Some algebraic properties of differential operators

Abstract

Abstract. First, we study the subskewfield of rational pseudodifferential operators over a differential field K generated in the skewfield K((∂ −1 )) of pseudodifferential operators over K by the subalgebra K[∂] of all differential operators. Second, we show that the Dieudonnè 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 2 × 2 matrix with entries in A[∂] whose Dieudonnè determiant does not lie in A.

A proof of the conjecture by Carpentier-De Sole-Kac

“…. h is a majorant, which implies that not only is it an optimal majorant, but that M ′′ is non-degenerate, and thus that its determinant is the determinant of the characteristic matrix [CDSK12,Thm.4.7]. To finish the proof, we now show that the determinant of the characteristic matrix is a multiple of c 1 .…”

“…., where A is the leading coefficient matrix of terms of degree N − h, B contains the terms of degree N − h − 1, and so on. Now, we know that since M ′ is degenerate, its characteristic matrix has determinant 0 (see [CDSK12,Thm4.7]). In this case, the characteristic matrix is exactly A.…”

“…if dd(A) ≥ 1, then det(Ā(λ)) = 0 for any majorant iv. if dd(A) = 0, then det(Ā(λ)) = 0 for any majorant which is not optimal, and det(Ā(λ)) = detA for any majorant which is optimal It is obvious that det 1 A ∈ R if dd(A) = 0, and it was shown in [CDSK12] that det 1 A always lies in the integral closure of R in K. However, in general det 1 A / ∈ R; there is a counterexample in [CDSK12] with dd(A) = 2. It was conjectured in [CDSK12] that det 1 A ∈ R when dd(A) = 1.…”

Singular Degree of a Rational Matrix Pseudodifferential Operator

“…In the present paper we continue the study of the algebra Mat ℓ×ℓ K(∂) of ℓ × ℓ rational matrix pseudodifferential operators that we began in [2,3,4].…”

“…Any rational pseudodifferential operator L ∈ K(∂) admits a fractional decomposition h = ab −1 , with a, b ∈ K[∂] (see e.g. [2]).…”