An algebraic approach to the ellipticity of linear differential operators

TL;DR

This paper introduces an algebraic method to define and analyze the ellipticity of linear differential operators, demonstrating its invariance under algebra homomorphisms and providing examples on real affine varieties.

Contribution

It presents a purely algebraic framework for ellipticity of differential operators, extending classical concepts and establishing invariance properties.

Findings

Ellipticity is preserved under surjective algebra homomorphisms.

Every real affine variety admits an elliptic differential operator.

The method connects principal symbols at K-points with algebraic modules.

Abstract

We demonstrate a method of associating the principal symbol at a $K$-point with a linear differential operator acting between modules over a commutative algebra, and we use it to define the ellipticity of a linear differential operator in a purely algebraic way. We prove that the ellipticity is preserved by a surjective homomorphism of algebras. As an example, we show that for every real affine variety there is an elliptic linear differential operator acting on the algebra of regular functions on this variety.