitrary characteristic.

Connected higher derivations across different coordinate systems.

Provided a formula for the inverse of polynomial automorphisms in any characteristic.

Abstract

The multidimensional chain rule formula for analytic functions and its generalisation to higher derivatives perfectly work in the algebraic setting in characteristic zero. In positive characteristic one runs into problems due to denominators in these formulas. In this article we show a direct analog of these formulas using higher derivations which are defined in any characteristic. We also use these formulas to show how higher derivations to different coordinate systems are related to each other. Finally, we apply this to polynomial automorphisms in arbitrary characteristic and obtain a formula for the inverse of such a polynomial automorphism.