An algebraic approach to the ellipticity of linear differential operators

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.