Some algebraic properties of differential operators

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.

💡 Research Summary

The paper is divided into two main parts, each addressing a distinct algebraic aspect of differential operators in a non‑commutative setting.

In the first part the authors consider a differential field (K) equipped with a derivation (\partial). The full skew‑field of rational pseudodifferential operators (\mathcal{K}(\partial)) consists of formal expressions (\sum_{i=-m}^{n} a_i\partial^{,i}) with coefficients (a_i\in K) and allows both positive and negative powers of (\partial). Within this large non‑commutative field they isolate the smallest sub‑skew‑field (\mathcal{S}) generated by the ordinary differential operator algebra (D=K