Some algebraic properties of differential operators

TL;DR

This paper investigates algebraic properties of differential operators, focusing on the structure of rational pseudodifferential operators and the behavior of the Dieudonne' determinant in relation to differential subrings.

Contribution

It introduces new insights into the subskewfield of rational pseudodifferential operators and analyzes the Dieudonne' determinant's properties over differential subrings.

Findings

The subskewfield of rational pseudodifferential operators is generated by differential operators.

The Dieudonne' determinant of matrix pseudodifferential operators lies in the integral closure of the coefficient ring.

An example shows the determinant may not lie within the original differential subring.

Abstract

First, we study the subskewfield of rational pseudodifferential operators over a differential field K generated in the skewfield of pseudodifferential operators over K by the subalgebra of all differential operators. Second, we show that the Dieudonne' determinant of a matrix pseudodifferential operator with coefficients in a differential subring A of K lies in the integral closure of A in K, and we give an example of a 2x2 matrix differential operator with coefficients in A whose Dieudonne' determiant does not lie in A.